Stability of LIDAR measurement buoys for registration of ......de beperkingen van het auteursrecht,...
Transcript of Stability of LIDAR measurement buoys for registration of ......de beperkingen van het auteursrecht,...
Fen Van Liefferinge, Dennis Van Eecke
of offshore wind velocity profilesStability of LIDAR measurement buoys for registration
Academiejaar 2011-2012Faculteit Ingenieurswetenschappen en ArchitectuurVoorzitter: prof. dr. ir. Joris DegrieckVakgroep Toegepaste Materiaalwetenschappen
Master in de ingenieurswetenschappen: werktuigkunde-elektrotechniekMasterproef ingediend tot het behalen van de academische graad van
Begeleider: Kameswara VepaPromotoren: prof. dr. ir. Wim Van Paepegem, prof. dr. ir. Joris Degrieck
Stability of LIDAR measuring buoy for
offshore wind profile registration
Students
Fen Van Liefferinge
Dennis Van Eecke
Promotors:
Prof. Dr. Ir. Wim Van Paepegem
Prof. Dr. Ir. Joris Degrieck
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis
Master thesis – Mechanical Engineering - Permission for use of content Page a
Permission for use of content
The authors give the permission to use this thesis for consultation and to copy parts of it
for personal use. Every other use is subject to copyright law, more specifically the source
must be extensively specified when using this thesis.
Dennis Van Eecke,
Fen Van Liefferinge
June 2012
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis
Master thesis – Mechanical Engineering - Toelating tot bruikleen Page b
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Dennis Van Eecke,
Fen Van Liefferinge
Juni 2012
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis
Master thesis – Mechanical Engineering - Foreword Page c
Foreword
For any Engineering student, the graduating year is an exciting chapter in life. The
thesis is an important part of this. Especially this thesis where we get an
opportunity to assist in the development of a new and interesting engineering
application. We had the opportunity to put a lot of our own ideas and our own
reasoning into this document. Instead of citing dozens of references we could
spend a lot of time finding out new things for ourselves. Although we did read
books there were still a lot of questions unanswered afterwards. Especially
concerning the practical implementation of the SPH models. Even getting simple
wave tank simulations to work was a big challenge. Every stride forward we took,
was the result of a lot of thought and effort.
Happily we could count on great support of the staff of the Mechanics of Materials
and Structures dept. of the University of Ghent. Especially our thesis coordinator
Kameswara Sridhar Vepa and our thesis promotor Prof. Dr. Ir. Wim Van Paepegem.
Without the perseverance and insights of K. Vepa and the good advice of W. Van
Paepegem we would have had a much tougher time. We would like to express our
most sincere gratitude to them and to the other helpful people at MMC: Joren
Pelfrene, Ives De Baere, and everyone we forgot to mention.
Of course this work was commissioned by 3E environmental consultancy. We
collaborated with their representative Ir. Thomas Duffey. We would like to thank
him as well for the continued support, the opportunity to visit the workshop and
his constructive and helpful attitude towards us and the thesis.
Last but not least, we would like to thank our parents and family as well for
providing the support we needed to devote so much of our time on our education
in general and this document in particular. They cared for our every need and bore
the considerable financial strain students impose upon parents all for our
happiness and for a fruitful ending of our education.
Sincerely,
The authors,
Fen Van Liefferinge
Dennis Van Eecke
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis abstract
Master thesis – Mechanical Engineering –Introduction Page A
Stability of LIDAR measuring buoy for
offshore wind profile registration
By
Fen Van Liefferinge
Dennis Van Eecke
Masterproef ingediend tot het behalen van de academische graad Master in de
ingenieurswetenschappen: Werktuigkunde - Elektrotechniek
Promotoren: Prof. Dr. Ir. Wim Van Paepegem, Prof. Dr. Ir. Joris Degrieck
Scriptie begeleiders: Kameswara Sridhar Vepa
Vakgroep toegepaste materiaalwetenschappen
Voorzitter: Prof. Dr. Ir. Joris Degrieck
Faculteit Ingenieurswetenschappen en architectuur
Universiteit Gent
Academiejaar 2011-2012
Abstract
The goal of this thesis is to assist in the development of a stabilisation mechanism for
offshore meteorological measuring equipment (Lidar) and propose an optimal buoy design
to mount this upon. For this purpose, an extensive study on water wave mechanics, as well
as a study on Smoothed Particle Hydrodynamics (SPH), used to model the sea, was
conducted. The thesis consists of two parts. The first part is the modelling of the sea state
using SPH with LS-Dyna software, followed by an analysis of buoy performance. The
second part was a kinematic study in Universal Mechanism (UM) software of the
stabilization mechanism to understand its behaviour and optimize its performance. Both
studies were combined to formulate a final conclusion on the performance of the buoy and
stabilization mechanism combined. More specifically the maximum inclination of the Lidar
module. The stabilisation mechanism decreases the Lidar inclination up to 90%, increasing
accuracy of the measurements.
Keywords
SPH, water wave mechanics, Watercraft stability, stabilisation mechanism, gyroscope
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis abstract
Master thesis – Mechanical Engineering –Introduction Page B
Stability of LIDAR measuring buoy for offshore
wind profile registration
Fen Van Liefferinge, Dennis Van Eecke Supervisors: Kamerwara Sridhar Vepa, Prof. Dr. Ir. Wim Van Paepegem
Abstract — The goal of this thesis is to assist in the development of a stabilisation mechanism for offshore meteorological measuring equipment (Lidar) and propose an optimal buoy design to mount this upon. For this purpose, an extensive study on water wave mechanics, as well as a study on Smoothed Particle Hydrodynamics (SPH), used to model the sea, was conducted. The thesis consists of two parts. The first part is the modelling of the sea state using SPH with LS-Dyna software, followed by an analysis of buoy performance. The second part was a kinematic study in Universal Mechanism (UM) software of the stabilization mechanism to understand its behaviour and optimize its performance. Both studies were combined to formulate a final conclusion on the performance of the buoy and stabilization mechanism combined. More specifically the maximum inclination of the Lidar module. The stabilisation mechanism decreases the Lidar inclination up to 90%, increasing accuracy of the measurements. Keywords — SPH, water wave mechanics, Watercraft stability, stabilisation mechanism, gyroscope
I. INTRODUCTION
This thesis is commissioned by 3E environmental consultancy. 3E is developing a product for measuring offshore wind profiles to assess the profitability of prospective wind farm locations. A first buoy prototype has already been designed with the Lidar measuring module on top.
The quality of the Lidar measurements is inversely proportional to the misalignment of its laser from vertical. The original stabilization mechanism was built as a pendulum and the buoy consisted of two conventional steel buoys. This article researches different options on the choice of buoys and a new stabilization mechanism to increase the quality of the measurements.
II. BUOY CHOICE
A downside of the first prototype was the fact that it was very heavy and difficult to handle. The end product has to be more cost-effective. Manageability, transportability and costs are key elements. The PEM58 buoy by RESINEX meets these requirements best. The use of composite materials makes it lighter and more manageable, while its modular design leads to easier transport. The PEM58 is produced in series, most likely resulting in a lower price than the custom built prototype.
Figure 1: PEM58 [1]
III. MODELLING OF DEEP WATER WAVES
In simulations, the grid based methods are most popular. However, their applications are limited for many complex problems. These limitations become apparent when modelling free surfaces, deformable boundaries and extremely large deformations. All common in wave simulations. To be able to deal with these problems, meshfree particle methods are used. In particular the SPH-method. Unlike conventional techniques, the SPH-particles do not have a fixed connectivity, resulting in the capability to model water waves. Downsides are the increased calculation time and the possibility of clustering. [2]
Similar to real experiments, the model consists of a wave tank, a wave generator and water. This is comparable to the work of [3], only on a larger scale and in three dimensions. The model was tuned to produce waves with similar characteristics as the waves at the Thornton bank. It was calibrated using sea state data provided by GeoSea and mathematical models [4]. The results are pictured in table 1.
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis abstract
Master thesis – Mechanical Engineering –Introduction Page C
Sea state by GeoSea
Sea state in model
Wave period [s] 5.594 5.6
Wave height HM1 [cm] 224.46 244.6
Highest wave [cm] 275.69 276.8
Wavelength [m] 48.27 48.47
Table 1- comparison of sea state data
IV. MODELLING OF THE BUOYS
Rigid finite element models of the chosen buoys were produced. Specifications of the 3E prototype were provided by 3E themselves. Therefore this model was the most accurate. Unlike the PEM58 which was modelled only using the information available from the product catalogue. A SPAR buoy was modelled as well to serve as a reference for optimal stability. Each buoy was fitted with a platform, based on the current prototype, and simplified installations such as solar panels, battery box and Lidar.
Figure 2- 3E prototype model
Each buoy model was equipped with a mooring cable as well to prevent it from drifting away. Several options were explored to impose the effect of an undercurrent on the behaviour of the model. The most feasible option was to apply forces on the mooring cable itself. Unfortunately this was abandoned for reasons of calculation time.
V. BUOY PERFORMANCE
Buoy performance was determined by subjecting the buoy models to the modelled sea waves. The translational and rotational deviation measured relative to the starting position are tracked, but far more important is the rotational speed. The faster the buoy pitches, the less time the stabilisation mechanism has to counteract the movement. The results for the current 3E prototype (pictured in green) and the suggested alternative, the PEM58 (pictured in red), are visualised in figure 3.
Figure 3-Rotational speed versus time
Despite the fact that the PEM58 rotates 53% faster than the 3E buoy, the recommendation is still valid. The PEM58 is lighter, easier to transport, has a flat top to mount the installations on and will probably be cheaper.
VI. KINEMATIC STUDY
Figure 4: Model of stabilisation mechanism
To stabilise the Lidar, a system using a flywheel was preferred instead of an actively controlled system because of its simplicity. It only relies on a gyroscope to function. An extensive empirical study with many simulations has shown that the flywheel radius is best chosen as large as possible within practical limits. Making the flywheel thicker increases accuracy, but to a lesser extent. An excessively heavy flywheel should be avoided because of practical issues. A separate study with a differently shaped mechanism has shown that a ring shaped flywheel can yield better accuracy for a similar weight.
Any change resulting in higher flywheel capacity does increase accuracy in an ideal situation, but deteriorates the mechanism’s tolerance for disturbances. These cause a precession motion of which the amplitude is dominant compared to the amplitude excited by the buoy movement itself. The flywheel capacity should be high enough to yield sufficient accuracy, but making it too high can make precession motion too hard to damp out.
Jerking, both in rotation and translation, does not cause too big of a disturbance. Sudden acceleration or deceleration of the flywheel is most damaging.
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis abstract
Master thesis – Mechanical Engineering –Introduction Page D
VII. COMBINED RESULTS
Data extracted from Dyna wave simulations has been used as input to generate corresponding frame movement in UM. This way the Lidar inclination can be measured as before, but with the frame moving like the buoy would. The average and maximum absolute value of the Lidar rotation vector resulting from the UM simulations is an indicator of the accuracy of an actual mechanism on an actual buoy. The tested buoys were the 3E prototype and the PEM 58.
Figure 5: Maximum Lidar inclination for varying flywheel settings.
It is obvious the flywheel is of great benefit to accuracy. Although the 3E prototype buoy performs better in damping out wave movement, the difference in accuracy is negligible when using the gyroscope. It seems that increasing the flywheel speed from 3000 to 6000 rpm is not worth the effort, despite the theoretical conclusions of the kinematic study.
The final recommendation is that the PEM buoy in combination with the flywheel is a more manageable and cost effective solution compared to 3E’s original design. The flywheel has increased the accuracy with approximately 90% for the 3E design.
VIII. REFERENCES
[1] Resinex, Resinex, March 2012. [Online]. Available: http://www.resinextrad.com/
[2] M. B. L. G. R. Liu, Smoothed particle hydrodynamics, World Scientific Publishing Co. Pte. Ltd., 2003.
[3] J. Pelfrene, Study of the SPH method for simulation of regular and breaking waves, Gent: Universiteit Gent, 2011
[4] R. A. D. Robert G. Dean, Water wave mechanics, World scientific publishing Co. Pte. Ltd., 1992.
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Ghent University Master Thesis abstract
Master thesis – Mechanical Engineering – Table of contents Page E
Stability of LIDAR measuring buoy for offshore
wind profile registration
Fen Van Liefferinge, Dennis Van Eecke Begeleiders: Kamerwara Sridhar Vepa, Prof. Dr. Ir. Wim Van Paepegem
Samenvatting — Het onderzoek in deze thesis dient als hulp bij de ontwikkeling van een stabiliserend mechanisme voor een offshore meteorologisch meetinstrument (Lidar) en het voorstellen van een optimaal boei ontwerp om dit instrument te dragen. Ter voorbereiding werd eerst het golfgedrag bestudeerd alsook de ‘Smoothed Particle Hydrodymanics” (SPH) techniek. Deze laatste werd uiteindelijk gebruikt om de zee te modeleren. De thesis bestaat uit twee grote delen. Het eerste deel handelt over het modelleren van de zee gebruikmakend van SPH en LS-Dyna software, gevolgd door een analyse van de prestaties van de boeien. Het tweede deel is een kinematische studie in Universal Mechanism software (UM) van het stabiliserende mechanisme. Dit om zijn gedrag te doorgronden en zijn prestaties te optimaliseren. Tot slot werden beide delen gecombineerd om een besluit te trekken over de prestaties van het gecombineerde geheel van boei en stabilisatie mechanisme. Meer specifiek de maximale inclinatie van de Lidar module. Het stabiliserende mechanisme verkleint de maximale inclinatie met 90%, resulterend in een betere accuraatheid van de metingen. Trefwoorden — SPH, golftheorie, vaartuig stabiliteit, stabiliserend mechanisme, gyroscoop
I. INLEIDING
Deze thesis is geschreven in opdracht van 3E environmental constaltancy. 3E ontwikkeld een product om offshore wind profielen op te meten. Dit om de opbrengst van toekomstige windmolenparken te voorspellen. Een eerste prototype van de boei die het Lidar meettoestel draagt, is reeds gemaakt.
De kwaliteit van de metingen is omgekeerd evenredig met de afwijking die de laser van de Lidar heeft ten opzichte van de verticale richting. Het origineel bevat daarom ook een stabiliserend mechanisme dat uitgevoerd is als een pendulum. De boei zelf is opgebouwd uit twee conventionele stalen boeien. Dit artikel onderzoekt verschillende mogelijke keuzes van de boeien en het ontwerp van een nieuw stabiliserend systeem om de kwaliteit van de metingen op te drijven.
II. KEUZE VAN DE BOEI
Een groot nadeel verbonden aan het eerste prototype is zijn grote massa en het feit dat hij moeilijk handelbaar is. Het eindproduct meer kosteneffectief, handel- en transporteerbaar maken, is het voornaamste doel. De PEM58 boei van RESINEX voldoet het beste aan deze gestelde eisen. Het gebruik van composiet materialen maakt hem lichter en als gevolg ook beter handelbaar. Het modulaire design leidt tot een betere transporteerbaarheid. De PEM58 wordt tevens in serie geproduceerd. Dit resulteert hoogstwaarschijnlijk in een lagere prijs dan het op maat gemaakte prototype.
Figuur 1: PEM58 [1]
III. MODELLEREN VAN DIEP WATER GOLVEN
Methodes gebruikmakend van een rooster zijn zeer populair bij numerieke simulaties. Toch zijn hun toepassingen beperkt voor vele complexe problemen. Deze beperkingen zijn zeer duidelijk bij het modeleren van vrije oppervlakken, vervormbare grenzen en zeer grote vervormingen in het algemeen. Deze zijn allen vaak voorkomend bij het simuleren van zeegolven. Om dit toch te kunnen modelleren, wordt gebruik gemaakt van de SPH-techniek. In tegenstelling tot conventionele methodes, zijn de SPH-partikels niet vast verbonden, wat resulteert in de mogelijkheid om golven te kunnen simuleren. Er zijn echter ook nadelen aan verbonden, waaronder een sterk toegenomen rekentijd en de mogelijkheid tot clusteren. [2].
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis abstract
Master thesis – Mechanical Engineering –Introduction Page F
Net zoals bij praktische experimenten bestaat het model uit een golftank, een golfgenerator en water. Dit is vergelijkbaar met het werk van [3], maar dan opgeschaald en in drie dimensies. Het model is zodanig afgesteld dat de gegenereerde golven gelijkaardige karakteristieken hebben aan de golven bij de Thornton bank. Het model is gekalibreerd op basis van data over de zeegolven gemeten door GeoSea en wiskundige modellen [4]. De resultaten zijn weergegeven in tabel 1.
Sea state by GeoSea
Sea state in model
Wave period [s] 5.594 5.6
Wave height HM1 [cm] 224.46 244.6
Highest wave [cm] 275.69 276.8
Wavelength [m] 48.27 48.47
Tabel 1- vergelijking van data over de zeegolven
IV. MODELLEREN VAN DE BOEIEN
Vervolgens werden starre, eindige elementen modellen gemaakt van de geselecteerde boeien. Van de huidige boei van 3E werd voldoende data verschaft om een zeer representatief model te ontwerpen. De PEM58 daarentegen werd uitsluitend gemodelleerd op basis van informatie afgeleid uit de product catalogus. Ook werd een SPAR boei gemodelleerd die zou dienen als referentie voor optimale stabiliteit. Elke boei werd uitgerust met een platform, gebaseerd op hetgeen momenteel gebruikt door 3E, en een vereenvoudigde set van installaties zoals zonnepanelen, batterijen en de Lidar.
Figuur 2- Model van het 3E prototype
Daarbovenop werd elke boei voorzien van een kabel om te verhinderen dat deze zou afdrijven. Om de effecten van een onderstroom te incorporeren in het model werden verschillende denkpistes onderzocht. De meest haalbare optie was om de resulterende krachten te laten inwerken op de verankeringskabel. Spijtig genoeg is dit niet uitgevoerd vanwege de grote rekentijd die dit met zich zou meebrengen.
V. PRESTATIES VAN DE BOEIEN
De prestaties van de boeien werden begroot door ze te onderwerpen aan de gemodelleerde zeegolven. Zowel de translationele als de rotationele afwijking ten opzichte van de startpositie werd onderzocht. Veel belangrijker is echter de rotatiesnelheid van de boei om een as in het horizontale vlak. Hoe sneller de boei roteert, hoe minder tijd het stabiliserende mechanisme heeft om deze beweging te corrigeren. De resultaten van het huidige 3E prototype (in het groen) vergeleken met deze van de PEM58 (in het rood) zijn voorgesteld door figuur 3.
Figuur 3-Rotatiesnelheid in functie van de tijd
Ondanks het feit dat de PEM58 53% sneller roteert in vergelijking met de 3E boei, blijft deze aangeraden. De PEM58 is lichter, gemakkelijker te transporteren, heeft een vlak oppervlak waar de installaties kunnen gemonteerd worden en zal waarschijnlijk goedkoper uitkomen. Ook dient er nog op gewezen te worden dat de werkelijke PEM beter zal presteren dan het model. Dit laatste is namelijk enkel gebaseerd op informatie beschikbaar in de catalogus.
VI. KINEMATISCHE STUDIE
Figuur 4: Model van het stabiliserende mechanisme
Stability of LIDAR measuring buoy for offshore wind profile registration Master thesis abstract
Master thesis – Mechanical Engineering –Introduction Page G
Om de Lidar te stabiliseren werd een systeem gekozen dat gebruik maakt van een vliegwiel. Dit kreeg de voorkeur ten opzichte van een actief gestuurd systeem vanwege zijn eenvoud. Een uitgebreide empirische studie met vele simulaties wees uit dat de straal van het vliegwiel best zo groot mogelijk gekozen wordt, binnen praktische grenzen. Het dikker maken van het vliegwiel resulteert eveneens in een verhoogde nauwkeurigheid, maar dan minder uitgesproken. Een zeer zwaar vliegwiel dient vermeden te worden om praktische redenen. Een aparte studie met een vliegwiel van een andere vorm heeft aangetoond dat een ringvormig vliegwiel de metingen nog nauwkeuriger kan maken voor gelijkaardig vliegwielgewicht.
Elke verandering resulterend in betere prestaties van het vliegwiel, verbetert de accuraatheid van het systeem in ideale situaties. Het verlaagt echter de tolerantie van het mechanisme voor storingen. Deze veroorzaken een precessiebeweging met een amplitude die dominant is ten opzichte van deze van de Lidar-beweging geëxciteerd door de beweging van de boei. De capaciteit van het vliegwiel dient hoog genoeg te zijn om voor voldoende nauwkeurige metingen te zorgen, maar een te hoge capaciteit zorgt ervoor dat de precessiebeweging moeilijk is om uit te dempen.
Hevige bewegingen, zowel rotationele als translationele, veroorzaken geen sterke verstoringen. Plotse versnelling of afremming van het vliegwiel is schadelijker voor de accuraatheid.
VII. GECOMBINEERDE RESULTATEN
Data uit de Dyna golf simulaties werd gebruikt als input om corresponderende frame bewegingen te genereren in UM. Op deze wijze kan de hoekafwijking van de Lidar ten opzichte van de verticale gemeten worden, maar nu met de echte bewegingen die het op zee zou ondervinden. De gemiddelde en de absolute waarde van de rotatievector van de Lidar uit de UM simulaties dient als een indicator voor de nauwkeurigheid die het mechanisme en de gekozen boei kunnen bereiken. De geteste boeien waren het huidige prototype van 3E en de PEM 58.
Figuur 5: Maximale uitwijking van de Lidar voor verschillende vliegwiel
paramters.
Het is duidelijk dat het vliegwiel de nauwkeurigheid sterk verbetert. Ondanks het feit dat het 3E prototype beter presteert als het op uitdemping van de zee beweging aankomt, is het verschil in nauwkeurigheid klein als er gebruik gemaakt wordt van de gyroscoop. Het vliegwiel versnellen van 3000 tpm tot 6000 tpm levert niet veel meer voordeel op. Dit in tegenstelling tot de conclusies gemaakt in de kinematische studie.
Het uiteindelijke voorstel voor 3E is de PEM 58 in combinatie met het vliegwiel. Dit geheel is beter handelbaar en kosteneffectief vergeleken met het huidige ontwerp. Daarbovenop heeft het vliegwiel de nauwkeurigheid verhoogd met zo’n 90% ten opzichte van een identiek systeem met het vliegwiel uitgeschakeld.
VIII. REFERENTIES
[1] Resinex, Resinex, Maart 2012. [Online]. Available: http://www.resinextrad.com/
[2] M. B. L. G. R. Liu, Smoothed particle hydrodynamics, World Scientific Publishing Co. Pte. Ltd., 2003.
[3] J. Pelfrene, Study of the SPH method for simulation of regular and breaking waves, Gent: Universiteit Gent, 2011
[4] R. A. D. Robert G. Dean, Water wave mechanics, World scientific publishing Co. Pte. Ltd., 1992.
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Stability of LIDAR measuring buoy for offshore wind profile registration Master Thesis
Master thesis – Mechanical Engineering –Introduction Page I
Table of contents
1 Introduction ............................................................................................................................... 1
2 Study of commercially available hull designs ................................................................ 4
2.1 General considerations ................................................................................................................................ 4
2.2 Stability .............................................................................................................................................................. 4
2.3 Commercial solutions ................................................................................................................................... 5
2.3.1 Catamaran ................................................................................................................................................................... 5
2.3.2 Resinex PEM 58 Catamaran buoy [2] ............................................................................................................. 8
2.3.3 Resinex PEM 58 buoy [2] ................................................................................................................................... 12
2.3.4 Spar buoy .................................................................................................................................................................. 12
2.4 Conclusion ....................................................................................................................................................... 13
3 Numerical study ...................................................................................................................... 14
3.1 Preface [3] ....................................................................................................................................................... 14
3.2 Numerical simulations in general [3] .................................................................................................. 14
3.2.1 SPH: a meshfree particle method ................................................................................................................... 14
3.2.2 Clustering .................................................................................................................................................................. 17
3.3 LS-Dyna and LS-PrePost ............................................................................................................................ 18
3.4 Water wave mechanics [6] ....................................................................................................................... 24
3.5 Wave tank design and dimensions ....................................................................................................... 26
3.5.1 Basic wave tank dimensions ............................................................................................................................. 26
3.5.2 Boundary conditions of the paddle ............................................................................................................... 29
3.5.3 Boundary conditions of the wave tank ........................................................................................................ 32
3.5.4 Summary: Basic wave tank ............................................................................................................................... 35
3.5.5 Friction interface ................................................................................................................................................... 36 3.5.5.1 Undercurrent modelling with linear water duct and pistons ............................................................................. 36 3.5.5.2 Undercurrent modelling by closed loop water circuit ........................................................................................... 37 3.5.5.3 Friction interface with moving plate .............................................................................................................................. 37 3.5.5.4 Simulating undercurrent by applying resultant forces on the mooring cable ........................................... 39
3.5.6 Additional cards ..................................................................................................................................................... 40 3.5.6.1 INITIAL_STRESS_DEPTH ...................................................................................................................................................... 40 3.5.6.2 LOAD cards .................................................................................................................................................................................. 40
3.6 Model verification ........................................................................................................................................ 41
3.6.1 Mesh convergence ................................................................................................................................................. 41 3.6.1.1 1000 kg particles ...................................................................................................................................................................... 42 3.6.1.2 Conclusions ................................................................................................................................................................................. 43
3.6.3 Further model checks .......................................................................................................................................... 48 3.6.3.1 Buoyancy check ........................................................................................................................................................................ 48 3.6.3.2 Breaking wave check .............................................................................................................................................................. 49 3.6.3.3 Dam break check ...................................................................................................................................................................... 50
3.7 Modelling the buoys for LS-Dyna .......................................................................................................... 51
3.8 Calculation time ............................................................................................................................................ 56
3.8.3 SPH particle mass versus calculation time ................................................................................................. 59
3.8.4 Mooring cable versus calculation time ........................................................................................................ 60
3.9 Simulation results on buoy performance ........................................................................................... 61
3.9.1 General information on displacement of the buoys ............................................................................... 62
Stability of LIDAR measuring buoy for offshore wind profile registration Master Thesis
Master thesis – Mechanical Engineering –Introduction Page II
3.9.2 Performance of the different buoys in terms of rotation .................................................................... 63 3.9.2.1 3E prototype buoy ................................................................................................................................................................... 63 3.9.2.2 PEM58 buoy ................................................................................................................................................................................ 65 3.9.2.3 Spar buoy ..................................................................................................................................................................................... 67 3.9.2.4 3.9.2.4 Conclusion .................................................................................................................................................................... 70
4 Study of Lidar stabilization mechanism ......................................................................... 72
4.1 Introduction ................................................................................................................................................... 72
4.2 Model overview ............................................................................................................................................ 72
4.3 Model parts, joints and parameters...................................................................................................... 76
4.3.1 Part: Frame ............................................................................................................................................................... 76
4.3.2 Part: Outer gimbal ................................................................................................................................................. 76
4.3.3 Part: Inner gimbal.................................................................................................................................................. 76
4.3.4 Part: Lidar ................................................................................................................................................................. 76
4.3.5 Part: Flywheel ......................................................................................................................................................... 77
4.3.6 Gimbal joints ............................................................................................................................................................ 77
4.3.7 Overview and reference situation .................................................................................................................. 78
4.4 Measuring performance ............................................................................................................................ 80
4.5 Parameter influence .................................................................................................................................... 83
4.5.1 Flywheel speed ....................................................................................................................................................... 83
4.5.2 Amplitude.................................................................................................................................................................. 86
4.5.3 Flywheel dimensions ........................................................................................................................................... 88
4.5.4 Flywheel weight ..................................................................................................................................................... 95
4.5.5 Flywheel offset ........................................................................................................................................................ 96
4.5.6 Lidar offset ................................................................................................................................................................ 98
4.5.7 Conclusion of mechanism geometry design: ‘super mechanism’ .................................................. 102
4.5.8 Damping .................................................................................................................................................................. 107 4.5.8.1 Effect of damping in the reference mechanism ...................................................................................................... 107 4.5.8.2 Beware of precession .......................................................................................................................................................... 111
4.6 Reaction of the mechanism to irregularities ................................................................................. 112
4.6.1 Sudden deceleration of flywheel ................................................................................................................. 112
4.6.2 Sudden changes in buoy frame movement ............................................................................................. 122 4.6.2.1 Functions ................................................................................................................................................................................... 122 4.6.2.2 Rotational jerking ................................................................................................................................................................. 123 4.6.2.3 Translational jerking ........................................................................................................................................................... 126 4.6.2.4 Change of amplitude ............................................................................................................................................................ 127 4.6.2.5 Conclusion ................................................................................................................................................................................ 128
4.6.3 Restarts during operation ............................................................................................................................... 128
4.6.4 Influence of friction ........................................................................................................................................... 130
4.7 Frequency response ................................................................................................................................. 133
4.7.1 About UM and the reference model ............................................................................................................ 134
4.7.2 Excitation of frame for typical wave periods ......................................................................................... 134
4.7.3 Resonance .............................................................................................................................................................. 137
4.8 Motor requirements ................................................................................................................................. 140
4.8.1 Motor power in steady state .......................................................................................................................... 140
4.8.2 Flywheel start-up................................................................................................................................................ 142
4.9 Conclusion .................................................................................................................................................... 146
5 Combined results ................................................................................................................ 147
5.1 3E prototype ............................................................................................................................................... 149
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5.2 PEM 58 ........................................................................................................................................................... 151
5.3 Comparison ................................................................................................................................................. 152
6 Conclusion ............................................................................................................................. 154
7 References ............................................................................................................................. 156
8 Figures, graphs and tables ............................................................................................... 157
8.1 List of figures .............................................................................................................................................. 157
8.2 List of Graphs .............................................................................................................................................. 163
8.3 List of tables ................................................................................................................................................ 165
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1 Introduction
In recent times wind power is becoming increasingly important as a renewable energy
source and the subsequent demand for offshore wind farms has created a need to
accurately measure the wind profile on a particular location. A wind profile is a wind
velocity and direction distribution on a particular location in relation to the height above
the sea level. The knowledge of wind profile(s) is vital to determine the profitability and
feasibility of prospective offshore wind farm locations.
The wind profile can be measured with laser guided measuring devices such as the
‘Lidar’. A device similar to a radar, but it uses laser light instead of radio waves.
Figure 1: Lidar measuring equipment
To make a correct measurement the Lidar must remain stationary for an extended
period of time during which the unit must remain as level as possible. This is necessary
because the Lidar must take a great deal of measurements to determine the average
wind distribution and the scatter of wind velocity and direction. Typically, the device
must remain in place for several weeks.
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Figure 2: Schematic overview of measuring device operating conditions
To provide these operating conditions on open ocean the measuring device must be
installed on a stable platform or a pontoon of sorts. The goal of this master thesis is to
determine the requirements and geometry of such a vessel. In order of importance, it
must be able to fulfil the following demands:
The Lidar must remain as upright as possible during normal sea conditions in
which wind turbines can still operate so that the Lidar can take measurements.
The efforts and costs to relocate the vessel must be minimized as much as
possible. Typically it must be light and/or easy to tow through the water for cost
effective transportation. Autonomous propulsion is not necessary but can be
considered. When the vessel is made to fit in a standard shipping container, it
would be a big advantage.
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The vessel must be able to operate autonomously for an extended period of time.
It must have an independent, reliable and redundant on board power supply for
the Lidar and any additional equipment.
The vessel must be able to stay in the same position as much as possible despite
wind or ocean currents. There must be a means of anchoring the vessel to the
bottom reliably.
The vessel must be able to resist storm conditions within reasonable limits while
keeping the Lidar module and support system undamaged. Since wind turbines
do not operate during these conditions the Lidar inclination is no longer
important.
This thesis will consist of multiple studies.
The first study will focus on existing and/or commercially available vessels (buoys,
platforms, watercrafts,…). This study will summarize the possible solutions for the
vessel design. Later on it will be determined which design is the most stable and suitable
for our application.
In later stages there will be a study in which the selected vessels are tested and
compared to one another. This will be determined using a numerical model of a wave
tank simulating ocean waves and a CAD model of the buoys. The motion of several buoys
with different geometries will be tested and compared using this model. This study will
point out the best possible solution in terms of stability within practical limitations.
After a suitable buoy has been chosen an additional study must be made to optimally
design a system that keeps the LIDAR level given the theoretical motion of the craft
resulting from the numerical simulation. A theoretical study of a stabilising mechanism
will provide insights and should enable the reader to design a similar mechanism.
Finally the sea simulations as well as the mechanical simulations are combined to
quantify performance in actual working conditions. At that point an optimal design will
have been attained.
For easy reading, the whole Lidar stabilizing watercraft with all its peripheral
installations will be referred to as ‘the vessel’.
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2 Study of commercially available hull designs
2.1 General considerations
Despite the specific nature of the vessel it is unfeasible and impractical to create a
custom made hull from the ground up. A commercially available hull, such as the hull
from a watercraft, pontoon or buoy, must be chosen and adapted to its intended purpose
afterwards. Considering there are many commercial watercrafts and installations where
stability is also a primary design feature, because of passenger comfort or necessary
stable operating conditions, there were lots of options to explore.
2.2 Stability
To be able to evaluate a good design and to not be deceived by the manufacturer’s
advertisement, a literature study was done on vessel stability. A short overview is given
in the next paragraphs.
Stability is a measure for the tendency a ship has to return to its upright position when
brought out of balance. If the ship is considered to be in rest or the disturbing forces act
very slowly, it is the static stability that tries to bring the vessel back to its equilibrium
point. The dynamic stability has to be taken into account if the vessel is subject to
sudden changes of the occurring forces. As an example, these can be due to wind or
waves. Both types of stability are very important, but mainly performance in terms of
dynamic stability will tested in the this thesis. Because, for 3E’s applications, the buoy
has to remain very stable when subject to wave forces.
An important concept in ship stability is the metacentre. The metacentre of a vessel is
the intersecting point of the work lines of the upward forces (Archimedes force) in two
different situations where the ship has a slightly different inclination. The centre of
gravity (COG) and the centre of buoyancy (COB) are visualized by the red dots in figure
3. The COB is the geometrical centre of the submerged volume and changes position due
to the different inclination. Wide but shallow or slender but deep hull shapes result in a
high metacentre. Figure 3 is a simple sketch on the construction of the metacentre. The
distance between the COG and the metacentre is known as the metacentric height and is
seen as a measure for initial static stability. In other words, a ship with a larger
metacentric height offers more resistance against overturning. The metacentric height
can be increased by a change in form, resulting in a higher metacentre, or the use of
well-placed ballast, lowering the COG. However, the larger the metacentric height, the
shorter the natural period of hull rolling, the more uncomfortable for the passengers.
Since no passengers will be present on the buoy, this causes no limitations.
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Figure 3: Location of metacentre
Previous sections point out that good watercraft stability is obtained by good choice of
the relative position of the COG and the COB and the resulting metacentric height. As
state before, there are different ways to obtain this and they can be divided in a ‘weight
manner’ and a ‘form manner’. The first is due to the correct placement of ballast,
ballasting, while the latter is known as form stability. Both can be combined.
Another distinction is made between stability for the main axis of the vessel. One speaks
of transverse and long stability. However, since ocean waves are not unidirectional, the
writers of this thesis have a preference for axisymmetrical hull shapes.
In the next paragraphs the most promising commercial designs will be described and
discussed.
2.3 Commercial solutions
2.3.1 Catamaran
The well-known catamaran design is used for high speed ferries like the ‘seacat’. This
vessel was specifically designed to cross the channel at high speed with maximum
passenger comfort despite the high currents and rough seas. A lot of stability is achieved
due to the forward movement of the craft because of its hydrodynamic shape. The Lidar
vessel, however, will be stationary so this effect cannot be beneficial.
Metacentre
Waterline
COG
COB
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If a catamaran was considered, the so called SWATH type would be better still. This type
of catamaran is pictured in figure 4 and performs better in violent sea conditions.
Because of this, the old fleet of pilot ships in port of Zeebrugge, Belgium, has been
replaced by more versatile SWATH type catamarans which can still operate in wind
conditions up to 10 beaufort because of their stability. Unlike the old ones who could
only operate up to 7 beaufort. One of these new ships can be seen in figure 5.
Figure 4: Conventional catamaran versus SWATH type catamaran
Figure 5: New SWATH type catamaran pilot ship for the port of Zeebrugge [1]
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When the catamaran is stationary, it gets its stability due to form stability and almost
none due to ballasting. The catamaran is very performing in terms of static stability. As
stated before, dynamic stability is what will be tested and this is not that great at all for a
catamaran design. Unfortunately it pitches and rolls easily in waves, making it less
attractive.
However, the reason that the catamaran is included in this thesis, is because of its
practicality. Especially in calm seas or lakes. It could be beneficial to retrofit a small
catamaran hull and build a platform upon it to house the measuring equipment. There
have been similar endeavours in the past. Like this scientific research craft: figure 6.
Figure 6: Small catamaran research vessel
Additionally, the catamaran design offers lots of space for the equipment. Not to forget it
would be very light and, since it is designed to be a moving watercraft, very easy to
manoeuvre around. If desired, an outboard motor could be installed and make the vessel
self-propelling when necessary.
The motion of the Lidar is minimised by placing the sources of the movement, the
vessel’s hulls, further apart. The amplitude of the motion is smaller in between the hulls.
For example: A similar phenomenon can be observed in a passenger train. The
passengers sitting in the middle of a wagon experience less movement than the ones
sitting on either side. This is because the middle passengers sit as far from the motion
sources, being the train bogies, as possible. The amplitude of the motion is much smaller
in the middle of the train. This works the same for catamarans as the amplitude of the
motions is much smaller in between the hulls.
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Beside bad dynamic stability, there are some more downsides. The catamaran hull will
have to be prepared and retrofitted with metal frames to be able to receive the
necessary installations for the LIDAR such as its on-board power supply and any
mooring hooks or cable attachments. This means that holes will have to be drilled in the
composite hull and metal parts will have to be fastened to it. Fastening metals to
composites is a delicate operation and will require specialized workmanship and
processes. If the fastening is not done properly, the composite could delaminate or
matrix cracks could occur. Eventually the fasteners could shear out because of fatigue
and the frame could detach from the hull and the installation could be lost.
2.3.2 Resinex PEM 58 Catamaran buoy [2]
Since our vessel is not really a moving watercraft, other vessels such as buoys or
mooring pontoons can be considered too as long as they have a stable design.
Figure 7: Stable floating plastic modular buoy for mooring applications [2]
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Specifications: PEM43 – PEM58
Manufacturer Resinex
Dimensions 4300x2200mm – 5800x3000mm
Mass 8.85 tonnes – 20 tonnes
Table 1: Specification of Resinex Buoys
“The exceptional stability of the buoy is due to a particular mooring
stabilizing system which Resinex has studied to guarantee the
maintenance of a constant structure even during a 75 – ton mooring
traction.” [2]
“This particular modular system can also be transported and
positioned at a very low cost” [2]
“These ‘monsters’ with the capacity to face the most adverse meteo-
marine conditions without any difficulty” [2]
As pointed out above a catamaran definitely has some benefits. The Italian manufacturer
Resinex implemented these benefits into their buoy design thereby making a new type
of buoy. The manufacturer names these vessels a ‘catamaran’ type buoy. They are
constructed using a metal frame filled with plastic floaters. These floaters do not fill the
entire radius of the buoy, leaving a hollow area in the middle and keeping the cross
section, intersecting with the waterline, smaller than the diameter would suggest.
A great advantage the Resinex buoys offer, is that they are made from modular
components and can be (dis)assembled easily. As a consequence, handling and
transporting them is very practical.
They provide a useful platform to build upon and are purpose built to be anchored to the
seabed. The buoys are designed to remain as stable as possible since they are designed
to let passengers have access to offshore installations if needed. They can be used in
deep ocean water up to 6 km of depth. Which is more than sufficient considering the
depth limitations for windmills.
These buoys are used for mooring ships, but there was at least one case where the buoys
were successfully used to place radio transmitters at sea. Lots of preparations have
already been made to make it easier to install equipment such a micro windmills, solar
panels and perhaps measuring equipment. Also a towing hook is preinstalled. In other
words, this product has already covered a lot of practical issues.
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The manufacturer produces two types. The PEM 43 (figure 8) and the PEM 58 (figure 9)
with respective diameters of approximately 4.3 and 5.8 meters. Taking into account the
dimensions of the solar panels and other installations 3E implements on the buoy, the
PEM 58 is most suitable.
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Figure 8: Resinex PEM 43 Catamaran buoy. Diameter approximately 4.3m
Figure 9: Resinex PEM 58 catamaran buoy. Diameter approximately 5.8 m
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2.3.3 Resinex PEM 58 buoy [2]
In a consultation with 3E, a decision was made to contact the manufacturer of the PEM
buoys, Resinex. Without going in detail the required specifications for a buoy were
clarified to the Resinex engineers. In response a suggestion was made to use their
normal PEM58 Buoy instead of the catamaran model. The buoy offers all the benefits
noted in previous paragraph, but should be even more suitable for 3E’s application.
It looks exactly the same as the buoys displayed in figure 9, but there is a steel
centrepiece instead of the centre being hollow.
2.3.4 Spar buoy
The spar buoy is mentioned because its design is a classical solution of a stable floating
buoy for installing radio equipment.
It is a slender buoy, but with a large mass located deep under the waterline, at the
bottom. Its slenderness makes the cross-section intersecting the water small. Therefore
the response to wave forces is minimized. The mass, on the other hand, acts as a
counterbalance to keep the buoy upright. Both these features combined ensure that this
buoy does not follow the wave movement and make the spar buoy superior in terms of
dynamic stability.
Despite being very stable it also has multiple drawbacks. To be able to provide enough
space for all measuring equipment etcetera, the dimensions would have to be very big.
This and the fact that it was never designed to be transported, make it very unpractical.
Towing a spar buoy is difficult, so it could only reasonably be moved over a long
distance if it was lifted out of the water onto a ship. This means only larger and more
equipped ships could transport the vessel.
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Figure 10: Schematic of a typical spar buoy design.
2.4 Conclusion
The conclusion of the hull study is that there is no such thing as ‘the perfect buoy’. For
every practical use a trade-off will have to be made between dynamic stability and
practicality.
The buoy mostly meeting the needs of 3E (section 2.3.3) is the PEM 58 buoy. Not only it
has a proven stability, but due to the modular design it will be easy to handle and
transport. The large, flat top area offers lots of space for the installations and due to the
steel centrepiece their fixation can be realised with conventional techniques such as
welding, bolting,... All of this in combination with an enthusiastic manufacturer as
Resinex can produce great results.
Keep in mind that the Lidar will be suspended in a stabilizing unit. The extra movement
of the Lidar, due to the difference in stability between the spar buoy and the PEM 58,
will be minimised.
In next stages, a model of the current 3E prototype will be used in simulations and the
results will serve as a reference. Afterwards the PEM 58 and the spar buoy will be
subject of the test. The performance of the latter is useful as a measure for dynamic
stability.
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3 Numerical study
3.1 Preface [3]
Computer models using numerical simulations have increasingly gained significance for
solving practical problems in engineering and science. It offers an inexpensive and fast
alternative to test multiple situations. The correctness is evaluated afterwards by
comparing the numerical results to experiments. Due to a lack of physical experiments
on ocean waves and the undercurrents, the results in this thesis will be compared to
expectations based on theories and literature.
3.2 Numerical simulations in general [3]
The models in this thesis use a combination of grid-based and meshfree numerical
methods.
The grid-based methods are very popular. However, their applications are limited for
many complex problems. These limitations become apparent when modelling free
surfaces, deformable boundaries and extremely large deformations. All common in wave
simulations.
3.2.1 SPH: a meshfree particle method
To be able to deal with these problems, meshfree particle methods (MPM’s) are used. In
particular the smoothed particle hydrodynamics (SPH) method. This technique is
capable of dealing with all previously mentioned challenges. On top, the SPH method is
also the oldest MPM, thus numerous improvements have been implemented. This results
in an accuracy and stability that have reached an acceptable level for practical
engineering applications.
The grid based methods consist of two fundamental frames for describing the physical
governing equations. These are conservation laws for mass, momentum and energy.
The Eulerian description is a first frame and is a spatial description, typically
represented by the finite difference method (FDM). Eulerian grid-based methods are
widely used in computational fluid dynamics (CFD).
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The second fundamental frame is a material description named the Lagrangian
description. Mostly represented by the finite element method (FEM). The Lagrangian
methods are very popular for computational solid mechanics (CSM). All the modelling
done in this thesis is based on Lagrangian methods. A Lagrangian grid is fixed to the part
throughout the entire computation and therefore moves with the part as it deforms.
Because of this, the entire time history of all the field variables at a fixed point on the
part can easily be tracked and obtained. For this reason, a Lagrangian method is used for
all parts. The seawater in particular is modelled using a Lagrangian MPM.
This is excluded because the accuracy would be severely affected due to the earlier
mentioned limitations grid based methods impose. A possible option is to rezone the
mesh or re-mesh the problem domain. The rezoning techniques are quite popular for
simulations for impact, penetration, explosion, turbulence flow and fluid-structure
interaction problems. On the other hand, the rezoning techniques can be very time
consuming and material history can be lost.
As mentioned before, the MPM’s can offer an alternative solution. Here the problem
domain is discretized in particles without a fixed connectivity. That is why, in this thesis,
water is modelled using SPH-particles, a meshfree Lagrangian method. Treatment of
large deformations is relatively easy. The most significant advantage SPH especially has
over traditional grid-based numerical methods is its adaptive nature. This adaptability
of SPH is implemented at the very early stage of the field variable approximation. The
latter is performed at each time step based on a current local set of arbitrarily
distributed particles. It is, in fact, due to this adaptive nature, that SPH formulations are
not affected by the randomness of the particle distribution. More on this can be found in
following paragraphs.
Each SPH particle represents a part of the problem domain, with attributes such as mass,
position, momentum and energy concentrated on the mass or geometric centre of this
sub-domain. The key idea of mesh free methods is to provide accurate and stable
numerical solutions for integral equations or partial differential equations (PDE’s) with
all kinds of possible boundary conditions with a set of arbitrarily distributed nodes, the
particles, without using any mesh that provides the connectivity of these nodes.
The SPH method itself was invented originally for the modelling of astrophysical
phenomena. In essence the interaction of stars. It uses an integral representation
method for field function approximation. In the SPH method, this is defined as kernel
approximation and is shown underneath.
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This integral is then approximated using particles. This is called particle approximation.
It is done by replacing the integration in the integral representation, of the field function
and its derivatives, with summations over all the corresponding values at the
neighbouring particles in a local domain. This area called the support domain. In this
domain the smoothing function has a non-zero value.
Figure 11: Support domain
Another important concept is the influence domain. The difference between both is
that the latter is the area in which a particle influences others, while the support domain
is an area around a field point. The use of support or influence domain leads to different
approximations: respectively the gather and scatter model. Since the smoothing length
of two particles may not necessarily be the same, a violation may occur as pictured in
figure 12. It may happen that the red particle falling within the influence domain of the
blue particle does not influence the blue particle. Therefore it is possible for the blue
particle to exert a force on the red particle, without the red particle exerting one on the
blue particle. This is an obvious violation of Newton’s Third Law. This fault is overcome
by the use of a mean value of the smoothing length for the two interacting particles. This
mean value becomes the radius of both domains. Therefore the support domain and the
influence domains for an SPH particle are practically the same.
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Figure 12: 2D representation of support and influence domains for illustrating possible problems when using
different smoothing lengths.
The particle approximation is performed at every time step and depends on the current
local distribution of the particles. This mechanism grants SPH its adaptive nature. It was
initially developed as a probabilistic MPM, but was modified later on as a deterministic
MPM. This use of particle summations to approximate the integral is, in fact, a key
approximation that makes the SPH method simple without using a background mesh for
numerical integration. In this particle approximation mass and density of the particles
are introduced into the equations and therefore is attractive for hydrodynamic
simulations in which the density is a field variable in the system equations.
In mesh free particle methods each particle can be either directly associated with one
discrete physical object or be generated to represent a part of the continuum problem
domain. The latter applies to the water wave simulations.
3.2.2 Clustering
A common flaw when using SPH for numerical simulation is the occurrence of ‘clustering’. The paragraph explains the clustering phenomenon in a few sentences, since the term will be used in section 3.5.3, 3.5.5.3 and 3.6.3.
1
1
Blue influence domain Red influence domain
Force by blue on red
Support domain of (1,1)
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The clustering of SPH particles is the result of a tensile instability. Parts that have an EOS are particularly sensitive to this effect. When compressed, the SPH particles repel each other. This is comparable to repulsive forces between atoms in reality. If the pressure on the SPH particle is negative, however, this can result in an unphysical clumping of the SPH particles due to an instability in the exerted attracting forces. This clumping is defined as ‘clustering’. According to [4], there is a relation between the said instability and the sign of the second derivatives of the SPH interpolating kernel when the particles are under negative pressure. A clear image of the clustering phenomenon can be observed in figure 38.
3.3 LS-Dyna and LS-PrePost
The modelling and simulating of the ocean waves and the subsequent movement of the
buoys is achieved using the LS-Dyna software. LS-Dyna is a software package developed
by Livermore Software Technology Corporation (LSTC). The software is used for
advanced multiphysics simulations by numerous industries such as automotive (figure
13), aerospace and so on.
Figure 13: Crash test [5]
One way to create, import and adjust models is with help of LS-PrePost. In LS-PrePost
the code used by the LS-Dyna Program can be created in a more user friendly way. One
can also type the code manually, however this is only recommended for the more
experienced user.
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The specifications (materials, boundary conditions, …) of a model are defined in LS-Dyna
with the use of cards. For example, a material can be defined by the use of the MAT-card.
The MAT-card itself has subtypes such as MAT_ELASTIC, MAT_RIGID,… which represent
material models. When a complete model is generated it can be saved as a file with a ‘.k’
file extension or more simple a ‘k-file’. This k-file, a text file with code containing all the
defined cards, is the input for the LS-Dyna Program. The latter can interpret the cards
and their coherence and run the simulation. Afterwards, the results can be viewed and
interpreted by using the post-processing tools offered in LS-PrePost.
To create a model one usually starts with creating the geometry of the parts that have to
be included in the model. The geometry itself can be generated using the associated
tools in LS-PrePost, but they can also be imported in a STEP or IGES file format. This last
option gives the user the opportunity to make the geometries with CAD-software.
A geometry is made up out of points, edges and planes. When meshing the geometry one
creates a part, represented by a PART card. A part consists of elements (ELEMENT)
while the elements themselves are defined by a group of nodes. This hierarchy is shown
in figure 14. Different types of elements exist. For example, one node can represent an
SPH-element, while four nodes can make up a shell element or eight nodes could define
a solid element. Please remember, almost every object in Dyna is represented by a card.
Figure 14: From geometry to useable LS-Dyna code by meshing
Geometry
Points
Edges
Planes
Part
Elements
Nodes
Created in LS-PrePost or imported out of CAD software
Useable code for LS-Dyna
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The PART card needs a reference to at least a material card (MAT) and a section card
(SECTION). The former specifies all the material specifications such as density, Young’s
modulus, Poisson coefficient and so on. While the latter offers the ability to define the
part as a solid, a shell, an SPH particle or a beam. There are more section definitions, but
only those used in this thesis are mentioned. When defining an SPH-part, an equation of
state card (EOS) might also be necessary. For clarity, the structure for a rigid shell part
is given in figure 15 while the corresponding extract of the k-file can be seen in figure 16.
Figure 15: Hierarchical structure used in the LS-Dyna code for a rigid shell part.
Part
Mat
Rigid
Density
Young’s modulus
…
Section
Shell
ElForm
Thickness
…
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Figure 16: Extract of k-file
Just as in an experimental wave study the performance of the buoys will be tested in a
wave tank. A wave generator, named ‘the paddle’, will produce the waves. More over the
exact design and dimensions can be found in sections 3.4 and 3.5.
As an example, the working method of wave tank creation is explained brief and to the
point. For parts with a simple geometry it is easier to define the nodes and elements
directly instead of creating a geometry and meshing it. To create the paddle, two planar
elements are constructed. Both elements together can be interpreted as the centre plane
of the paddle. It seems impossible to make a paddle with a rotational axis at the bottom
edge. The software does not allow to select an axis, but places it at the centreline of the
part. For this reason two elements are used instead of one.
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Figure 17: Paddle
When adding the SECTION_SHELL card to the part ‘paddle’ a shell thickness has to be
defined. A value of 0.2 means the shell thickness is 20 centimetres. One can think of it as
10 centimetres added to each side of the centre plane. This can be seen in figure 18. Here
the centre plane of the paddle can be seen in top view, represented by the green line.
Finally the MAT_RIGID card is added.
Figure 18: Spacing between the parts and particles
Node
Element
Rotation axis
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As discussed in section 3.2, SPH-particles are used to model the water. One way to create
an SPH part in LS-PrePost is by defining a box that contains all SPH-particles. The outer
dimensions of this box have to be specified. Once this is done, the spacing of the
particles, named pitch length, between neighbouring particles in all major directions has
to be determined. It was decided to take the same pitch length in each direction.
Together with the initial, user definable, density this pitch length determines the
particle mass. This mass is fixed during the entire simulation. This in contrast to the
volume and the density of the particles.
The SPH-particles all together can represent a piece of glass, a human being or in this
case seawater. The physical behaviour of the SPH-part is also derived from a material-
and a SECTION-card. When using NULL_MATERIAL, as in this thesis, an EOS_card is
needed as well. In this case EOS_GRUNEISEN. It is very important to fine-tune the
parameters in these card to make sure the modelled water acts and reacts just as real
water. The used values can be found in table 2.
Material card
Density ρ 1000 kg/m3
Viscosity 0.001 m/s2
Section card
CSLH 1.3
Table 2: Material and section card parameters for SPH
Due to the salinity of the seawater the real density is around 1020 kg/m³. Instead of this
value a density of 1000 kg/m3 was chosen, as seen in table 2, to be able to easily
calculate the SPH-particle mass. Smaller particle masses generally give more stable and
more realistic simulation. However, as said before, calculation time is a limiting factor. In
section 3.6.1 a mesh convergence test is done to determine and optimum in calculation
time and representativeness.
A very important card to implement is the CONTACT-card. This makes sure the water
stays in the wave tank, reacts to the paddle movement and keeps buoy afloat. The
particular contact card used is CONTACT_AUTOMATIC_NODES_TO_SURFACE in
combination with CONTROL_CONTACT. The contact gives repelling forces between
nodes of the parts where contact is defined. These forces increase when the nodal
distances decrease.
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To avoid very large contact forces at the start of a simulation, one should avoid nodes
penetrating into other parts. This would result in an unphysical behaviour. However, the
SPH-particles do have to be closely packed. The correct method to do so is shown in
figure 18. In this figure the green line is the top view of the centre plane of the paddle
while the red box is the SPH-box. The blue circles in the box represent the SPH-particles.
When using this method in combination with small SPH-particle weights, say 8 kg or
less, contact will not fail. The use of heavier particles will most likely result in a contact
failure. One can use the thickness override options, but this does not guarantee a
satisfying result.
3.4 Water wave mechanics [6]
To be able to develop representative models of realistic sea states an extra study was
done on water wave mechanics. Multiple books and articles ( [7] [6] [3] [8] [9]) have
been studied, but also sea state data on the Thornton Bank provided by 3E and Geo Sea
was inspected thoroughly. An extract of this study can be found in following paragraphs.
Gravity and surface tension tend to maintain a level water surface. Due to forces acting
on the fluid, and counteracting gravity and surface tension, waves are formed. These are
so called wind generated waves and will be simulated in the models. Another category,
the tides, which are the longest known water waves, are the result of the gravitational
attraction of the moon and the sun.
The most important parameters to describe a wave are its wavelength λ0, wave height H
and the wave period T. Other parameters can be determined theoretically from these
quantities. Using figure 19, the wavelength can be defined as the horizontal distance
between two successive wave crests. However, in this thesis, wavelength is represented
by λ0 instead of L. The period T is the time required for two successive crests to pass a
particular point. The speed of the wave, called the celerity C, is defined as C= λ0/T. The
wave amplitude a is H/2. The wave tank dimensions, the boundary conditions of the
paddle will be deduced, and the correctness of the simulations will be checked using
these parameters.
Figure 19: Wave characteristics [6]
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A real sea wave is a superposition of a large number of sinusoids moving in different
directions. The use of this linear wave theory makes it possible to simulate the sea in a
number of different simulations followed by a superposition of the results. This has been
shown to be reasonably accurate for some purposes. At first this method was preferred
to simulate the sea in a representative manner. However, due to factors as calculation
time, multiple simulations with changing orientation of the buoys would be very time
consuming. For this reason only one simulation per buoy, and thus no superposition,
will be made. Because only the movement of the buoy in the water is of interest and
waves in deeper water appear to be rather random, this has no effect on the reliability of
the results.
In table 3 one can find an extract of the sea state data as provided by GeoSea.
Measurements of every hour of every day in September, October and November in the
year of 2011 were given. These were taken at the Thornton bank, which is a sandbank in
the North Sea and the location of a wind farm. It goes without saying that this is a very
representative to the working environment of the 3E buoy. To subject the buoys to a
tough test, a selection of only the roughest sea states over the three months was made. It
is this selection that is displayed in table 3. The wave height HM1 gives the mean of the 1
% highest waves measured in a time span of one hour. The last row gives the mean
values of all selected measurements.
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Date and time Wave height HM1 Highest wave Wave period
(highest energy)
[cm] [cm] [s]
11.09.2011 (21u00) 253.53 311.40 6.67
11.09.2011 (22u00) 307.17 377.29 6.67
21.09.2011 (18u00) 242.33 297.64 5.88
21.09.2011 (19u00) 262.64 322.59 5.88
07.10.2011 (02u00) 483.01 593.26 7.69
07.10.2011 (03u00) 457.87 562.39 7.69
17.10.2011 (22u00) 191.80 235.59 6.67
17.10.2011 (23u00) 212.48 260.98 6.25
01.11.2011 (03u00) 106.66 131.00 3.57
01.11.2011 (04u00) 111.98 137.54 3.57
01.11.2011 (11u00) 106.74 131.11 4.55
03.11.2011 (16u00) 110.11 135.24 3.85
03.11.2011 (17u00) 110.25 135.42 3.85
04.11.2011 (08u00) 164.08 201.53 4.17
04.11.2011 (09u00) 172.55 211.94 4.35
06.11.2011 (17u00) 265.11 325.63 7.14
06.11.2011 (18u00) 257.45 316.22 6.67
Mean values 224.46 275.69 5.59
Table 3: Sea state date on the Thornton bank
3.5 Wave tank design and dimensions
3.5.1 Basic wave tank dimensions
This wave tank was designed for testing purposes after the example of Joren Pelfrene.
The wave tank dimensions are chosen to model a so called ‘deep water’ wave.
As stated before, the dimensions of the wave tank will also be determined based on the
values of table 1 and relations found in [6].
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The authors of this thesis had a meeting with someone from the department of maritime
technologies at the University of Ghent. It was said that 25 metres is a common
wavelength in the North Sea at Zeebrugge. This value will serve to determine the length
of the wave tank. This length has a linear relation with the wavelength. More concretely,
wave tank length should be equal to three times the wave length. The dimensions of the
wave tank will be determined based on a wavelength of 25 metres, although the
wavelength derived from GeoSea will be longer. The wave tank will be 75 meters long in
total. This is done to limit the amount of SPH-particles needed to fill the tank and
consequently reduce calculation time. A more detailed overview follows in the next
paragraphs.
The buoy will be placed in deep water, so the simulations should be representative for
deep water conditions. ‘Deep water’ is a relative concept. In [6], a relation between the
tank depth h and the wave length λ0 is found for deep water waves. Here it was
proposed that the wavelength of a deep water wave expressed relatively to the tank
depth is:
λ0: Wavelength of wave
h: Tank depth
When using 25 metres for λ0, a lower value to limit tank dimensions, it becomes:
When substituting twenty five meters for λ0 the tank depth h of the wave tank becomes
16 meters. Again, the only reason the value of 25 metres is used for λ0 to make sure the
wave tank is not excessively large.
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To extract the real wavelength out of the sea state data by GeoSea a relation between
wave period and wavelength is needed. This was found on the website of the Flemish
Hydrography [9] and can be written as follows.
There is a large dispersion for the occurring wave periods and also the water depth at
the Thornton bank varies between 12 and 27 meters. For this reason there will also be a
large variation in discernible wavelengths. When substituting the mean value of 5.59
seconds, as found with the GeoSea data, as the wave period and taking h as the average
depth at the Thornton bank, which mathematically is 19.5 meters, in the formula above,
the wavelength is 48.27 meters.
The generated waves in the simulations should have this value for the wavelength. The
boundary conditions for the paddle will be set to do so.
Otherwise, when using a value of 25 metres for the wavelength and the tank depth h
of 16 a value of the wave period T of 4.003 seconds was calculated. This value fits right
in the wave data provided by GeoSea. This proves once more that the chosen value to
determine the depth and length of the tank is also pretty accurate.
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As a summary table 4 gives some of the design parameters.
Wavelength to determine tank dimensions 25 m
Wave tank depth 16 m
Wave tank length 75 m
Target wavelength of the generated waves 48.27 m
Target wave period 5.594 s
Target average wave height 224.46 m
Target largest wave height 275.69 m
Table 4: Summary the design parameters for basic wave tank
3.5.2 Boundary conditions of the paddle
In the book [6], it is said that the flap type wave maker is the more efficient for deep
water waves. This power efficiency is the main reason the flap type wave generator is so
popular in experimental wave studies. A picture of a real wave tank can be seen in figure
20, while a schematic representation of a flap type wave maker can be seen in figure 21.
The displacement S of the wave generator, which can be seen in figure 21, is known as
the stroke.
Figure 20: Wave Tank CSI Chicago [10]
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Figure 21: Flap type wave maker
The water displaced by the wave maker should be equal to the crest volume of the
propagating wave form. The volume of a wave crest can be calculated like this:
‘k’, in the formula above, is known as the wave number. This is calculated as λ0/(2∙π).
The volume displaced by the paddle, with h the water depth and S the stroke of the
paddle, is:
Which makes, for a flap type wave generator:
h
S
S: Paddle stroke
h: tank depth
Flap type wave maker
or paddle
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This simple correlation for the ratio of wave height to stroke is accurate only for low
values of .
When extracting some values out of the GeoSea measurements the stroke of the wave
generator S can be calculated. As previously mentioned 48.47 meters will be used as a
typical wavelength λ0 in the North Sea and 2.757 meters as target maximum wave
height. The depth of the wave tank h was chosen to be 16 meters in section 3.4.
However for values of greater than two, this simple correlation is not that accurate.
In this case equals 2.07408. So another correlation will be used for reasons of
accuracy. The correlation suggested for k∙h greater than 2 becomes, with wave height H
and stroke S, according to [6] is:
When substituting the mentioned values the stroke S can be calculated and becomes
S=2.573m
This value will initially be used as the stroke of the wave maker. This is done by setting
an appropriate value for the amplitude of the sinusoidal function used as a boundary
condition in LS-Dyna. Further fine-tuning will be done using the output of the simulation
as feedback.
The sinusoidal curve is defined in DEFINE_CURVE and is used as load curve (LC) in the
PRESCRIBED_MOTION_RIGID card.
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The used sine function is:
The value of 1.123 for the angular speed ω is a characteristic value for North Sea waves.
This value is attained with the use of the GeoSea data. The mean value for the period at
rougher sea conditions, extracted out of table 3 is 5.594 seconds.
The amplitude A is set to 0.1, but will be fine-tuned using the scale factor SF in the
PRESCRIBED_MOTION_RIGID card. In the final model this value is set to 0.895. The
amplitude A itself yields no displacement in meters, but the maximum deflection angle
in radians. With the rotational axis set at the level of the seabed the used stroke S’ is
As can be seen the value of S’ is somewhat larger than the theoretical stroke S of 2.573
metres, found with the theoretical formulations. However, the used stroke S’ gives
waves in best agreement to the sea state data. In addition, a more violent sea is even
better to test the performance of the buoys.
3.5.3 Boundary conditions of the wave tank
A big problem when simulating ocean waves, are the limited dimensions of the wave
tank. In practice, the sea can be regarded as an infinitely big domain, but due to our
restricted computer calculation power the dimensions in the numerical simulation are
very small in the relative sense. Because of this, waves will ‘see’ boundaries and interact
with them. The result is that the waves will reflect on the boundaries. The interest of the
simulations lies only in the movement of the buoys so the reflections themselves would
not be dramatic. However, because the reflections might cause a standing wave instead
of a travelling wave, they should be minimized.
An infinite domain can be simulated by taking special measures to absorb wave energy
at the boundaries. There are different ways to implement a wave energy absorber. Some
suggestions found in literature and some personal ideas of the writers of this thesis are
discussed below.
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One of them, as discussed by Milgram [7], is based on the flap type wave generator. The
idea is that the paddle moves in such a way that it is invisible to the incident wave. So to
generate a wave, the fluid is pushed forward by the paddle and thus making a crest. In
this case, an absorbing paddle moves backwards when the wave arrives, making it
appear as if the wave has passed through. To make it an efficient absorber, the paddle
motion has to be just right, making it insufficiently robust for our purposes.
Another technique, used in Boussinesq wave modelling [7], is the use of a sponge layer.
This is a more user friendly manner to prevent reflection and prevent a standing wave.
The sponge layer is an extra layer of SPH particles at the boundary opposite to the wave
maker, but with added artificial damping. The damping is applied using the card
*DAMPING_PART_MASS. The damping coefficient has to be determined by trial and
error and will differ from case to case. It has been proven that the values of the damping
coefficients increase with increasing depth. The sponge layer is an extra part of SPH
particles with the same particle spacing as for the particles representing the water. A
sponge layer that is one wavelength λ0 long in the wave direction should be sufficient.
Just as for the sea water NULL-material is used in combination with the GRUNEISEN
equation of state (EOS). In the EOS, the value for the speed of sound c is set to 100 m/s.
This is done to reduce the calculation time. The sponge layer is visualized with a red
colour in figure 22. A Dyna model can be observed in figure 23.
Figure 22: Wave tank design with sponge layer from [7]
Figure 23: Model with 3E prototype, cable and sponge layer
Sponge layer
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Due to the large dimensions of the wave tank, the use of a sponge layer would increase
the amount of nodes with about 33%. The consequence is an even longer calculation
time. Which is unsurpassable considering the limited calculation power.
For this reason an SPH_SYMMERTY_PLANE is used. If this is implemented, the model
behaves as though there is an imaginary mirror image of the wave tank against the right
side. LS-Dyna does not mirror the entire wave tank, but only the three rows of SPH
closest to the SPH symmetry plane. At the symmetry plane it seems as if two waves with
the same amplitude, but different propagation directions meet. Reflection is not fully
eliminated, but calculation time is greatly reduced in comparison to the use of a sponge
layer.
As a proof of the performance of the SPH_SYMMETRY_PLANE a sequence of screenshots
is shown in figure 24, visualizing the travelling wave. In other words, the use of an
SPH_SYMMETRIE_PLANE was successful.
Figure 24: Travelling wave in Dyna model
However, for longer simulations it can be seen that a travelling wave evolves into a
standing wave. This phenomenon has only been observed for a 37.5 m long wave tank,
though. For the termination times set, no standing wave is formed for the 75 m long
wave tank.
An SPH symmetry plane was also chosen at the bottom of the wave tank instead of a
rigid wall, which is used at the sides of the tank. Clustering is reduced heavily because of
this. In LS-Dyna symmetry planes are used to model a continuous domain, so a
symmetry plane on the bottom makes the water behave like it would in a deeper tank.
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3.5.4 Summary: Basic wave tank
A schematic representation of the modelled basic wave tank is shown as a summary in
figure 25:
Figure 25: Boundaries and dimensions
The final model of the wave tank has:
Flap type wave generator, known as the paddle, on the left side
Representative domain with a length of three wavelengths where wave
behaviour can be observed
At the side opposite to the wave generator an SPH-symmetry plane is used to
prevent the generation of a standing wave.
The water is modelled with SPH. This technique and the materials were discussed
earlier.
The tank width is chosen to be 10 meters. This is more than sufficient to make sure the
buoy will not hit the walls.
SPH-symmetry plane (Back)
SPH- symmetry plane
(Bottom)
Rigid wall (Sides)
3∙λ0 = 75m
10m Water depth = 16m
Wave generator
(Moving)
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3.5.5 Friction interface
3.5.5.1 Undercurrent modelling with linear water duct and pistons
Now that the tank and the wave generator have been created, the model can be
expanded to incorporate the effects of the undercurrents typically encountered in
oceans. The speed of this undercurrent can rise op to 2,5 meters per second [8]. A
current speed of 1,5 meters per second is believed to be representative.
This undercurrent could be modelled using a moving stream of SPH particles. However,
LS-Dyna supports neither particle generators nor destructors. However, a flow of
particles could be generated by using a pseudo generator and a pseudo destructor. How
this system was implemented, is explained below.
Figure 26: Scheme of undercurrent duct
The wave tank sits on top of a narrow water duct. The water in this duct is forced to flow
by pistons pushing it along. The translational speed of the pistons determine the speed
of the undercurrent. The duct is much longer than the wave tank because the duct
volume before and after the wave tank serves as a particle buffer . The minimal buffer
length is easily calculated:
Minimal buffer length = termination time ∙ current speed
This setup understandably succeeds in creating an underwater flow and creating a
friction interface with the bottom particles of the wave tank. Obvious disadvantages are
that a lot more particles are needed and the geometry is more complicated.
Accumulation of outflowing particles, unable to enter the outflux buffer, was also an
issue.
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3.5.5.2 Undercurrent modelling by closed loop water circuit
To escape the limitations of the buffer size on the termination time a closed loop SPH
particle circuit was considered. A geometry with a closed loop circuit was modelled and
is pictured in figure 27.
Figure 27: Undercurrent circuit model
The current circulates in the water duct on the bottom of the wave tank (red part). The
undercurrent is propelled by a rotor (grey part) that keeps the water circulating. This
model has indeed the advantage that very long buffers are no longer needed but the
geometry and contact definitions are so complex that this idea was abandoned.
3.5.5.3 Friction interface with moving plate
One could think that modelling of the undercurrent is not about the undercurrent itself,
but about the friction interface between the upper body of water, moving under
influence of the wind and exhibiting the wave motion, and the lower layer of water
flowing at the undercurrent speed. Even more important is the effect of the
undercurrent on the mooring cable . A very efficient way to model this is the use of a
moving plate which slides against the bottom particles. The translation itself was done
with the use of the PRESCRIBED_MOTION_RIGID card combined with a unity vector as
LC and VAD set to tree.
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The first model that was made to realize this friction effect used protrusions. A
screenshot can be seen in figure 28. Due to the compression of particles between the
paddle and the closest lump, very strange phenomena could be observed.
Figure 28: Lumped plate friction model
Another attempt to avoid these phenomena was to use a plate with friction. This is
pictured in figure 29. The friction was set using the DEFINE_FRICTION card and making
small changes for the associated parameters in the contact definitions. The plate would
represent the interference of an undercurrent layer and a top layer moving at a different
speed. Friction added to this plate is in agreement with the relative speed of the
undercurrent that is desired to be modelled. The plate had the desired effect, but due to
rapid clustering (section 3.2) it was inappropriate to use in longer simulations.
Figure 29: Friction plate model
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3.5.5.4 Simulating undercurrent by applying resultant forces on the mooring
cable
Eventually a concept was thought out to impose the effect of the undercurrent without
disturbing the SPH particles. Instead of trying to model the undercurrent itself, the
resulting forces on the mooring cable could be applied. This can be done using the
LOAD_BEAM_ELEMENT card since there is only one beam element used to represent the
mooring cable. This option is the most feasible, but due to a lack of time it will not be
implemented. The implementation itself is not the problem tough. A practiced Dyna user
can incorporate the needed card in a few minutes and so it has been done. Again the
calculation time was the killjoy. The final model has a buoy anchored with a mooring
cable, without resultant undercurrent forces. With the use of a supercomputer owned by
the University of Ghent it still takes about a thousand hours to run the simulation. For
some unexplainable reason the cable has an immense effect on calculation time. Adding
forces to the cable would only worsen the this influence.
Figure 30: Buoy model with mooring cable and two forces: downwards and sideways
Buoy
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3.5.6 Additional cards
3.5.6.1 INITIAL_STRESS_DEPTH
Another card that was tried is the INITIAL_STRESS_DEPTH card. This card initializes
solid element stresses where stress is a function of depth. The card was added hoping it
would not only work on solids, but also set the hydrostatic pressure before the
simulation starts.
For heavy particles, in the early stages of the research, strange effects were observed
due to the settling of hydrostatic pressure and the effects on the particle spacing
between the bottom SPH-particles. In essence, it was a transient phenomenon.
Initially all SPH-particles are arranged in a grid with equal spacing in all major
directions. This can be seen in figure 31. At the start of the simulation the hydrostatic
pressure is set and the particle spacing changes, while particle mass is fixed. Therefore
volumes and, as a consequence, the reaction forces of the contact change. Because of the
large mass of the SPH particles in early simulations this entailed unphysical effects.
Later on, this transient phenomenon was minimized as detailed models were used with
heavily reduced particle weights. In retrospect this card had no or just a minimal effect.
Figure 31: Initial grid of the SPH-particles
3.5.6.2 LOAD cards
An absolute necessity is the inclusion of gravity in the simulation. Gravity is not present
as standard, but with the use of LOAD_BODY_Z and the correct LC this can easily be
solved. This card affects every part in the model.
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To simulate the launch of the buoy in the water as if executed with a crane an additional
card is used. This is LOAD_BODY_GENERALIZED_SET_PART en works only on the buoy.
The load curve is defined as piecewise function and yields the following features.
The first 2 seconds of the simulation, the buoy hovers above the water. This to make
sure the buoy does not interact with the water when the transient phenomenon of the
SPH is still in progress. Gravity is completely counteracted.
In the next stage the buoy is lowered slowly into the water. This is implemented to make
sure the buoy does not slam too hard into the water and the SPH particles do not get
compressed too much. Gravity is counteracted up to 75%. More on this in section 3.8.3.
After 7 seconds from the start, the upward forces defined by the
LOAD_BODY_GENERALIZED_SET_PART card are all cancelled.
3.6 Model verification
3.6.1 Mesh convergence
It is important that the modelled water has a steady behaviour and behaves just as
actual water. Apart from using the correct values in the MAT- and SECTION-card and
implementing the appropriate EOS, as mentioned before, the convergence of the mesh
should also be checked.
As explained before, large pitch lengths result in high particle masses. In turn this can
result in unphysical behaviour. The behaviour of the SPH-water in terms of stability,
contact, wave period, wave height and wavelength are checked for increasingly smaller
pitch lengths until convergence occurs and a stable simulation is guaranteed.
The pitch lengths were chosen in a way the associated particle masses were 1000 kg,
500 kg, 250 kg, 100 kg, 50 kg, 25 kg, 12 kg ,10 kg and 8 kg. Even smaller particle mass
simulations were executed, up to 1 kg. However, the long simulation times made this
very unpractical.
The measurements are made using the post processing tools in LS-PrePost. Nodal time
history of the z-displacement is used to check the wave height and the wave period. The
selected nodes are all in the vicinity of where the buoy would be placed. The latter is not
yet included in these simulations. Wavelength is measured by selecting two nodes at the
crest of two consecutive waves. All measurements were done seven times to be able to
average them out.
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3.6.1.1 1000 kg particles
A huge benefit of working with heavy particles is that the required computation power
and calculation time are very small. A ninety second simulation of the sea can run in
under ten minutes. Nevertheless, this is also the only benefit. Due to the immense mass
of the ‘water droplets’ contact failure is inevitable. In figure 32 failing contact is pictured.
The water goes through the paddle and splashes up a rigid wall, left to the paddle. The,
non-visible, rigid wall is included in the model to ensure the SPH-particles stay enclosed
in a relatively small volume. As can be seen in following subsections, this contact
problem fades when particle mass decreases while contact definitions remain
unchanged.
Figure 32: Failing contact
Rigid wall
Contact failure
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Measurement [#] Period Wave height Wavelength
[-] [s] [m] [m]
1 5.89 1.47 46.07
2 6.42 2.44 56.56
3 5.40 1.57 41.74
4 5.30 1.64 55.77
5 6.10 0.92 47.03
6 5.59 2.33 22.85
7 5.71 1.59 50.23
average 5.77 1.71 45.75
max 2.44 56.56
Standard deviation 0.40 0.52 11.40
Table 5: Table of wave measurements in Dyna models for 1000 kg particles
3.6.1.2 Conclusions
It would be a waste of paper to include all tables, for all listed particle masses, in this
thesis. Only relevant conclusions have been documented.
It can be concluded that simulations run more stable when particle weight is decreased.
The amount of particles leaking out of the wave tank decreased and for 25 kg particles,
this was negligible compared to the 1000kg simulations. This can be seen in figure 33.
Figure 33: Wave tanks with 25kg (upper) and 1000kg(lower) particles.
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This leakage and the settling of the hydrostatic pressure at time step 0 have a great
effect on the water level. For the 1000 kg particles, the sea level was 2 metres lower than
intended. This poses a problem, because the buoys would be lowered into the water as if
performed by a crane. The first two seconds of the simulation the buoy hangs steady
above the water. Between two and seven seconds the buoy drops slowly into the water.
This is achieved by adding an extra force on the part set ‘buoy’, which counteracts
gravity. If the sea level is lower than intended, the buoy will not be in place when the
extra force is no longer present and the buoy falls in at full speed.
As the pitch length decreased this effect on the sea level decreased as well, as can be
seen in figure 34. The sea level is appointed with the red line.
Figure 34: Sea level for 1000 kg(upper) and 25 kg(lower) particles
Sea level at -2m
Sea level at -0.5m
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The average wavelength, period and wave height however do not really change for
decreasing particle weights. One thing that can be seen is that the deviation between the
different measurements decreases and for small particle masses each wave is similar.
This can be seen in table 6, included underneath, by noting the decrease of standard
deviation (stdev) for smaller particle weights.
1000 kg
Measurement period wave height wavelength
[s] [m] [m]
1 5.89 1.47 46.07
2 6.42 2.44 56.56
3 5.40 1.57 41.74
4 5.30 1.64 55.77
5 6.10 0.92 47.03
6 5.59 2.33 22.85
7 5.71 1.59 50.23
average 5.77 1.71 45.75
max 2.44 56.56
stdev 0.40 0.52 11.40
100 kg
Measurement period wave height wavelength
[s] [m] [m]
1 5.50 3.93 47.68
2 5.70 4.30 48.81
3 5.50 4.43 48.80
4 5.70 4.50 47.08
5 5.60 4.21 48.59
6 5.80 4.70 47.81
7 5.70 4.69 46.41
average 5.64 4.39 47.88
max 4.70 48.81
stdev 0.11 0.27 0.92
Table 6: wave measurements in Dyna models for 1000 kg and 100kg particles
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Finally, the chosen pitch length was a trade of between detail and calculation time. An
optimum was found for 8 kg particles. As said before, simulations with 1 kg particles had
very promising results, but were very unpractical as well for file size as for calculation
time.
3.6.2 Sea state accuracy
Now a stable model has been established and particle mass is set to 8 kg, it is important
to check whether or not our SPH-modelled sea is representative for a real sea state.
In section 3.4 sea state data provided by GeoSea was combined with theoretical
relations extracted out of [6]to quantify what would be a common sea state to simulate.
For the reader’s convenience the conclusion is summarised in table 7.
Wave period T 5.594 s
Wave height HM1 224.46 cm
Maximum wave height 275.69 cm
Wavelength λ0 48.27 m
Table 7: Experimental wave data by GeoSea
The accuracy of the sea model will be verified by comparing these theoretical values to
the measured values. This is achieved using the post processing tools in LS-PrePost.
Using the nodal time history of the z-displacement, the graphs of figure 35 and figure 36
were made. The graphs are included with additional information in the following
paragraphs. Only the first 25 seconds of a 90 second simulation are pictured to clarify
the graphs. However, the transient phenomenon has already damped out, so these
results are representative for the entire simulation.
The first parameter that was checked is the wave period T. The calculated average value
was 5.594 seconds. The mathematical average of the pictured wave periods is 5.6
seconds and thus in very good agreement with the theoretical value.
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Figure 35: Wave period; Z-displacement (in meters) versus time (in seconds)
Another very important parameter is wave height. The measurements were done using
the trace function in LS-PrePost. The wave height was measured as in figure 36.
Figure 36: Wave height; Z-displacement (in metres) versus the time (in seconds)
According to the wave data provided by GeoSea a mean value of the top 1 % highest
waves is 224.46 centimetres while mean value of the recorded highest waves is 275.69
centimetres. The average wave height in the simulation is 244.6 centimetres, while the
highest is 276.8 centimetres. In section 3.5 the paddle stroke S was determined using
theoretical formulations out of [6]. The stroke S was said to be 2.573 metres. As said, in
the simulations a higher value was used and this is also the one used for the wave
generation represented by figure 36.
Wave height
5.9 s 5.3 s
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The desired wavelength λ0 to simulate was 48.27 metres, as seen in figure 37. As said
before the wave tank itself is 75 metres long now but if the ‘rule of thumb’ (length is tree
wavelengths) was followed it had to be twice as long. This would result in a tank of
about 150 metres in length and an extremely high amount of SPH particles. To limit the
calculation time a length of 75 metres was chosen. However the wave generator should
still produce waves with a wavelength of 48.27 metres. This was checked by subtracting
the x-position of two selected nodes at the crest of two adjacent waves. This is pictured
in figure 37. The average wavelength that was measured was 48.4746 metres.
Figure 37: Measured wavelength
3.6.3 Further model checks
3.6.3.1 Buoyancy check
Another very important property to check, is the buoyancy. This is done by adding a
solid buoy with simple geometry to the simulations. By varying the density of this solid
buoy and visually checking whether or not it floats, sinks or, with a density equal to this
of the water, stays at a giving depth the model was approved. A screenshot of a
simulation can be seen in figure 38. Please note that in order to be able to repeatedly do
these tests a smaller wave tank was used. This to suppress calculation time. This subject
will be handled later on in detail.
≈48.4746 metres
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Figure 38: Buoyancy check
3.6.3.2 Breaking wave check
Another way to check if our SPH-modelled water behaves as real water is to simulate
breaking waves. In order to do so the motion of the wave generating paddle is enlarged.
By raising the scale factor in the card BOUNDARY_PRESCRIBED_MOTION_RIGID the
paddle will have a larger amplitude and the generated waves will become increasingly
violent. As can be seen in figure 39 LS-Dyna can handle simulation of breaking waves.
Also a splash-up can be seen in the screenshot at the right.
Figure 39: Screenshots of breaking waves
Clustering
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3.6.3.3 Dam break check
As a final verification of the model, a dam break test was performed. The results will not
be discussed in depth, since a great deal on this topic can be found in [7].
In figure 40 the evolution of the water level is visualised for the dam break tests. The
results on the left are those extracted from the performed dam break test. The volume of
the water held by the dam before breaking was 6000m3, about 12000 times larger than
the one in the test on the right. This explains the difference of one order of magnitude on
the vertical axes. Please note the similarity of the curve on the left with the experimental
data, visualised by the red dashed line on the right in figure 40. This again confirms the
accuracy of the model.
Figure 40:- Z-coordinate of water level for LS-Dyna Dam Break test (left) and experimental and Joren’s data (right).
Right picture from [7]
Screen shots of the test are pictured in figure 41. The huge volume of water in the dam at
t=0 explains the violent reaction and thus the huge splash against the left wall.
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1 2
3 4
5 6
7 8
Figure 41: Series of screenshots of the executed dam break test
3.7 Modelling the buoys for LS-Dyna
From the initial phase of the thesis three buoys were selected for testing. As a reference,
the first working prototype by 3E would be tested. The others were the PEM-58 buoy by
RESINEX and an a spar buoy. The last one because of its proven stability and the PEM
because of its modular design which makes it perfect for transportation. The final choice
will be a trade-off between the performance in terms of dynamic stability in the wave
tank and practicality. Keep in mind that not the entire movement of the buoy will be
transferred onto the LIDAR because of the extra stabilizing system which will come on
top of the buoy. The latter will be discussed in chapter 4.
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In a first step detailed drawings were made in CAD-software. Materials, shell
thicknesses, etc. were defined to be able to locate the centre of gravity and the exact
mass of the different parts. 3E has also provided CAD-drawings of their current design.
Based on this, platforms for the buoys were drawn. These detailed 3D-drawings could be
imported in LS-Dyna.
However, to reduce calculation time and to be able to run multiple simulations, a
simplified design was made. A visual representation of this design can be seen in figure
42. This simplified design is used to debug and fine-tune the models further. Special
consideration was given to make sure these simplified models would be good
representatives for the real buoys. The LIDAR and its stabilizing structure are
represented by one part. This part has the combined weight, and the centre of gravity
was chosen to be at approximate the same location, as the combination LIDAR and
frame. The battery box has remained unchanged in terms of dimensions and weight.
Figure 42: Detailed CAD drawing (left) and simplified mesh (right) for the PEM58 buoy
Of course, when buoy-water interaction was brought to an appropriate level, more
detailed versions of the buoys were created. This to be able to simulate the movement of
the whole more accurately. The gimbals are now introduced as well as the solar panels.
The frames and platforms are built using I-beams. The platform where all installations
are fixed upon is based on the current design by 3E, but its base is now a square instead
of the original rectangular form. This fits the axisymmetrical shape of the spar buoy and,
in some ways, of the PEM58, better. A CAD-drawing can be seen in figure 43.
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Figure 43: Platform drawn with CAD
The more detailed and meshed designs for the 3E buoy, PEM 58 and spar buoy
respectively, can be seen in the figures below. The design is made based on a
combination of CAD-files provided by 3E, photographs taken on a visit to the
construction area of the 3E buoy and data of weights of the different parts. Some
photographs are shown in figure 44.
Figure 44: Detailed 3E buoy Dyna model
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Figure 45: Detailed PEM58 buoy Dyna model
Figure 46: Detailed spar buoy Dyna model
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Figure 47: 3E working prototype
The buoyancy of the modelled buoys has been checked in a wave tank of 10 deep, 10
meters wide and 10 meters long. Fine-tuning was necessary because information on
shell thicknesses of the buoy and inner structures were not available on the internet and
manufacturers keep this information classified. Standard a shell thickness of two
centimetres was chosen as standard and good buoyancy was guaranteed with small
changes in the density of the used materials.
Figure 48: Buoyancy check
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3.8 Calculation time
LS-Dyna and its PrePost are undoubtedly very powerful programs. The ability to model
a real sea-situation as truthful as possible and the accuracy of the simulations are not
limited by the program itself, but by the computer used to run it. When using more
detailed models, calculation time easily rises up to unfeasible values. If changes were
made to the model, a part in it or a value in one of the cards, it literally takes hours or
even days to see which effect the changes had and whether or not it was a good idea to
make them in the first place. This makes the optimisation a lengthy and frustrating task.
Again, this explains why in some of the model checks above smaller wave tanks and
simplified buoy-designs were used.
To limit the calculation time to a practical value a trade-off was made between the
calculation time itself and the model detail. To give an example, the simulation of most of
the models discussed in this thesis cannot be done on a ‘normal’ laptop (Intel i7
processor and 8GB RAM), where the RAM is the limiting factor. When a more detailed
simulation is run, LS-Dyna demands up to 70 GB of RAM. The calculation times noted in
the following paragraphs are these when using a high performance workstation owned
by University Of Ghent (Intel Xeon processor and 24GB RAM).
3.8.1 Material choice versus calculation time
An easy way to reduce calculation time is to use Null-material in combination with a
Gruneisen-equation of state instead of using the fluid-elastic-fluid-material. The latter
does not require an equation of state. To reduce calculation time the speed of sound,
which has to be specified in the Gruneisen-EOS, is set to a lower value than normal. It
was said, as a rule of thumb, speed of sound can be reduced up to 10% of the actual
value. Even though a value of 100m/s is even less, it works great and its accuracy has
been proven in earlier sections and in [7]. In this way the simulations run significantly
faster while staying representative. The fluid-elastic-fluid material also offers the
opportunity to change the value for the speed of sound, although in a more impractical
way. Null-material offers multiple other advantages too. One of them being a better
behaviour for compressibility. Other ideas to reduce calculation time are discussed
below.
The parts of the buoys are all modelled as rigid. This means that mass and mass
distribution is equal to reality, but the parts cannot deform. This rigidness offers a huge
benefit in calculation time over deformable materials.
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3.8.2 Wave tank design versus calculation time
A first idea was to make a water tank with the following dimensions (see table 8)
Knowing that the use of a small pitch length for the SPH-particles for the whole domain
would result in an insanely high amount of nodes it was chosen to split the domain in
two. One section would be a fine top layer with a more coarse layer underneath, named
the depth layer. This can be seen in figure 49. Because of the difference in weight the
more heavy depth layer would always stay on the bottom and intermixing between the
two layers would be negligible. Or so it was assumed.
Dimensions of original model
Width 20 m
Length 75 m
Depth 66 m
Pitch length between SPH
Top layer 0.25 m
Depth layer 1 m
Table 8: Original dimensions (deep sea)
Figure 49: Two layer model
Top layer
Depth layer
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Multiple approaches were tested. One of them was to implement the top and depth layer
as two different parts as seen in figure 50. The two parts would have the same material-
and section-card and the same equation of state. The only difference, besides the part
number of course would be the spacing (pitch length) between the SPH-particles which
make up the part. Many models and a thorough inspection of the resulting simulations
later the conclusion could be made that this was not possible. Due to the difference in
pitch length and weight of the particles very unphysical behaviour was observed.
Contact, even though this does not have to be defined explicitly between two SPH-parts,
kept on failing.
Another problem was a transient phenomenon due to the hydrostatic pressure which is
set on time equals 0+ seconds. The bottom particles will be pushed closer together
under influence of the weight of the other particles resting on top of them. The result is a
drop of the water level, as discussed earlier. When one keeps in mind that the depth
layer had a pitch length of one meter in every direction it is easily found that for each
cubic meter there is exactly one SPH-particle. With the density of water, being
, this means that every ‘droplet of water’ weighs one ton. The result is that
the transient phenomenon is actually very violent. After the drop the particles are more
closely packed than they should be. Therefore there are great repulsing forces due to the
contact launching the top layer multiple meters above water level. A card called
INITIAL_STRESS_DEPTH was used, hoping it would set the hydrostatic pressure before
the first time step. However, the effect of this card is negligible as discussed earlier.
Figure 50: Dyna two layer model: Two layers, two parts
Instead of implementing the two layers as two different parts they could also be placed
under the same part number. A screenshot of such a model is demonstrated in figure 51.
Contact between paddle and SPH or wave tank and SPH is defined by using node sets
representing the distinctive layers. The results were unsatisfying and roughly the same
as those of previous paragraph.
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Figure 51: Dyna two layer model: Two layers, one part
A smaller wave tank was designed because it is apparently impossible to work with a
fine top layer and a coarse bottom layer and because filling the whole tank with the
small SPH-droplets would result in millions and millions of nodes. To be able to run a
stable simulation the weight of the SPH-particles should be fairly low. In section 3.6.1, 8
kg was chosen as an optimum. Even the powerful desktop could not run this simulation,
but the UGent clustered super computer (cluster) provided the necessary computational
power. Eventually it was chosen to set the dimensions of the wave tank to be
representative for a sea state at the Thornthon bank. The work method in order to
determine the wave tank dimensions was already been shown in paragraph 3.5.1. The
90 second simulations for the final model takes an estimated 1000 hours to complete on
the cluster. In this model the buoy and the mooring cable is, of course, included.
3.8.3 SPH particle mass versus calculation time
To be able to check the effect of different cards or values in the LS Dyna input files it is
necessary to run multiple simulations. A normal reflex is to make a simple file, in this
case a file with a coarse sea, so the simulation does not take long.
At a first attempt the pitch length for the SPH particles was set to be one metre. As
mentioned before these results in modelled water droplets with a fixed weight of one
tonne. The whole simulation can run in under 5 minutes. Nevertheless, it has already
been mentioned before that the benefit of low calculation time is heavily tempered by
the instabilities in the models.
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It has been shown that particle weight has to be decreased to assure a stable situation.
When adjusting the pitch length with a quarter, up to 0.25 metres, the particle weight
decreases to 15.625 kg. On the other hand, simulation time rises from a couple of
minutes to an approximate 16 hours. Contact can be fine-tuned relatively easy when
using the slave surface thickness override (SST) in the contact card. Leakage is
eliminated and the hoped-for waves are attained. The transient phenomenon due to the
setting of the hydrostatic pressure is reduced to an acceptable level, tough still present.
To deal with this, the buoy hovers above the water level until the sea has reached its
steady state condition. This is done by adding a upward force to the buoy, which
counteracts gravity for the first two seconds. However, when the buoy slowly drops in
after two second, another unwanted phenomenon presents itself. The density of the SPH
particles decreases significantly. Due to this drop, and keeping in mind the constant
mass of the particles, the individual SPH volume increases. The particles are too closely
packed in the wave tank and the repulsing forces of the contact become very high. The
effect is that particles get launched out of the tank, again.
The only solution is to decrease the particle weight once more. When using a particle
mass of one kg the wave tank contains seven and a half million particles and the memory
demand is too high to run on an ordinary desktop (octacore with 24 GB of ram) and use
of the UGent cluster is needed. The simulations with one kg particles were set on hold,
because of the immense demand for RAM. Even the supercomputer could not handle
these files. For the small period these simulation have ran, it was noted they were very
stable and the interaction with the buoy was just as hoped. This explains the settlement
for the use of 8 kg particles that was made in paragraph 3.6.1. It offers great detail for
reasonable calculation time.
3.8.4 Mooring cable versus calculation time
The buoy should be moored to the seabed. In theory two nodes are enough to be able to
define the cable. However, as explained before, to simulate the effects of an
undercurrent we defined more beam elements to model the cable behaviour with more
detail and to be able to impose ‘undercurrent forces’ on these beam elements. Two
effects will be taken into account. The first is a constant force, representing a drag force.
The second will be a sudden pull to represent cable snagging.
The cable material is CABLE_DISCRETE_BEAM and also a BEAM-section is used. Using
these a steel cable is modelled with a diameter of 0.01 m.
To investigate the influence of adding a cable to our model on the calculation time tree
easy models were created. A first model only had a wave tank and a coarse sea. In the
second one an solid cubical buoy was added. This buoy was moored by a two-node cable
in the third model. A screenshot of these models can be seen in figure 52.
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Figure 52:Wave tank (left), wave tank and buoy (centre), wave tank and moored buoy (right)
Remarkably the cable was very heavy to process by LS-Dyna. The simulation of the first
two models took only 4 minutes to run. By adding the cable only three extra nodes were
incorporated, but calculation time rose up to 10 hours. In other words a factor of 150
more. A comparable result was found when the same procedure was repeated in more
detailed simulations and the calculation time went up to a 1000 hours. This represents
an increase with a factor 5.
3.9 Simulation results on buoy performance
At this time, good operation of the wave tank and its wave generator has been achieved.
The wave generator’s boundary conditions are calculated and set to create a sea state
comparable to where the buoys will actually operate. Meanwhile, the selected buoys are
modelled and tested in smaller wave tanks and cable behaviour is optimised. In other
words, everything is done to be able to check buoy behaviour in realistic circumstances.
In this section the performance of the buoys at sea is discussed and compared. To be
able to do so, ‘nodout’-data is used in LS-Dyna. This option makes it possible to retrieve
information on the position, velocity, acceleration and rotation of selected nodes. In this
case, only one node had to be selected since the whole structure is defined as rigid. The
cable‘s attachment point to the buoy has been especially chosen for every buoy. The
reason for this is that special care was taken to place the attachment point exactly
underneath the COG of the entire buoy.
Due to problems regarding computational time, only a very limited amount of data has
been obtained from the simulations. The termination time of the simulations is set to be
90 seconds. However, the simulations have only ran for a maximum of 30 seconds.
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3.9.1 General information on displacement of the buoys
The translation of the buoy does not really affect the Lidar vessel’s performance. For the
accuracy of the Lidar’s measurements, it is only important that it stays as level as
possible. The position will vary of course, within reasonable limits. The cable will
prevent the buoy from drifting away. The displacement of the 3E buoy is only included
to clarify some statements made earlier. The graphs showing the displacement of other
buoys are very similar and for that reason they are not included in this thesis.
In figure 53, the displacement in all major directions is shown for the current prototype
of the 3E buoy. It is chosen to work in a right-handed coordinate system using:
X-axis in the length direction of the wave tank, pointing away from the paddle
Y-axis in the width direction of the wave tank
Z-direction in the ‘upward’ sense
Figure 53: Displacement in x- (red), y- (green) and z-direction (blue) for the 3E buoy
The x-displacement, pictured in red, shows the back and forth motion in the direction of
the waves the buoy exhibits when subject to the wave forces. The buoy is moored and
thus this motion is limited.
The y-displacement is shown in green. As can be seen, the buoy drifts a maximum of
20cm in the y-direction from its starting position. This is mentioned to prove that the
suggested 10m tank width is indeed sufficient, as stated in 3.5.4.
X direction Y direction Z direction Translation [m] Time [s]
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The z-displacement is shown in blue. For the first 2 seconds the displacement is zero
followed by a slow lowering of the buoy, because of the used LOAD-cards as mentioned
in 3.5.6.2. Afterwards, the buoy follows the wave motion. It can be seen that the wave
height is not exactly the same as the z-displacement of the buoy. This is due to the
rotation of the buoy itself and the fact that the selected node is at the lowest point of the
buoy (greatest effect of the rotation).
3.9.2 Performance of the different buoys in terms of rotation
As discussed in previous paragraphs, calculation time is a serious issue. Because of this,
a smaller wave tank was designed to be able to interpret the results on buoy
performance. This wave tank has a length of only 35 m. Other dimensions were kept the
same.
Even though the dimensions were smaller, the simulation still took a long time to run. So
long in fact, that the simulation of the PEM58 could only run for 17 seconds and this
already took it 94 hours, 28 minutes and 45 seconds. While LS Dyna’s projected run time
was 10 hours.
3.9.2.1 3E prototype buoy
Figure 54 displays the rotation of the buoy around all mayor axes. Since the waves are
unidirectional, only the rotation around the y-axis, perpendicular to the wave direction,
is noteworthy. This is why in following graphs only the this rotation will be pictured.
The maximum inclination is observed at 13.33 seconds and is 0.24 radians (13.75°). A
screenshot of the simulation at the moment of maximum inclination is shown in figure
55.
Figure 54: rotation around x- (red), y- (green) and z-axis (blue) for the 3E buoy
around X axis around Y axis around Z axis Rotation [rad] Time [s]
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Figure 55: side view of 3-buoy model at 13.33 seconds showing the maximum inclination
Limitation of the maximum inclination is important, but far more important is the
rotational speed. This is pictured in Figure 56. The slower the buoy tilts, the more time
the Lidar stabilisation mechanism has to counteract this movement. Detailed
information on this topic in found in 4.6.2. The maximal rotational speed for the 3E buoy
is 0.183 rad/s (10°/s) at 19.95 seconds. This is so slow that for the first two waves in the
simulation, will not follow the wave motion entirely, but stays tilted with a positive
inclination.
Figure 56: rotational speed around y-axis for the 3E buoy
13.75°
Rotation speed[rad/s] Time [s]
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It can be seen there is a peak at the end (after 26 seconds). This has nothing to do with
an error in the simulation or a strange movement of the buoy, but is merely a
consequence of an imperfection in LS-Dyna itself. If one terminates the simulations or
extracts date from it before it has ran completely, strange effect are noticeable for the
final seconds.
3.9.2.2 PEM58 buoy
Same discussion of the results can be made for the PEM58 buoy.
The maximum inclination, seen figure 57, is found at 16 seconds and is 0.25 rad (14.32°). This is
visualised in figure 58. It can be seen that the PEM58 follows the wave motion more than the 3E
buoy.
Figure 57: rotation around the y-axis for the PEM58
Rotation [rad] Time [s]
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Figure 58: side view of PEM58 model at 16 seconds showing the maximum inclination
The maximal rotational speed is 0.2639 rad/s ( 15.12°/s) at t equals 8.9 seconds.
Figure 59: rotational speed around y-axis for the PEM58 buoy
14.32°
Rotation speed [rad/s] Time [s]
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3.9.2.3 Spar buoy
Many problems have occurred during the spar buoy tests. Even though the total mass
divided by the volume that should be submerged in the water is equal to the density of
the water, the spar buoy does not stay afloat. A possible solution would be to decrease
the densities of the used materials of the spar buoy up to the point where the buoyancy
of the spar buoy is acceptable. However, this only led to two extreme scenarios: or the
spar buoy sinks or the spar buoy is barely submerged (figure 60).
Figure 60: Barely submerged, but floating SPAR buoy
Some possibilities for the reason the spar buoy sinks are listed below.
First of all, some parts of the SPAR Buoy shown in figure 60 will be defined. The pink
part will be described as ‘the cylinder’, while the blue part is named ‘the buoy’ and the
entire construction is named ‘the spar buoy’. This will make the following explanation
more readable. The dotted red line denotes to where the buoy should be submerged.
Just as the 3E buoy and the PEM58, the spar buoy would be lowered into the water as if
lowered by a crane. When the buoy contacts the water, it pushes it away. However,
when the buoy is totally submerged, a suction effect is present at the time the cylinder
enters the water. This creates vortices pushing the buoy down, making it sink even
deeper. This theory is visualized in Figure 61.
Desired location of the water level
Buoy
Cilinder
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Despite the fact that the SPH water model behaves well for wave modeling, it cannot be
denied that the particle size is immense in comparison to real water droplets. Because of
their size and mass, the particles do not surround the spar buoy as completely as real
droplets would. Their size makes the particles react slower as well. Consequently the
submerged volume of the spar buoy is not always equal to the displaced water volume.
This has an obvious effect on buoyancy according to the law of Archimedes. This
imperfection would be less apparent when using smaller sized particles.
Both effects push the spar buoy deeper into the water than expected. Because the
buoyancy tests are executed in smaller wave tank (10m deep), due to the calculation
time (see 3.8), it quickly reaches the bottom and interacts with the SPH-symmetry plane.
However, since both mechanisms can be regarded as transient phenomena, the buoy
might reach a representative steady state motion if the simulations were longer and the
tank was deeper. This could not be tested, due to a lack of time.
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Figure 61: sketch showing vortices when placing the spar buoy
Because the performance of the Spar Buoy would be a great benchmark, a lot of effort
was made to find another solution. The following model enhancements or changes have
been tried:
Cili
nd
er
Buoy
Cili
nd
er
Buoy
L L
Large water particles react slower
than they should. Impulse laden jets
of water rush to the low pressure
zones. Too much and too late.
The subsequent vortices push the
buoy down. The Spar buoy moves
further downward than it is
supposed to. It takes some time for
this phenomenon to disappear.
Cili
nd
er
Buoy
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A boundary condition was used that moved the buoy into place, left it there for 3
seconds in order for the transient phenomena to damp out. After this, the
boundary condition was eliminated.
Using the fill algorithm of the SPH-particles to fill the wave tank with SPH water
surrounding the buoy, when the buoy was held into its correct position.
Filling the wave tank with SPH water, placing the spar buoy in its correct place
(before the start of the simulation) and deleting all intersecting nodes.
Unfortunately all of the proposed solutions above did not have the desired outcome.
3.9.2.4 Conclusion
In figure 62 and figure 63, a combination of the results for the 3E buoy and the PEM58 buoy is
made.
Figure 62: rotation around y-axis. Red for the PEM58 and green for the 3E buoy
Figure 63: rotational speed [rad/s] versus time [s] for the 3E and the PEM58 buoy
PEM 58 3E prototype Rotation [rad] Time [s]
PEM 58 3E prototype Rotation speed[rad/s] Time [s]
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A summary of the data can be found in table 9. Especially the relative numbers are easy
to interpret.
The PEM58 model has been constructed based on plans and pictures. However,
materials used, were not specified, neither was the mass distribution nor information on
the inside structure of the buoy. The actual buoy will be quite different and will most
likely perform better than the model used in these simulations.
3E buoy PEM58
Inclination
Maximum [°] 13.75 14.32
Relative [3E buoy = 1] 1 1.04
Amplitude [°] 18.91 25.21
Relative [3E buoy = 1] 1 1.33
Rotational speed
Maximum [°/s] 9.74 14.89
Relative [3E buoy = 1] 1 1.53
Amplitude [°/s] 19.48 26.36
Relative [3E buoy =1] 1 1.35
Table 9: Summary of the simulation results for 3E and PEM58 buoy
It can be seen that the PEM58 buoy scores worse on every performance parameter.
However, as stated in section 2, the PEM58 buoy is by far the most practical. Mostly due
to its lighter weight and the modular design. On top of this, the stabilization mechanism,
discussed in chapter 5, will damp out most of the movement and keep the Lidar level. It
remains to be seen how the vessel, buoy plus stabilizing unit, performs.
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4 Study of Lidar stabilization mechanism
4.1 Introduction
One of the main parts of this thesis is the design of a system that keeps the Lidar module
facing upwards despite the motion imposed by the sea. Looking at existing sea craft, one
could get inspiration for such a mechanism. Travelling by sea has been the only method
to cross the Atlantic Ocean for a very long time. Since a ‘smooth passing’ was vital for
passenger comfort and profits; systems were thought out to stabilise the ocean going
ships despite rough sea conditions. Two of the most important solutions engineers came
up with were gyroscopic stabilisers down in the ship’s lower decks and hydrodynamic
stabilising ‘wings’ on the ship’s hull under the waterline. Since the latter can only be
used for moving vessels it’s the former that might provide a solution for the stationary
Lidar buoy. Sources on the internet reveal that wealthy yacht owners can buy gyroscope
‘modules’ which can be bolted down to the ship’s frame to reduce rolling and pitching of
the ship with surprising results. Although today the hydrodynamic wing option is the
most popular, some ocean liners have successfully relied on gyroscopes since the 1930’s.
So a solution based on a gyroscope was chosen to stabilise the Lidar.
4.2 Model overview
Unlike the yachts, passenger comfort is not an issue for the Lidar vessel. To reduce the
necessary size and power requirements of the gyro mechanism and to increase its
performance, a dual gimbal linkage system was thought out, similar to the gyroscopes
one could have seen in a physics lesson. The movement of an inner Lidar supporting
module is made independent of the motion of the sea craft using two rotating joints. The
inner frame or gimbal, holding the Lidar, rotates in an outer frame or gimbal which
rotates again in an external subframe fixed to the buoy. Both axes are perpendicular to
each other.
The shape and construction of the mechanism as well as the position of the parts was
inspired on a prototype mechanism. Later on it shall be explained that some aspects of
this are not necessarily ideal.
The inner, and independently moving, frame with the Lidar is stabilised by a gyroscope
much smaller than the one that would be needed to stabilise the entire craft. CAD
drawings of such a linkage system are shown in figure 64 with artificial colouring to
distinguish the separate components.
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Figure 64: CAD drawing of linkage system to make the Lidar move independently
Rotation axis: Inner gimbal – outer gimbal
Rotation axis: Outer gimbal – Frame
Lidar module
Inner gimbal
Outer gimbal
Frame; fixed to buoy
Fixed
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The dimensions of the components in the CAD drawing are based on visual clues of the
first Lidar prototype. This model however is made only for simulations and empirical
research. But the actual dimensions may differ slightly. Also, for simplicity, physical axes
are not modelled. Motion is described with mathematically formulated joint definitions.
This is why the components appear to be floating while, in fact, they have been fixed in
space with a mathematical formulation. Please note that both rotation axes are not in
the same plane for this design. This is not advantageous since the position of the
point around which the Lidar rotates – designated as ‘the rotation point’ – is
dependent on both axis’ rotational coordinate. This will be discussed later on.
The software used to simulate this mechanism is ‘Universal Mechanism’ by Universal
mechanism Lab. For readability the software will be referred to as ‘UM’. The CAD
components were imported in UM as ‘images’ and their mechanical relations were
carefully defined. What this looks like in the software animation window is shown in
figure 65.
Figure 65: UM animation window: representation of Lidar stabilizing mechanism model
Please take careful notice of the fact that the Y-axis is upwards instead of the Z-
axis. All components from the CAD drawing can be recognized. The flywheel, shown in
red, was added in UM with parameterized dimensions. This means that the flywheel is
modelled as a simple disc and its height, density and radius can be changed easily to
allow for quick optimization of flywheel dimensions.
Z-axis X-axis
Y-axis
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All components shown are considered to be made of steel with a density of 7800kg/m³
for most simulations. Except for the Lidar box which has a weight imposed by a weight
parameter. For this study, that weight is assumed to be 45kg, which is the current
weight of the actual system.
An important reference point is the centre point of the inner gimbal’s rotation axis with
the outer gimbal. Both the position of the Lidar module and the flywheel can be set with
parameters in relation to this point.
Figure 66: Offset parameters of mechanism model in UM and reference points.
The frame will also rotate relative to a stationary reference frame in UM just like the
buoy moves because of the waves relative to a stationary reference in real life. The
centre of rotation is also a reference point and it is located 1.18 meters downward
relative to the first reference point. The real buoy will rotate around its centre of gravity
which will be lower than the upper reference point. This implies that buoy rotations also
involve translations and translational accelerations for the Lidar. The offset between
said points might be important to mechanism performance. If the Lidar-flywheel
subsystem is suspended like a pendulum, then the fact that the ‘centre of rotation of
frame movement’ (figure 66) is below the Lidar will have more effect.
Lidar offset
Flywheel offset
Reference point for offset parameters
Center of rotation of frame movement
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4.3 Model parts, joints and parameters
4.3.1 Part: Frame
As mentioned before the frame moves with respect to an inertial reference frame. It’s
rotation around the rotation reference point is expressed in radians and has a sinusoidal
expression for the X and Z axis. The time variable is ‘t’.
So the frame has four parameters concerning its movement.
ax Amplitude of x-axis rotation [rad]
az Amplitude of z-axis rotation [rad]
Tx Period of x-axis excitation [s]
Tz Period of z-axis excitation [s]
Table 10: Parameters of Frame movement.
Amplitude values of 0.1 rad have been chosen for most of the simulations because it is a
representative value for measurement conditions of the Lidar. The excitation periods
(inversely proportional to angular frequencies) are studied more intensively and its
relation to the maximum inclination of the Lidar will be carefully looked at later on.
4.3.2 Part: Outer gimbal
No parameters. Weight: 49.53 kg.
4.3.3 Part: Inner gimbal
No parameters. Weight: 35.73 kg.
4.3.4 Part: Lidar
Box shaped part. It represents the Lidar module. Its offset in relation to the central
reference point is also a parameter.
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oLidar Offset of Lidar module [m]
Table 11: Lidar offset parameter
The mass of the Lidar is considered 45kg for most simulations. The offset is typically
zero but can be changed to tune performance.
4.3.5 Part: Flywheel
The flywheel image is, unlike the CAD imported images of other parts, a simple cylinder
determined by parameters. Its rotational speed is a parameter as well.
Rflywheel Radius of flywheel [m]
hflywheel Height of flywheel [m]
Ωflywheel Flywheel angular speed [rad/s]
[rpm]
oflywheel Flywheel offset [m]
Table 12: Flywheel geometry and movement parameters
R and h are parameters that must be optimized. The best values within practical limits
are 1cm height and 0.5m radius. The flywheel angular speed is taken to be 3000 rpm or
6000 rpm. These speeds are achievable with invertor fed induction motors. Its offset is
also a parameter. These parameters will be discussed further. Naturally, the flywheel
weight can be calculated:
4.3.6 Gimbal joints
The rotational joints between the gimbals are very simple 1 degree of freedom
rotational joints. However, an expression was implemented defining a restoring
moment:
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Figure 67: Relative movement and restoring moment.
A damping and stiffness coefficient is added to create the restoring force as a function of
the relative speed and position of the joint. The zero position reference is the position
when all parts’ Y-axis are perfectly aligned. These coefficients are relevant because a
spring or a damper could be installed if necessary to tune performance. Friction will not
be taken into account for this research because it is unclear what type of bearings and
lubricants will be used. Friction is considered as a disturbance and will be studied
separately later on in section 4.6.4.
Cstiff Stiffness coefficient [N.m/rad]
Cdiss Damping coefficient [kg.m/rad/s]
Table 13: Joint restoring force parameters
Of course one could define axis specific coefficients for each axis. But research has
shown that the effort of tuning each axis separately is really not worth the time because
both axis behave so similar. The units for these values will be identical throughout this
thesis and they will not always be mentioned. When stated ‘a damping value of 20’ one
should interpret this as ‘a damping value of 20 kg.m/rad/s’.
4.3.7 Overview and reference situation
The reference situation will now be defined. It is a set of parameters that has been
optimized given specific input. The reference values are given in the right column. This
set of parameters shall be referred to as the ‘reference mechanism’ and is very
important.
Part 1
Relative position: x
Restoring moment: M
Part 2
Relative speed: v
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Excitation parameters
ax Amplitude of x-axis rotation 0.1 rad
az Amplitude of z-axis rotation 0.1 rad
Tx Period of x-axis excitation 5 s
Tz Period of y-axis excitation 5 s
Optimization parameters
Ωflywheel Flywheel angular speed 3000 rpm
6000 rpm
Rflywheel Radius of flywheel 0.5 m
hflywheel Height of flywheel 0.01 m
oflywheel Flywheel offset 0.2 m
oLidar Offset of Lidar module 0 m
Cstiff Stiffness coefficient 0 N.m/rad
Cdiss Damping coefficient 20 kg.m/rad/s
Table 14: Summation of reference mechanism model parameters
This reference mechanism is quite optimized. So any change made to the optimization
parameters for the given environment parameters should lead to a decrease of
performance or to an impractical or unfeasible geometry. In almost all of the simulations
the values Tx and Tz will be chosen equal as well as the ax and az values. This makes it
look like the frame rotates around the bisector of the axes. This is the ‘worst case’ since
both gimbals will have to rotate and the restoring force of both pairs of bearings is
exerted. The next paragraph will discuss the influence of parameters on the model.
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4.4 Measuring performance
The mechanism performance is inversely proportional to the maximum Lidar inclination
angle. The smaller the maximum inclination angle, the better the system is doing its job.
For illustration purposes, a graph showing the X and Z inclination angles of the Lidar
part is shown below for the reference mechanism (flywheel speed is 6000 rpm) in a 120
second simulation of the mechanism with a sinusoidal moving frame as described
earlier. It shows the satisfying performance of the system with maximum inclination
values as low as 0.07 degrees for 6000 rpm, which can barely be seen with the naked eye
in the graphical representation. It looks like the reference mechanism performs very
well.
Figure 68: Result of 120 sec simulation of the reference mechanism. X and Z inclination is shown with max. inclination
value of 0.07 degrees. Speed: 6000 rpm
Z axis X axis Inclination [rad] Time [s]
6000 rpm
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Figure 69: Sine function of angular movement definition of frame.
The difference in amplitudes for the rotation of X and Z axis is caused by the difference
in rotational inertia for the outer and inner gimbal respectively. It does not matter which
axis is observed for system optimization. For further research the Z-axis was chosen
since this axis shows the largest inclinations.
The inclination functions look like a superposition of two sinusoids: one with a high
frequency equal to the excitation frequency (f=2∙π/T) and one with a lower frequency of
40 mHz (with a flywheel speed of 6000 rpm). The lower frequency sinusoid is caused by
a so called precession motion. This motion is exhibited by all gyroscopes if its initial
position differs slightly from vertical or if it is disturbed. Unlike the excitation
component it damps out over time. The precession motion should not be present in a
theoretically ideal situation yet it still is. This is due to the fact that the frame joint
definition function has a discontinuity in the derivative at t equals zero as shown in
figure 66.
The calculations are made with a numerical iterative solver and values preceding the
time origin are zero. This error will not affect the conclusions made in this theoretical
study because any detrimental or beneficial influence of parameter changes will still be
clear in the increase or decline of the max amplitude and in the excitation component,
despite any precession. Yet is it still worthwhile to keep track of both the output signal’s
components. From now on they will be defined as the precession component and
the excitation component for the higher frequency and the lower frequency
component respectively.
Non-physical discontinuity of angular speed at t=0 is the cause of the high frequency oscillations in the Lidar inclination angles
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Figure 70: Inclination signal component definition
If the gyroscope was aligned exactly vertical and the Lidar and flywheel were perfectly
held steady during flywheel acceleration there should not be any precession motion. The
precession decreases over time due to the damping in the bearings but only very slowly.
A simulation result of the reference mechanism for the Lidar Z-axis inclination angle is
shown below with a termination time of 10 minutes. Precession becomes much more
important in physical situations and shall be looked at very extensively later on.
Figure 71: 600 sec simulation of reference mechanism highlighting the slow decline in precession
For further analysis the absolute maximum of the Lidar inclination will be measured.
The graph above shows that representative maximum values can be measured with only
60 seconds of simulation time.
Excitation component
Excitation frequency
Precession component
Low frequency
Total inclination
Z inclination [rad] Time [s]
First representative maximum.
Precession motion decreases over time
6000 rpm
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4.5 Parameter influence
Now that the mechanism’s performance can be evaluated, the influence of the
mechanism’s parameters on its performance can be made.
4.5.1 Flywheel speed
The easiest parameter is discussed first. To put it simply: a high flywheel speed yields
lower maximum amplitudes. The reference mechanism has a flywheel angular speed of
3000 or 6000 rpm. These speeds are achievable with an invertor fed induction motor. A
fast DC motor could also serve well but would impose more demanding maintenance.
Figure 72: Simulation of Lidar inclination for varying flywheel speeds. Lower amplitude curves represent higher
speeds and vice versa.
Relevant numbers from figure 72 are summarized in another graph below. As shown, a
power function fits the empirically generated graph quite conveniently. It could be
concluded that it is worthwhile to increase the speed to 6000 rpm.
But it looks like beyond that point it is no longer desirable to increase speeds further
since it will not change the results that much for the better. Very high speeds, such as
9000 rpm may give rise to practical issues. Speeds lower than 3000 rpm are really not
desirable for the mechanism performance.
Please observe that both amplitudes decline with decreasing speed; both the sine with
the excitation frequency and the precession motion.
1500 rpm 3000 rpm 4500 rpm 6000 rpm 7500 rpm 9000 rpm Z inclination [rad] Time [s]
1500 - 9000 rpm
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Graph 1: Maximum Lidar inclination versus flywheel speed and the interpolating power function.
Speed Maximum inclination Relative to 0 rpm Relative to 3000 rpm
Rpm Radians Degrees % %
0 0.02108 1.208
602.67%
1500 0.00613 0.351 -70.92% 104.33%
3000 0.00300 0.172 -85.77%
4500 0.00188 0.108 -91.08% -37.33%
6000 0.00143 0.082 -93.22% -52.33%
7500 0.00124 0.071 -94.14% -58.83%
9000 0.00101 0.058 -95.21% -66.37%
Table 15: Results of simulation series for varying speeds
In order to gain more insight, longer simulations were carried out for 3000, 6000 and
9000 rpm. The results show a shift in the amplitudes of both sine components:
excitation component and precession component. The period of the precession motion
changes as well.
y = 0.3437x-1.008
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
1500 3000 4500 6000 7500 9000
Max
imu
m in
clin
atio
n [
°]
Flywheel speed [rpm]
Practical:
Using an invertor to feed the motor is very interesting for this application. It can be powered directly from the batteries and it makes high speeds possible. The most important is its ability to let the induction motor start the flywheel without draining excessive current. Since overcoming the inertia at startup requires the most power; a less powerful motor can be chosen because maximum torque is also available at the lowest speeds.
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Figure 73: 20 minute simulations of Lidar inclination for varying flywheel speeds. Lower amplitude curves represent
higher speeds and vice versa.
Graph 2: Lidar inclination components versus flywheel speed (primary vertical axis) and precession period versus
flywheel speed (secondary vertical axis).
The results in graph 2 show that the decline in maximum amplitude observed earlier
was mostly due to a decline in precession amplitude. In the simulations, the sine
functions component’s heights were measured. This is the amplitude multiplied by two.
The ‘excitation amplitude’ (the amplitude of the sine component with the same
frequency as the excitation functions) does not change much. The fact that the increase
in precession period versus the flywheel speed seems to be a perfectly linear relation is
noteworthy as well.
0
50
100
150
200
250
0
0.001
0.002
0.003
0.004
0.005
0.006
3000 6000 9000
Pre
cess
ion
pe
rio
d [
s]
Am
plit
ud
e [
rad
]
Flywheel speed [rpm]
Ex. ampl.
Prec. ampl.
Prec. per.
3000 rpm 6000 rpm 9000 rpm Z inclination [rad] Time [s]
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At this point it is instructive to remind the reader that the precession motion is the
result of the relatively violent starting discontinuity and should not be present. The
maximum Lidar inclination graphs could be lower if only the excitation component were
present, such as in a theoretically ideal situation. The authors have decided to keep track
of the influence of changes on the precession motion because, later on, it will be
established that the precession motion is the biggest cause of mechanism inaccuracy and
will be common with practical applications. The starting discontinuity is of benefit to the
research.
Although it seems that one should always strive for ultra-high rpm’s it is not always a
wise choice. A fast flywheel increases the gyroscopes capacity to fight inertia but it also
makes it increasingly difficult to damp out the precession motion. Precession with a long
period is tougher to damp out than precession with a lower period.
4.5.2 Amplitude
How will the components of Lidar inclination, precession and excitation, react if the
excitation amplitude changes? Obviously an increase in amplitude means a decrease in
performance.
Please remember that the Lidar inclination function is composed of two components: an
excitation component with the same period as the frame movement and the precession
period with a much larger period. The components are called excitation and precession
component respectively. (See 4.4)
A series of simulations with the reference mechanism were carried out with the
following amplitudes: 0.025, 0.05, 0.1, 0.2 rad.
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Figure 74: Simulation of reference mechanism with changing amplitudes Smaller amplitudes yield lower maxima.
The simulation’s result in graph 3 shows clearly that changing the amplitude does not
affect the period of precession. The maximum inclination rises linearly as expected.
Graph 3: Lidar inclination versus excitation amplitude.
It looks like the rise of maximum amplitude is linear as well as the increase in both of its
components. The excitation amplitude rises the quickest and the precession rises
somewhat slower. This behaviour was to be expected since the excitation period
remains the same.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0 0.05 0.1 0.15 0.2 0.25
Lid
ar in
clin
atio
n [
rad
]
Excitation amplitude [rad]
Max ampl
Exc ampl
0.025 rad 0.05 rad 0.1 rad 0.2 rad Z inclination [rad] Time [s] 6000 rpm
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4.5.3 Flywheel dimensions
What dimensions should the flywheel have to make the mechanism perform at its best?
This paragraph will look into this problem.
Figure 75: Scheme of flywheel dimensions and offset.
(Flywheel)
x
y
o
R
h
Lidar pivot point, Reference point, Simulation origin
Global reference system
Cartesian
Flywheel reference system
Cylindrical
ζ
θ
ρ
Iζζ: Inertia of rotation axis Iρρ: Inertia of radial axis
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Figure 76: Reference mechanism. Observe the thin flywheel with a large diameter.
In this paragraph the influence and the optimization of the flywheel geometry and its
offset will be discussed. The flywheel material is steel with a density of 7800 kg/m³.
Other relevant parameters can be calculated using these expressions.
R: Flywheel radius
h: Flywheel height
mflywheel: Flywheel mass
Iζζ: Inertia around rotation axis
Iρρ: Inertia around radial axis
During the research of the influence of the flywheel radius it became clear very quickly
that it should be as large as possible. This is why the reference mechanism has
approximately the largest possible flywheel radius within practical limits: a radius of
0.5m. The reference mechanism shows a flywheel thickness of just 1 cm to keep the
weight down to a practical value. The question is: how much will the performance suffer
if the flywheel is given a smaller diameter when the height is kept constant? This would
also mean that the weight decreases as well. Perhaps a lighter mechanism would be
more convenient?
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Figure 77: Simulation of Lidar inclination for changing flywheel radius with constant height. Speed is 6000rpm.
From figure 77 can be deduced that the performance suffers greatly when the flywheel
radius decreases. The precession component amplitude does not change much. The
excitation component amplitude seems very dependent on the flywheel diameter. If one
looks carefully at the red curve (0.2 m) it can be seen that the precession motion is still
noticeable but only in the beginning. It damps out very quickly. This is to be expected
because with a radius of 0.2 the flywheel is very small and its capacity to fight the
restoring force of the dampers and inertia is reduced. (More on precession: see section
4.5.8 on damping.) Looking at the table 16 it seems that both inertia’s rise with
increasing radius and decreasing inclinations.
Var Radius Height Weight Iζζ Iρρ Iζζ ∙ Iρρ Max.incl Exc. Incl Relative
Unit m m kg kg∙m² kg∙m² kg²∙m4 rad rad %
0.20 0.01 9.80 0.20 0.10 0.02 0.039840 0.030715 2929.66%
0.30 0.01 22.05 0.99 0.50 0.49 0.006630 0.005625 404.18%
0.40 0.01 39.21 3.14 1.57 4.92 0.002535 0.001949 92.78%
0.50 0.01 61.26 7.66 3.83 29.32 0.001315 0.000960 0.00%
Table 16: Results and calculated values of simulation with changing radius and constant thickness
0.5 m 0.4 m 0.3 m 0.2 m Z inclination [rad] Time [s] 6000 rpm
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Graph 4: Lidar inclinations (primary axis) and flywheel weight (secondary axis) versus changing radius
It looks as though the graph exhibits exponential behaviour for the inclination values.
The excitation component is also shown above. (Please remember: The difference
between the maximum inclination and the excitation component is the precession
component.) It seems that precession becomes slightly worse for lower radii, but as
mentioned before, it damps out quicker. For this flywheel geometry optimisation
research the excitation amplitude is most relevant. A good flywheel will keep this value
down. Still, it is good to keep an eye on precession. A flywheel can perform really well in
theory but can be useless in practice if it responds very bad to precession induced
disturbances.
To try and understand how important other parameters such as weight and height
matter more simulations will be undertaken. Flywheel capacity has a lot to do with
inertias. This will be the focal point of the oncoming simulations.
The flywheel thickness’ effect on performance is next. The flywheel thickness is changed
while keeping its radius at the reference value. Please read the results below.
0
10
20
30
40
50
60
70
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.2 0.3 0.4 0.5
Fly
wh
ee
l we
igh
t [k
g]
Max
imu
m in
clin
atio
n [
rad
]
Flywheel radius [m]
Max. incl.
Exc. Incl.
Weight
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Figure 78: Simulation of Lidar inclinations of Z axis for varying flywheel thickness. Lower radii yield higher maxima.
Performance can still be increased by making the flywheel heavier, it seems, with
minimal influence on precession period.
Var Radius Height Weight Iζζ Iρρ Iζζ ∙ Iρρ Max. incl Exc. Incl Relative
Unit m m kg kg∙m² kg∙m² kg²∙m4 rad rad %
0.50 0.005 30.63 3.83 1.91 7.33 0.001066 0.000659 32.02%
0.50 0.010 61.26 7.66 3.83 29.32 0.001116 0.000736 0.00%
0.50 0.020 122.52 15.32 7.66 117.34 0.001315 0.000960 -15.13%
0.50 0.040 245.04 30.63 15.35 470.12 0.001736 0.001787 -18.94%
Table 17: Results and calculated values of simulations with changing thickness and constant radius
Relevant measurements are visualised below.
0.005 m 0.01 m 0.02 m 0.04 m Z inclination [rad] Time [s] 6000 rpm
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Graph 5: Maximum Lidar inclination versus flywheel thickness for a constant radius.
It looks as if more accuracy can be achieved by making the flywheel heavier, but the
performance increase comes at a high price. A flywheel of almost 250 kg is not exactly
practical. A much bigger motor would be needed as well which is unfeasible because of
the Lidar buoy’s limited power supply. The tables show that the parameter ‘Iζζ ∙ Iρρ’ has a
strong correlation with performance. This is because a high inertia around the rotation
axis increases flywheel performance and because a high inertia around the radial axis
increases the force needed to tilt the flywheel from vertical. So the mechanism’s
performance is expected to be proportional to the value of ‘Iζζ ∙ Iρρ’. This hypothesis is
tested by making more graphs.
If all the results so far in this paragraph were rearranged and an XY-plot was made in
which the product of inertias was plotted on the x-axis and the maximum Lidar
inclination was plotted on the y-axis, then a monotonously descending curve would
appear if the hypothesis were true.
Graph 6: XY-plot of the product of inertias and the maximum Lidar inclination.
0
50
100
150
200
250
300
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
0.002
0 0.01 0.02 0.03 0.04 0.05
We
igh
t [k
g]
Incl
inat
ion
[ra
d]
Flywheel height [m]
Exc. Incl.
Exc. Incl.
Weight
0.000000
0.010000
0.020000
0.030000
0.040000
0.050000
0.00 100.00 200.00 300.00 400.00 500.00
Max
imu
m in
clin
atio
n
Product of inertias [kg^2*m^4]
Reference
flywheel
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Not exactly what one would expect. Looking at graph 6 it seems that, for lower values,
the performance increases drastically (the Lidar inclination decreases). But for very high
values the performance remains constant. So there is a point where additional increase
in inertia is of no benefit anymore. It looks as if the hypothesis is wrong. A final series of
simulations was made each with a constant product of inertia equal to the value
calculated for the reference mechanism. If the hypothesis were true the mechanism
performance would be more or less constant. It is not.
Var Radius Height Weight Iζζ Iρρ Iζζ ∙ Iρρ Max. incl Exc. Incl Relative
Unit m m kg kg∙m² kg∙m² kg²∙m4 rad rad %
0.2000 0.2967 290.82 5.82 5.04 29.32 0.009380 0.005470 613.31%
0.3000 0.0763 168.37 7.58 3.87 29.32 0.003940 0.002090 199.62%
0.4000 0.0244 95.67 7.65 3.83 29.32 0.001918 0.001242 45.86%
0.5000 0.0100 61.26 7.66 3.83 29.32 0.001315 0.000960 0.00%
Table 18: Results and calculated values of simulations with constant product of inertias.
Graph 7: Lidar inclination versus flywheel radius. Product of inertias is constant at the reference value.
Although the product of inertias has remained constant, performance still increases
rapidly with increasing radius. It is all about the radius, it seems. Finally a couple of
design rules for a cylindrical flywheel can be formulated:
1. Increase radius as much as possible considering space limitations.
Large performance gain. Consider changing design for additional space.
0.000000
0.001000
0.002000
0.003000
0.004000
0.005000
0.006000
0.007000
0.008000
0.009000
0.010000
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
Lid
ar in
clin
atio
n [
rad
]
Flywheel radius [m]
Max. incl.
Exc. Incl.
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2. Increase thickness as much as possible considering weight and power limitations.
Smaller performance gain. Changing weight/power limitations should not
require much more resources or it is not worth it.
4.5.4 Flywheel weight
The flywheel could be manufactured out of a lighter or a heaver material if needed.
Looking at the formulae in section 4.5.3, it is obvious that changing the density ρ will
have a similar effect to changing the thickness h. Nevertheless the flywheel material
should be looked at. A series of simulations was undertaken with a flywheel made of
aluminium (2700 kg/m³), steel (7800 kg/m³ (ref)) and lead (11340 kg/m³).
Figure 79: 90 second simulation of reference mechanism with a flywheel made from varying materials. Heavier
materials yield lower maximum inclinations.
The relevant measurements are summarized in Graph 8:
Aluminium Steel Lead Z inclination [rad] Time [s] 6000 rpm
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Graph 8: Inclination for varying flywheel density
The weight of the flywheel only seems to affect the amplitude of the excitation
component. It is very clear it is beneficial to make the flywheel heavy, just as expected.
The performance gained from an increase in density of 45.38% (steel to lead) only leads
to a performance gain of about 10%. (Table 19) This is not great considering the fact
that both the extra weight and the soft lead can cause additional practical issues. It can
be concluded that that steel is best kept as the material of choice for the flywheel.
Var Radius Height Weight Iζζ Iρρ Iζζ ∙ Iρρ
Max.
incl Exc. Incl Relative
Unit m m kg kg∙m² kg∙m² kg²∙m4 rad rad %
(Aluminum) 0.50 0.01 21.21 5.30 1.33 7.03 0.00222 0.00192 71.17%
(Steel) 0.50 0.01 61.26 15.32 3.83 58.69 0.00129 0.00095
(Lead) 0.50 0.01 89.06 22.27 5.57 124.04 0.00117 0.00081 -9.97%
Table 19: Results of simulation series with different flywheel materials.
4.5.5 Flywheel offset
Another very important parameter to consider is the flywheel position. More specifically
the distance of the flywheel to the Lidar.
0.00000
0.00050
0.00100
0.00150
0.00200
0.00250
0 2000 4000 6000 8000 10000 12000
Lid
ar in
clin
atio
n [
rad
]
Density [kg/m³]
Max. incl.
Exc. Incl.
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Figure 80: Simulation of Lidar inclinaton for varying flywheel offset. Smaller amplitudes represent smaller offset and
vice versa.
Var Radius Max. Incl Exc. Incl Relative
Unit m rad rad %
0.20 0.001315 0.000960 0.00%
0.25 0.001452 0.001043 10.42%
0.30 0.001599 0.001147 21.60%
0.35 0.001726 0.001236 31.25%
Table 20: Results and calculated values of simulations with changing offset
Relevant results are summarized below.
0.2 m 0.25 m 0.30 m 0.35 m Z inclination [rad] Time [s] 6000 rpm
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Graph 9: Maximum Lidar inclination versus flywheel offset for the reference mechanism.
Mounting the flywheel lower down worsens the precession motion and deteriorates
performance. This can be seen by looking at the graph of the simulation results. (Graph
9) Precession amplitude remains constant, its period shortens. Excitation amplitude
rises. The decrease in performance is linear and predictable. So if the flywheel must be
positioned slightly lower for some reason it is not very bad.
From a practical point of view a large offset is definitely not desirable. Larger offsets
limit the available space for installations under the Lidar module. Typically a Lidar and
flywheel assembly is easier to build with a smaller offset. Since smaller offset values
perform better as well, it can be concluded that is wise to put the flywheel as close as
possible to the Lidar within practical limits. However, no special measures should be
undertaken to mount it especially close.
The flywheel offset is only part of the more extensive research of the optimal positioning
of components. Much more conclusions will be made in the next paragraph.
4.5.6 Lidar offset
The position of the Lidar module in the reference mechanism is inspired on its position
in the prototype of the actual mechanism. This doesn’t mean that performance can’t be
enhanced by changing that position. Two experiments will be undertaken: one with the
Lidar positioned higher and one with the Lidar positioned lower. Just as in figure 81.
0.000000
0.000200
0.000400
0.000600
0.000800
0.001000
0.001200
0.001400
0.001600
0.001800
0.002000
0.15 0.20 0.25 0.30 0.35 0.40
Lin
dar
incl
inat
ion
[ra
d]
Offset [m]
Max. incl.
Exc. Incl.
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Figure 81: Mechanism configurations respectively with the Lidar moved downwards and upwards.
Figure 82: Comparison of the reference mechanism and a mechanism with the Lidar positioned lower.
It looks like moving the Lidar down has had a detrimental effect on performance. The
excitation component amplitude and the precession component amplitude increase.
Because the Lidar was moved down, the flywheel has moved down as well to keep it
from intersecting the Lidar. So the change of flywheel offset can account for some of the
loss in performance, especially for excitation amplitude gain, but not all of it. Actually,
comparing the earlier conclusions and these can be confusing. One would think that
moving the Lidar down would create a pendulum of sorts of which the mass distribution
would be helpful to achieve an upright position of the system.
Reference Lidar down Z inclination [rad] Time [s]
6000 rpm
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In practice, moving the mass that needs stabilizing further from the rotation point
increases precession dramatically. The increase in excitation component amplitude can
be explained by pointing out that the buoy’s centre of gravity lies way below the offset
reference point (and centre of gimbal rotation). (See 4.2) So a rotation of the buoy’s
frame implies a significant translation of the Lidar. This translation means translational
acceleration. If the Lidar subsystem is suspended as a pendulum, this acceleration
means that it will start swaying and, despite the efforts of the flywheel, lead to an
increase in Lidar inclination angle of the excitation component. The same reasoning
could be done for the flywheel, since it has also a mass.
Figure 83: Illustration of Lidar subsystem swaying.
It can be concluded that the whole ‘pendulum’ idea is misleading. To make a system
work without a flywheel while suspended as a pendulum it could only work with
carefully tuned active dampers.
Subsystem COG in rotation point.
OPTIMAL
Subsystem COG below rotation point.
PENDULUM
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Figure 84: Comparison of the reference mechanism and a mechanism with the Lidar positioned higher. The reference
position yields higher excitation amplitude.
The simulation above was undertaken with the Lidar moved upwards. Because the
flywheel would intersect the gimbals if it were moved up with the Lidar, it was kept in
the same position. As shown in the previous mechanism pictures. This means that the 45
kg Lidar is 200mm above and the 61 kg flywheel 200mm below the offset reference
point. The COG of the Lidar-flywheel subsystem is not far from the offset reference point
and the current rotation point. If the theory explained in the previous paragraph is true
this configuration should lead to better performance and it does. The simulation results
shown in figure 84 prove that both the precession component and excitation component
have decreased in amplitude. The precession period has increased, though. This is
caused by the fact that the pendulum, harmful as it may be for mechanism performance,
does help damp out precession a little (very little in fact) because its mass distribution
opposes the precession misalignment.
The Lidar position was not the only aspect of the research mechanism inspired by the
prototype. The shape of the gimbals was as well. This shape means that the outer
gimbals’ X rotation axis is 40mm above the inner gimbals Z rotation axis. This makes the
position of the inner subsystem’s rotation point dependent on the rotation. (The inner
subsystem consists of Lidar, inner gimbal and flywheel) During the previous discussion,
the offset reference point was always regarded as the rotation point while this was not
completely true. The real rotation point was not far away but its actual position was
unknown (See 4.2) If a new mechanism was to be designed it would definitely be worth
the trouble to make sure that said gimbal axes are in the same plane and that the centre
of rotation was independent of the movement.
Lidar up Reference Z inclination [rad] Time [s] 6000 rpm
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4.5.7 Conclusion of mechanism geometry design: ‘super mechanism’
The previous parts contained lots of numbers and simulation results. The most
important things to remember are:
It is good to strive for a high rotational speed for the flywheel. The higher the
better. From 6000 rpm onwards the performance gain becomes very little. When
disturbances are expected, it’s better to choose lower rpm to reduce settling time.
(See section 4.5.8.2)
A flywheel must have a large inertia around the rotation axis for its weight,
within practical limits. The inertia is calculated in a local coordinate system with
the flywheel COG as origin.
The entire subsystem that must be kept upright, consisting of Lidar, flywheel and
inner gimbal, must have its centre of gravity as close as possible to the point
around which the gimbals and outer frame rotate to prevent swaying.
It is good to emphasise the fact that the gimbals are a no more than an unfortunate
necessity to make the mechanism physically possible. Their inertia should be kept as
low as possible for good mechanism performance because then their influence (more
specific: inertia) becomes less of a nuisance.
To prove all this, a ‘super mechanism’ was made in CAD design software and imported
into UM. Figure 85 shows what it looks like.
Figure 85: Screenshot of ‘Super mechanism’ in UM. Would be difficult to build. Physical axes not drawn. Frame and
Lidar module drawn in wireframe.
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Figure 86: ‘Super mechanism’. From below. The flywheel is a ring around the Lidar itself with a hub at the bottom.
Figure 87: Side view of ‘super mechanism’. Look at the Lidar module’s orientation which is, as far as the eye can see,
perfectly upright while the mechanism is moving.
The gimbals are hollow box-sectioned steel tubes. The flywheel is no longer a disc, but a
ring. (The construction of a parameterized ring in UM would be very difficult.) The
flywheel weight is approximately the same as the reference mechanism’s flywheel:
about 61kg. All other specifications and parts are exactly the same as for the reference
mechanism, such as frame and Lidar.
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Figure 88: Flywheel of the ‘super mechanism’. A ring instead of a disc. A central hub was added to make it possible to
attach a motor.
So the ‘super mechanism’ has:
Rotation speed of 3000 or 6000 rpm. Unchanged.
Flywheel with high rotational inertia Iζζ
Stabilized subsystem COG exactly coincident with rotation centre.
Lighter gimbals with coplanar rotation axes
It can be expected that this mechanism performs much better than the reference
mechanism. And it does. As shown in the simulations at 6000 rpm.
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Figure 89: Simulations of Lidar inclination comparing the super mechanism to the reference mechanism
It seems that the super mechanism’s excitation component amplitude is much smaller
than the reference mechanism. The maximum inclination is 56% smaller, the excitation
component is 67% smaller. However the precession period of the super mechanism
seems very long. And it is.
Figure 90: Simulations of Lidar inclination comparing the super mechanism to the reference mechanism. Simulation
time is 1200 seconds or 20 minutes.
Super mechanism Ref. mechanism Z inclination [rad] Time [s] 6000 rpm
Super mechanism Ref. mechanism Z inclination [rad] Time [s] 6000 rpm
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In the 1200 second, or 20 minutes, simulation the precession amplitude becomes
apparent as well as the scale of the performance increase. Relevant numbers are
compared below.
Graph 10: Comparison of the reference mechanism to the hypothetical ‘super mechanism’.
The development of the ‘super mechanism’ has verified the conclusions made. But, and
there is a big but, the mechanisms performance comes at a high price. The gyroscopes
capacity is now so big precession motion becomes much harder to damp out. Especially
at 6000 rpm. Good performance means nothing if a slight disturbance induces a
precession motion that goes on for hours. On the other hand, if the mechanism was well
protected, it could deliver similar accuracy to an actively controlled system with none of
the worries and costs that active components (pistons, active dampers, electronics …)
would bring.
For the remainder of the thesis the old reference mechanism will be used, since it is
more representative.
0
10
20
30
40
50
60
Inner weight Outer weight
Ref
Super
0
2
4
6
8
10
12
Inertia ofrotation axis
Inertia of radialaxis
Ref
Super
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
Excitation inclination Maximum inclination
Ref
Super
(Gimbals)
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4.5.8 Damping
4.5.8.1 Effect of damping in the reference mechanism
The reference mechanism has some damping to control precession. The next study
covers the mechanisms behaviour when the damping is changed. Speed is now 3000
rpm to make damping more obvious.
Figure 91: Simulation of Lidar inclination for changing damping. Higher amplitudes mean higher damping values.
Flywheel speed is 3000 rpm.
Graph 11: Lidar inclination versus damping for the reference mechanism.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0.005
0 20 40 60 80 100
Incl
inat
ion
an
gle
[ra
d]
Damping [kg*m/rad/s]
Total Incl
Exc Inclin
20 kg.m/rad/s 40 kg.m/rad/s 80 kg.m/rad/s Z inclination [rad] Time [s]
3000 rpm
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cdiss Total Incl Exc Inclin Relative
kg.m/rad/s rad rad %
20 0.002598 0.001913 0.00%
40 0.00315 0.002412 21.25%
80 0.004605 0.003774 77.25%
Table 21: Results and calculated values of simulations with changing offset.
It looks as though the damping decreases performance and it does so seemingly in a
linear fashion. So why does the supposedly optimized reference situation have a
damping value of 20? Because, if there were no damping, precession would not diminish
at all (from a purely theoretical point of view). This is unacceptable since precession is
destined to occur at some point. Almost anything that can go wrong has precession as a
result. This is discussed in section 4.6.
To quantify how fast the precession is damped out. It is deliberately excited by starting
the simulation with no frame movement and an initial mechanism inclination value of
0.1 rad for both rotation axes. As shown in figure 92:
Figure 92: Mechanism with misaligned start-up position. Frame is fixed.
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Figure 93: Simulation results for verying damping with misaligned startup of 0.1 rad and stationary frame. 1200 sec
simulation. 3000 rpm
The simulation results seem to show that the intersection of the curves and the half life
line consistently seem to coincide with local maxima. This is due to the fact that the
chosen damping coefficients are multiples of each other. Half-life times are multiples of
the precession period. It is also obvious that the precession period remains constant
when damping is changed. The period is about 82 seconds. Half-life values are plotted in
graph 11.
Graph 12: Half-life of precession motion.
0
100
200
300
400
500
600
700
800
900
1000
0 20 40 60 80 100
Hal
f Li
fe [
s]
Damping [kg*m/rad/s]
Half-life line
20 kg.m/rad/s 40 kg.m/rad/s 60 kg.m/rad/s 80 kg.m/rad/s Z inclination [rad] Time [s] 3000 rpm
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The half-life diminishes with increasing damping. This is a good thing, but it comes at
the cost of an increase in excitation amplitude which should be kept as low as possible.
Especially if absolute accuracy is important. If this is the case, one could opt for a
flywheel speed of 6000 rpm. But how will the precession behaviour fare?
Figure 94: Simulation results for changing damping with misaligned startup of 0.1 rad and stationary frame.
Graph 13: Results of 6000 rpm simulation series compared to 3000 rpm.
~16 min
~8min ~5.5min ~4min
~1hour 3min
~32min
~21min 16min
0
500
1000
1500
2000
2500
3000
3500
4000
0 20 40 60 80 100
Hal
f Li
fe [
s]
Damping [kg*m/rad/s]
3000 rpm
6000 rpm
20 kg.m/rad/s 40 kg.m/rad/s 60 kg.m/rad/s 80 kg.m/rad/s Z inclination [rad] Time [s]
6000 rpm
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It was already mentioned that increasing the flywheel’s capacity means it becomes more
difficult to damp out precession motion and this shows in graph 13. Half-life results for
6000 rpm are much higher. If the mechanism builder opts for 6000 rpm it might be
worthwhile to use more damping to increase robustness and sacrifice some accuracy in
ideal situations because the increase in excitation component in relation to damping is
less severe than the decrease in half life.
It may be worthwhile considering that actual bearings dissipate energy and damp
motions due to viscous shearing of its lubricants. So arguably a reference situation with
damping present is more representative for a physical situation. If the naturally
occurring damping is enough to fight precession remains to be seen, but it is not to be
expected. Additional damping can be effectuated by adding rotational dampers in any
case.
4.5.8.2 Beware of precession
From the previous paragraph one can deduce that precession motion, whatever the
cause, can be the most dominant effect for Lidar misalignment. So perhaps sometimes
more effort should be put into fighting the precession motion than into getting the,
already very small, maximum amplitude down in a theoretical and ideal situation. Nit
picking in the ideal situation provided insight in flywheel optimization, but in a real
situation, with a semi optimal flywheel, it is the precession motion that should be
reduced as much as possible.
Prone to disturbances
• Lower speed
• Higher damping
Free of disturbances
• Higher speed
• Lower damping
Practical:
Damping can be effectuated by installing rotational dampers on the axes, like in the prototype. This thesis is a ‘pioneering’ research in the sense that its aim is to gain a thorough insight into the basics of the behavior of this mechanism. When designers become more experienced in the design of such a mechanism though it might be worthwhile to use computer controlled dampers with variable damping coefficient. When the computer detects precession motion, the dampers could be adjusted accordingly. When the system is stable, the dampers needn’t be switched on.
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4.6 Reaction of the mechanism to irregularities
Since the mechanism will be used in a hostile environment and has to operate
autonomously it is important to check the response of the theoretical model to
disturbances. If the theoretical model shows no important decline in performance for a
certain disturbance, no special measures have to be undertaken in the design of an
actual mechanism. If the model responds violently, countermeasures for the specific
disturbance should be built in to the actual mechanism.
In the following paragraphs certain disturbances will be described and the mechanism
model will be extended to allow for parameterized definition of the more complex
disturbance movements.
4.6.1 Sudden deceleration of flywheel
If something were to happen to the motor or if the flywheel was struck by an object it
would decelerate abruptly. The reference mechanism is simulated like before but the
flywheel speed has a sudden dip. This requires a more complex position description in
the flywheel-Lidar joint. Although it is quite difficult to slow down a 61 kg flywheel
spinning at high speed this must be looked into because the flywheel angular
momentum is one of the key elements for the mechanism behaviour. The mechanism is
expected to react very badly to this.
The flywheel angular speed is described by figure 95:
Figure 95: Flywheel angular speed function with parameters composed of constants and cosines
a
d
b
ω
t
c
spee
d
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ω Normal speed rpm
d Speed dip height rpm
a Dip start time s
b Dip duration s
c Spring back duration s
b + c Total dip duration s
Table 22: Parameters of flywheel speed function
The function is smooth, the derivative (angular acceleration) has no discontinuities. It is
actually composed of constants and cosine functions. The position function, which is
implemented in UM, is the integral of the speed function. To see the detrimental effect of
this phenomenon, simulations were configured with the following parameters and
executed. Since the disturbance is expected to cause precession motion the flywheel
speed is set at 3000 rpm. The first series is representative for small changes while a
second series is representative for ‘disasters’. Simulation results are printed for the most
spectacular changes. Please note that for the first ‘a’ seconds the situation is ‘normal’ and
is exactly like the reference situation at 3000 rpm.
ω 3000 rpm
d
50 100 200
rpm
400 800 1600
a 120 s
b 1 s
c 10 s
Table 23: Table with parameters for testing of detrimental speed dip: changing speed difference
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Figure 96: Detail of flywheel angular speed for the parameters described above for the ‘disastrous’ speed dips.
Figure 97: Simulation of Lidar inclination for large speed dips of 200, 400 and 800 rpm. Before t equals 120 the
situation is the reference situation. Total duration is 240 sec.
400 rpm dip 800 rpm dip 1600 rpm dip Z inclination [rad] Time [s]
3000 rpm
400 rpm dip 800 rpm dip 1600 rpm dip Angular speed [rad/s] Time [s]
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Graph 14: Lidar inclination for simulation with different speed dips.
A few pointers:
The proposed disastrous speed drops are very high and unlikely to occur. This is
fortunate because the induced precession motions are 313%, 671% and 1781%
worse than the reference situation at 3000 rpm for a speed dip 400, 800 and
1600 rpm respectively.
It looks as if the maximum inclination has an almost linear correlation with
height of the speed dip for reasonable values. One could conclude that the
induced precession motion will be approximately doubled in amplitude when the
speed drop is twice as severe for a quick drop in speed.
The half-life of the exponential decay is the same for all simulations above
because the damping was not changed.
In general: Slowing down the flywheel abruptly causes precession motion and
should be avoided.
Next up is the investigation of the influence of the speed dip duration. Simulate with the
following parameters:
0
0.01
0.02
0.03
0.04
0.05
0.06
0 500 1000 1500 2000
Lid
ar in
clin
atio
n [
rad
]
Speed dip height [rad/s]
Excitaton ampl.
Max ampl.
Ref max.
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ω 3000 rpm
d 200 rpm
a 120 s
b 1 2 4 s
c 10 s
Table 24: Table with parameters for testing of detrimental speed dip: changing dip time.
Figure 98: Simulation of Lidar inclination with a dip in the speed at 120 seconds. Varying speed dip duration.
1 sec dip 2 sec dip 4 sec dip Z inclination [rad] Time [s] 3000 rpm
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Graph 15: Lidar inclination for a constant spring back and changing dip duration.
1 sec dip 2 sec dip 4 sec dip Time [s] 3000 rpm
Left: Flywheel speed [rad/s] Right: Flywheel acceleration [rad/s²]
Figure 99: Flywheel angular speed and acceleration for said simulations.
It seems mechanism performance decreases first with increasing dip duration and then
increases again. The results above may seem surprising. One would think that less
abrupt speed dips would be less detrimental, but the opposite is true when dip time
rises from 1 to 2 seconds. What if the dip time was changed like in the last simulations,
but the total period with a speed difference remained constant (or b+c=10) for every
value of b?
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0 1 2 3 4 5 6
Lid
ar in
clin
atio
n [
rad
]
Dip duration b [s]
Excitaton ampl.
Max ampl.
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ω 3000 rpm
d 84 Rad/s
a 120 s
b 1 2 3 4 5 6 7 8 9 s
c 9 8 7 6 5 4 3 2 1 s
Table 25:Table with parameters for testing of detrimental speed dip: changing dip time with constant total dip period.
For brevity, the simulation result graphs will be omitted because they look similar to the
last ones. Only the acceleration graph will be shown. The relevant results are
summarized below and compared to the previous graph.
Figure 100: Flywheel angular acceleration for speed dip simulations for constant dip time.
Flywheel acceleration [rad/s^2] Time [s]
3000 rpm
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Graph 16: Lidar inclination versus dip duration for simulation series with constant speed dip time.
It seems that the mechanism’s performance is influenced by the shape of the speed dip.
If the speed dip period of the speed dip function is mirrored the mechanism reacts
similarly. It can certainly be said there are differences in the negative effects depending
on the speed dip characteristics.
A small and quick speed dip is not very bad. (Figure 101) An example of a physical
phenomenon that could cause this is an object impacting the flywheel. The mechanism
parts’ inertia makes sure the mechanism is unflustered.
A large but very gradual change in speed is not so bad either. (Figure 103) This could
happen when an invertor fed motor’s input frequency is gradually changed.
Changes of which the integral of the acceleration function for the dip period is large and
of which the speed is under 8 seconds are the worst, it seems. These could only happen
because of a severe motor or invertor failure of some other sort of mechanical failure.
Acceleration is just as bad as deceleration.
The first two theories are tested first. (The last one was deduced from the previous
experiment) The simulation below is configured with parameters representing a very
small and fast dip similar to the one the flywheel would experience if it was struck by a
small object. It can be observed that the precession motion is not very noteworthy.
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0 2 4 6 8 10
Lid
ar in
clin
atio
n [
rad
]
Dip duration [s]
Max. ampl.
Excitaton ampl.
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ω 3000 rpm
d 50 rpm
a 120 s
b 0.1 s
c 3 s
Table 26: Parameterset for fast speed dip simulation
Figure 101: Simulation of Lidar inclination with very fast but small speed dip at t equals 120 seconds.
The simulation below is held with parameters representing a gradual change of speed
form 3000 to 6000 rpm as shown in figure 102. Like an invertor output frequency being
changed from 50 to 100Hz for a fast induction motor. The speed change causes nothing
more than a minor transition effect. After the flywheel speed is stable at 6000 rpm, the
Lidar inclination quickly settles to the regime seen earlier at the constant speed
simulation for 6000 rpm.
Dip
Z inclination [rad] Time [s] 3000 rpm
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ω 3000 Rad/s
d -3000 Rad/s
a 120 s
b 120 s
c 10000 s
Table 27: Parameterset for gradual speed change.
Figure 102: Flywheel speed for experiment of gradual variation in flywheel speed.
Practical:
As mentioned before, increased flywheel capacity means increased performance, but makes the mechanism less forgiving for disturbances. This is why it is worthwhile to protect the flywheel and motor assembly both mechanically and electrically. A casing around the flywheel and motor can protect it from disturbances. Well protected power electronics and cables minimize the chance for electrical issues. Motor bearings would be protected against the corrosive environment as well, eliminating the need for special bearings and lubricants with lower friction coefficient and motor requirements as a result. Any protective covers should be kept as light as possible for mechanism performance and must possess no large flat surfaces since it might be subject to wave slamming. One must keep in mind to create the protection in such way that it does not affect the motor cooling capabilities negatively.
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Figure 103: Simulation result for Lidar inclination with gradually changing flywheel speed. Transition from 3000 to
6000 rpm. Transition time is 120 seconds.
One can conclude that small mishaps and gradual changes may happen without
compromising performance. If a remote operator wished to change the flywheel
speed/capacity he or she may do so without harming accuracy much as long as the
change of speed is very gradual.
4.6.2 Sudden changes in buoy frame movement
A floating buoy attached to a cable is subject to jerking when the current or waves act to
move the buoy. How will the mechanism react to this? First, the jerking motion is split
up into two components: a pitching and a translational component. The first will be
modelled by suddenly changing the amplitude of the sinusoidal motion. The second can
be modelled by translating the entire mechanism in a representative way.
4.6.2.1 Functions
To test the effect of an excitation amplitude change the original model is extended, just
like in the last paragraph. This time however, it is not the flywheel position function that
is extended but the excitation functions that make the frame move in a sinusoidal
fashion. The functions used to be simple sinusoids, but have been extended with a
parameterized and more complex envelope function as shown figure 104:
Z inclination [rad] Time [s]
3000/6000 rpm
3000 rpm Transition 6000 rpm
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Figure 104: Excitation function with envelope and its parameters.
The same envelope function can also be used as a single input function without the sinus
movement to keep the frame steady and impose a single and sudden rotation at time a. It
is similar to the flywheel speed dip function, but imposed on the frame as a rotation. The
amplitude function could look like the red line shown figure 104 without the sine.
4.6.2.2 Rotational jerking
a1 First amplitude value Rad
a2 Second amplitude value Rad
a3 Third amplitude value Rad
a Start time s
b First transition time s
c Second transition time s
sin_on Control parameter:
=0: Only envelope
=1: Sine with envelope
-
Table 28: Parameters of amplitude function of sudden change model
a1
a3
a2
a b c
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At first a simple test will be carried out. Imagine the mechanism upright when suddenly
it is jerked to one side and back again. The Lidar would start swaying (if the COG of the
Lidar subsystem has offset to the rotation point. See 4.5.6 Some offset will typically be
present.) but the gyroscope will counteract this motion. It is expected that the
mechanism will exhibit precession motion after the disturbance. Several simulations will
be made according to the following table:
a1 0 Rad
a2 0.05 0.10 0.15 0.20 Rad
a3 0 Rad
a 2 s
b 2 s
c 2 s
sin_on 0 -
Table 29: Rotational jerking simulation parameters: amplitude of impulse
Figure 105: Plots of frame movement functions for amplitudes 0.05, 0.10, 0.15, 0.20
0.05 rad 0.10 rad 0.15 rad 0.20 rad Frame inclination [rad] Time [s]
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Figure 106: Simulation results for Lidar inclination for a disturbance amplitude of 0.05 , 0.1, 0.15, 0.2 rad.
Graph 17: Results for simulation of mechanism reaction to sudden rotational jerking with the set of parameters
described above. The depicted values are the amplitude of the precession motion in steady state and not of the violent
transition maxima. The same experiment was carried out of 6000rpm.
The rotational jerking for the given duration does not disturb the mechanism too much.
Only a very violent rotational ‘tug’ can cause a noteworthy precession. An unlikely event
considering the buoy weighs several tonnes.
It is possible to include more results, but that would be a waste of time. More
simulations were undertaken but they all pointed out that rotational jerking is nothing
to worry about.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0 0.05 0.1 0.15 0.2 0.25
Lid
ar in
clin
atio
n [
rad
]
Max frame deviation [rad]
Max ampl 3000
Ref 3000
Max ampl 6000
Ref 6000
0.05 rad 0.10 rad 0.15 rad 0.20 rad Z inclination [rad] Time [s] 3000 rpm
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4.6.2.3 Translational jerking
Now research analogous to the previous paragraph will be undertaken with similar
functions, only for translation instead of rotation. The translation movement is parallel
to direction (1,4,1). This is to create a representative cable snagging motion on one hand
and a worst case scenario on the other hand. Pulling directions aligned with one of the
axes or perpendicular creates less of a disturbance.
Figure 107: Direction of translational motion (1,4,1)
The following parametersets represent a normal situation where, suddenly, the frame is
tugged down.
a1 0 m
a2 -0.05 -0.125 -0.25 -0.5 m
a3 0 m
a 120 s
b 2 s
c 2 s
(Normal frame movement with amplitude 0.1 and
period of 5 seconds. Similar to the reference situation)
Table 30: Translational jerking simulation parameters
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Figure 108: Simulations of Lidar inclination. Sudden translation at t equals 120 seconds.
The results above show that translational jerking is nothing to worry about. The only
effect of the translation is a slight transition phase. Whet it has passed, the movement is
exactly the same as seen in the theoretically ideal reference mechanism simulations.
Further analysis is not needed. When the dip duration is varied the results are similar: a
calm transition phenomenon and a steady state identical to the theoretically ideal
situation.
4.6.2.4 Change of amplitude
As mentioned earlier; the functions can also be sinusiodal with the step like function as
an envelope. This can simulate a change in amplitude. The simulation (figure 109) shows
the results (Lidar z-angle) of the reference mechanism starting with an excitation
amplitude of 0.1 rad and switching to 0.2 in two different time intervals: 120 seconds
and 15 seconds:
Z inclination [rad] Time [s] 3000 rpm
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Figure 109: Simulation of Lidar inclination for gradually changing amplitude from 0.1 to 0.2 rad.
There is absolutely no transition phenomenon nor dynamic effect visible in the results.
Even when testing with very sudden envelope functions. This means that all changes can
be regarded as quasi instantaneous. All maximum Lidar angles correspond with the
amplitude at the same time. Changes in amplitudes don’t cause problems as well. A
sudden tug inducing a pitch motion such as a cable snag was more detrimental, but still
not a real threat. Especially at a realistic transition time.
4.6.2.5 Conclusion
It can be concluded that jerking is not a threat to the accuracy of the measurements the
Lidar will take. Only when the mechanism is pitched very violently, the effect is
noticeable. Very sudden pitch movements are, considering the buoy’s bulkiness, very
unlikely.
4.6.3 Restarts during operation
If a problem occurs the flywheel may stop spinning. How bad will the performance be if
this happens? And what happens if the flywheel is restarted? To find out, simulations
will be carried out in which the flywheel is stationary for 120 seconds and is then
accelerated to 6000 rpm in a specified time.
15 sec transition 120 sec transition Z inclination [rad] Time [s] 6000 rpm 0.1 - 0.2 rad
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Figure 110: Simulations of Lidar inclination at start-up for varying start-up durations.
First things first. The mechanism’s performance with the flywheel inactive is 1868%
worse than the reference mechanism at 6000 rpm. An inactive flywheel means a
maximum inclination of 1.61°. When the flywheel is accelerated a violent transition
phenomenon is observed followed by a new steady state regime with an induced
precession motion. The transition regime is not really important. However, the newly
induced precession motion is. Its relationship with the transition time is shown graph
18.
Graph 18: Maximum Lidar inclination in steady state after flywheel acceleration from 0 to 6000 rpm.
2min
4 min
6min
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0 50 100 150 200 250 300 350 400
Ste
ady
stat
e m
axim
um
incl
inat
ioo
n
afte
r ac
cele
rati
on
[ra
d]
Flywheel acceleration time [s]
Max SS inclination
0 - 6000 rpm
120 sec transition 240 sec transition 10 sec transition Z inclination [rad] Time [s]
Steady state Steady state
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It looks like it is worthwhile to restart the flywheel as gently as possible. In case the
flywheel drive’s power electronics were incapable of achieving this, the effect could be
very bad as shown in the simulation results. If the motor spins up the flywheel in 10
seconds, the resultant maximum inclination is 6.4° and it will take very long to damp
out.
Changing flywheel speed without compromising performance too much is possible if the
changes are very gradual.
4.6.4 Influence of friction
Of course friction should be reduced as much as possible but it is inevitable. To quantify
the detrimental effect of friction, the force function of the joints were changed to include
friction force:
A constant restoring moment was added working in the opposite direction of rotation.
The force is determined by the friction coefficient μ, the load on the bearings [kg], the
gravitational constant g [m/s²] and the radius of the axis [m] rotating in the bearing.
The bearings used are roller bearings. The friction coefficient of these bearings is around
0.001 for oil lubricated bearings. For greased and sealed bearings however friction is
expected to be higher and because the working conditions for these bearings will be
especially bad. The friction coefficient is estimated at 0.01. The axis diameter is
estimated to be 6 cm.
To check how much it would pay off to decrease friction to 0.001 and to see how bad it
would be if the friction coefficient rose to 0.03 a series of simulations will be
undertaken:
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μ 0 0.001 0.003 0.01 0.03 -
mload X 200 kg
mload Z 150 kg
g 9.81 m/s^2
raxis 6 cm
(Normal frame movement with amplitude 0.1 and
period of 5 seconds. Like in the reference situation)
Table 31: Parameters for friction research
Figure 111: Simulation of Lidar Z axis inclination for varying friction coefficient. Simulation time is 90 seconds.
The simulations show an increase in amplitude as expected. It can be seen that only the
excitation component increases and not the precession. The precession period remains
the same. The shape of the graphs seem to change for higher friction due to the force
discontinuity caused by the discontinuous ‘signum’ function.
6000 rpm
0.001 0.003 0.01 0.03 Z inclination [rad] Time [s]
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Graph 19: Graphs of friction coefficient research for both Lidar axes.
For the first three data points the relation between performance and friction coefficient
seems completely linear. Only for the highest value the linear relation is somewhat
broken, but only by a small margin. The graphs above show that the performance
decrease is entirely due to the excitation component. It may also be noted that the slope
of the graphs is steeper for the X axis than for the Z axis. This is because the X axis’
bearings carry more load and produce more friction moment. Let see how much
performance declines in the relative sense.
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
0.002
0 0.01 0.02 0.03
Lid
ar in
clin
atio
n [
rad
]
Friction coefficient [-]
Max ampl Z Exc ampl Z
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0 0.01 0.02 0.03
Lid
ar in
clin
atio
n [
rad
]
Friction coefficient [-]
Max ampl X Exc ampl X
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
0.002
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Lid
ar in
clin
atio
n [
rad
]
Friction coefficient [-]
Max ampl X
Exc ampl X
Max ampl Z
Exc ampl Z
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Mu Max ampl Z Exc ampl Z Relative Max ampl X Exc ampl X Relative
- rad rad % rad rad %
0 0.00129 0.000964 0.000838 0.000596
0.001 0.00132 0.000974 1.73% 0.000856 0.000609 2.15%
0.003 0.00135 0.000994 4.23% 0.000892 0.000636 6.42%
0.01 0.00148 0.00108 14.02% 0.00103 0.000744 23.09%
0.03 0.00190 0.00142 46.49% 0.00151 0.00116 80.07%
Table 32: Friction simulation results. Absolute numbers.
All together the mechanism responds well to the friction. It does not really affect
performance too badly. That said, it is still worthwhile to reduce friction as much as
possible. For example: the Z axis inclination is 14% worse for a friction coefficient of
0.01 but only 4.2% for a friction coefficient of 0.003.
4.7 Frequency response
Previously optimal flywheel geometry and component arrangement was discussed. The
reaction of the mechanism to disturbances has been covered as well. Now it is time to
check how the mechanism reacts to changes in excitation period by changing frequency.
This research is conducted with the flywheel speed set at 3000 rpm. At which
frequencies will the mechanism resonate? Are those frequencies in a relevant range?
Can the joint dampers and/or springs be tuned to improve performance in certain
frequency ranges?
Expectations from a theoretical standpoint:
Excitation amplitude should not affect resonance behaviour/frequencies.
Damping should not shift resonance peaks or only very slightly. Resonance peak
amplitudes should lower with increasing damping.
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4.7.1 About UM and the reference model
Please take careful consideration of the fact that the mechanism parts in UM are
infinitely strong and stiff. Deformability, which would give an extra damping effect, is
not simulated. When the mechanism resonates, UM may show extremely violent results.
The reaction forces for such a movement would have to be impossibly high, high enough
to deform any physical component. Any violent amplitudes or movements shown in
simulation results, when the mechanism resonates, must be interpreted wisely.
Typically the results will show whether or not the mechanism resonates. However, the
numbers will be useless to determine the actual movement if the mechanism were real,
since deformation would occur.
The axes connecting the bearings are not modelled. Of course these points would
represent an elastic point in the mechanism. They are not present in the model so
resonance frequencies can change compared to an actual mechanism.
At this point it is unclear what an actual mechanism will look like. The current reference
mechanism’s dimensions were inspired by the 3E’s prototype but even this mechanism
looks quite different. Its sole purpose was to provide insights to help develop such a
mechanism. It is unclear what decisions shall be made in the future. Will an entire new
mechanism be developed or will the older mechanism continue to serve but with a
flywheel added?
Despite the fact that some of the results are not representative in the absolute sense,
behaviour should still be similar to an actual mechanism and frequencies should be
within the same order of magnitude.
4.7.2 Excitation of frame for typical wave periods
Simulations were carried out in a period range of 2.5 seconds to 10 seconds. The most
relevant range lies between 4 and 7 seconds. 5 seconds has always been the reference
value up until now which is approximately the most common value.
The depicted values are maximum Lidar inclinations. This is enough information since
the most important factors are the global performance and the possible danger of
resonance.
The reference mechanism was tested in two configurations:
1. The standard reference mechanism with changing excitation period.
2. The standard reference mechanism with changing excitation period and a
stiffness coefficient of 50 N.m/rad for both axes.
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…And two amplitudes for excitation: 0.1 and 0.2 rad. Just to verify the theoretical
expectations.
Graph 20: Comparison of the results for both versions of the reference mechanism for both amplitudes. This confirms
the proposition that a change of amplitudes doesn’t cause a shift in the frequency behavior.
The results above show that the amplitude doesn’t change the behaviour of the system
in the measured period range. It looks like very high values, meaning resonance, are not
present in the relevant range. This is good news. This means that the reference
mechanism is in no danger of resonating for reasonable excitation periods. It becomes
really interesting when the graphs are rearranged to compare the normal version versus
the version with extra stiffness.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
2 3 4 5 6 7 8 9 10
Max
imu
m L
idar
incl
inat
ion
[ra
d]
Excitation period [s]
Ampl 0.2 + spring
Ampl 0.1 + spring
0
0.002
0.004
0.006
0.008
0.01
0.012
2 3 4 5 6 7 8 9 10
Max
imu
m L
idar
incl
inat
ion
[ra
d]
Excitation period [s]
Ampl 0.2
Ampl 0.1
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Graph 21: Same as the previous graphs but rearranged to show the different behaviour caused by the extra stiffness.
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
2 3 4 5 6 7 8 9 10 11
Max
imu
m L
idar
incl
inat
ion
[ra
d]
Excitation period [s]
Ampl 0.1 + spring
Ampl 0.1
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
2 3 4 5 6 7 8 9 10 11
Max
imu
m L
idar
incl
inat
ion
[ra
d]
Excitation period [s]
Ampl 0.2 + spring
Ampl 0.2
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The results above point out the interesting changes the extra restoring force has caused.
It seems that the added spring in the joints helps overcome inertia for high
frequency/low period excitations producing higher accuracy in the low period range.
The spring’s stiffness counteracts the flywheel at low frequency/high period excitations
producing bad results in the high period range. Since high periods are far more common
than low periods, the system without spring looks the most promising and it is less
complex (no spring needed). Also, exceptional periods outside the normal range are far
more likely to occur on the high period side than on the low period side thanks to the
buoy’s inertia. It looks like the stiffness coefficient shifts the point where the mechanism
is most accurate on the excitation period axis. One could impose a stiffness to put the
optimum exactly at 5 seconds. There would not be much difference compared to the
performance without spring and is not worth risking bad performance for large period
movements and even the cost of the spring. For the reference mechanism, this is the
safest option since the lowest excitation periods have the most accuracy.
It can be concluded that the mechanism cannot be brought into resonance because of the
motion imposed by the waves. Another source of movement is the vibration of the
motor. The motor’s frequency can be any value between 0-100Hz because of the
invertor. So it is worthwhile to discover the frequencies at which the mechanism does
resonate. From this point on only the reference mechanism without spring will be used
for testing.
4.7.3 Resonance
The resonance peaks are not discovered yet. So where do they lie? For the unsprung
reference mechanism one would think that the mechanism will not resonate at low
frequencies but at high frequencies and it does of course. The sinusoidal excitations
functions served nicely to research the mechanism’s reactions but they can also serve to
impose high frequency low amplitude oscillations. Amplitude for frame excitation is set
at about 0.5° or 0.009 rad.
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Simulation for 1000Hz:
Figure 112: Simulation of the reference mechanism excited with 25 ms of excitation period or 40Hz of excitation
frequency. Excitation amplitude is about 0.5°. Duration: 18 seconds.
While the excitation is barely noticeable the Lidar inclination has an order of magnitude
of 100° and rising. Clearly resonance. The graph may not show a realistic result. It just
indicates that the mechanism reacts very violently.
Subsequent research shows that mild resonance starts at an excitation frequency of the
frame of around 300 Hz for the reference model. The damping, as expected, makes the
mechanism respond to a broader range of frequencies. More specifically between 300
Hz to 1500 Hz (approximately). A precise measurement is unnecessary since absolute
numbers are not really significant here. It sufficient to conclude that wave excitation will
not bring the system into resonance.
The motor however, with its higher invertor frequency, just might be the problem. To
check this, the flywheel position function was expanded to incorporate a vibration term.
v: Vibration amplitude: set at about 0.5 degrees
f: Vibration frequency
If the system responds badly for frequencies between 50 and 100 Hz it would be bad.
Simulation results are shown for Lidar x axis inclination and z axis inclination.
3000 rpm
1000 Hz excited Z inclination [rad] Time [s]
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Graph 22: Simulation of Lidar z inclination for varying vibration frequencies.
Lidar z angle inclinations show that the inner gimbal does not come into resonance for
the measured frequencies. If that were the case the excitation amplitude would increase.
The max amplitude did increase however but only because of an increase in precession
component. This component damps out like previously described, so the precession is
not enforced. It seems to be caused by a start-up phenomenon that intensifies when the
vibration frequency rises.
Graph 23: Simulation of Lidar x inclination for varying vibration frequencies.
0
0.005
0.01
0.015
0.02
0.025
0 200 400 600 800 1000 1200
Lid
ar z
incl
inat
ion
[ra
d]
Vibration frequency [Hz]
Max ampl
Exc ampl
0
0.1
0.2
0.3
0.4
0.5
0.6
0 500 1000 1500 2000
Lid
ar x
incl
inat
ion
[ra
d]
Vibration frequency [Hz]
Max ampl
Exc ampl
Difference is maximum precession amplitude.
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The x angle shows that there is nothing to worry about. It can be seen that the outer
gimbal responds to frequencies starting from 400Hz. But these frequencies will not be
attained by the motor drive. One can see that, because of the damping, the system
responds to a broad range of frequencies. The increase in maximum amplitude is now
caused by both the increasing precession phenomenon and resonance responsible for
the rise in precession and excitation component respectively. The actual peak values of
the resonance amplitudes in the graphs above could not be uncovered because from
2000Hz onwards it becomes very difficult to measure the excitation amplitude reliably.
Please note that when the outer gimbal rotates around its axis more mass is displaced
than when the inner gimbal rotates. The outer gimbal holds the inner gimbal and the
Lidar-flywheel subsystem. The inner gimbal only holds the Lidar-flywheel subsystem.
Because of the extra mass the resonance frequency for the outer gimbal is lower than for
the inner gimbal. In any case both of the gimbals’ resonance frequencies lie way above
100Hz.
4.8 Motor requirements
Since the buoy’s power sources will consist of solar panels and small windmills, it is
important to know the motor power requirements and more specifically if these
requirements are sufficiently low for the available power source.
Please keep in mind that all the following calculations have been made for the
unchanged reference mechanism.
4.8.1 Motor power in steady state
How much power will the motor need if the mechanism is spinning and the buoy is
moving? It is expected to be a very low figure. Bear in mind that in a system such as a
bicycle, the gyroscopic forces exerted by the wheels are quite low and yet they still
succeed in keeping the bike stable. This is because these forces act to keep the system
from tilting and try to keep it as level as possible. When the system is level this means
that it does not rotate (except for the spinning axis) and, as a consequence, no inertia
needs to be overcome. The forces are small, but they act as if controlled by an ultra-fast
feedback controlled system. Because the system reacts immediately, the correction force
needn’t be large. This means that a bicycle with wheels weighing as low as 1kg each can
keep itself and a 75kg rider up without difficulty.
In case of the reference mechanism, the most important forces to fight will be the
damping and the tilting force exerted by the Lidar subsystem’s inertia if it’s COG is not in
the rotation point. (See 4.5.6) The further the Lidar subsystem’s COG is from the centre
of rotation, the more torque the flywheel has to counteract.
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In UM, the speed is fixed with an expression at 3000 rpm. A certain torque is needed to
keep the speed at this rate. The reaction torque in the joint will be indicative of the
torque a motor would have to apply to keep the speed at 3000 rpm. In reality, a motor
will not react as fast as the infinitely fast responding theoretical simulation in UM. Still, it
should be quite representative. Because the flywheel is accelerated infinitely fast, a non
physical transition phenomenon was encountered. This is why only values between 10
and 20 seconds are analysed because this represents two oscillation periods in the
reference situation (2 times 5 seconds).
This was tested for the reference mechanism and for the reference mechanism with
changed flywheel offset. The latter will have a Lidar flywheel subsystem with changed
COG and this should need more torque to work. Results are shown in figure 113.
Figure 113: Reaction torque in Lidar-flywheel joint in UM model of reference mechanism and the reference
mechanism with doubled flywheel offset. Time window is 10 to 20 seconds in the simulation.
It looks like changing the Lidar-flywheel subsystem’s COG indeed increases the work the
motor, just as predicted. The transition phenomenon takes longer too for the changed
offset simulation, but that is not really important. The focus is on the torque for the
reference mechanism.
3000 rpm
Reference Double offset Torque [Nm] Time [s]
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Graph 24: Reaction torque in Lidar-flywheel joint in UM model of reference mechanism. Time window is 10 to 20
seconds in the simulation.
Torque Power
Average 5.28E-06 Nm 1.66 mW
Maximum 3.92E-03 Nm 1.23 W
Table 33: Statistics of torque shown in Graph 24
It looks like the torque needed to keep the flywheel turning is really low. A physical
mechanism will have more disturbances and drag, but the required power should still be
very low compared to the power needed to accelerate the flywheel. (See 4.8.2) Even if
the steady state requirements was a two digit number, it would still be acceptable.
4.8.2 Flywheel start-up
Accelerating the heavy flywheel is a far bigger challenge for the small motor than
keeping it up to speed once it is running. At this point it is not clear what type of motor
drive will be used so an energy equation will have to do.
The model’s flywheel is a simple disc with certain inertia. All the motor has to do is
overcome the inertial torque of the flywheel during a predefined start-up time.
Variables
ωend: Final speed of flywheel
Tfinish: Time span during which the flywheel must accelerate
t: Time variable
-0.00500
-0.00400
-0.00300
-0.00200
-0.00100
0.00000
0.00100
0.00200
0.00300
0.00400
0.00500
Torque [Nm]
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Calculated values
Izz: Flywheel inertia
ω(t): Instantaneous speed
α(t): Instantaneous flywheel acceleration
M(t): Instantaneous torque
P(t): Instantaneous power
Mmax: Max torque Mavg: Average torque
Pmax: Max power Pavg: Average power
Equations
The following values were substituted: 3000 rpm (~314 rad/s) for an acceleration time
of 240 sec (4 min). Figure 114 shows what the functions look like.
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Acceleration: α [1/s²] Speed: ω [1/s]
Torque: M [Nm] Power: P [Watt]
Figure 114: Function visualisation of motor start-up model
The main statistics are:
Torque [Nm] Power [W]
Average 10.02 1574.5
Max 15.75 3212.9
Table 34: Statistics of evaluated functions for flywheel start-up model.
It can be noted that quite a lot of power is needed for an acceleration time of 4 minutes.
3.2 kiloWatt to be precise. Please bear in mind that this power is only needed for a small
duration and not continuously.
Although the capacity of the vessel’s power generating systems is far below the
maximum power listed above, this is not a problem because of the battery. The motor
will run off the battery and, since it will be quite large, it can easily provide the power
needed for a short period of time. When the flywheel is accelerated, the power
requirements become very low and the battery can recharge.
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If the maximum power or maximum torque is imposed by specifications one can deduce
the approximate acceleration time using the model above.
These formulae can help determine the acceleration time when the motor is the limiting
factor. These functions are visualised in figure 115 and figure 116 for 3000 rpm.
Figure 115: Maximum torque for a final speed of 3000 rpm for varying acceleration time.
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Figure 116: Maximum torque for a final speed of 3000 rpm for varying acceleration time.
4.9 Conclusion
It can be concluded that the motor must be able to generate a peak power up to several
kilowatt depending on the desired acceleration time. Remember, in case the flywheel
has to be restarted at sea, it is beneficial to do so as gently as possible like discussed in
4.6.3. Once the flywheel has been accelerated the motor power requirements are very
low.
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5 Combined results
Now it is time to put all the preceding research together in a coupled simulation.
The process of the creation of LS-Dyna models in LS-Prepost was already discussed in
chapter 3. The Dyna model contains a wave tank with a body of water, agitated by a
paddle to create sea waves with a controlled period, wavelength and wave height. The
model is specifically tuned to represent a sea sate where a wind turbine will operate.
This sea sate can be regarded as violent, but not so that breaking waves occur. Of course
the model contains a buoy that floats in the body of water and is moored to the seabed
with a cable. How each of the three buoys performs is discussed in 3.9.
After the numerical simulation study, a kinematic study of the Lidar stabilisation
mechanism was carried out using Universal Mechanism software (UM). The Lidar
mechanism’s holding frame was agitated with a prescribed sinusoidal motion with a
specified period and amplitude. These and other input parameters, such as flywheel
speed, were varied to test the model and to help tune its performance. The result is a
stabilisation mechanism model with a supposedly optimal size and placement within
practical limits.
Now the moment has come to make the holding frame move as if its movement was
caused by the buoy attached to it while it is floating in the sea. This way the performance
of the mechanism can be assessed when it moves in a realistic way. Finally it will be
possible to determine how accurately the Lidar will be stabilised.
Please note that the following UM simulation is not a true coupled simulation. Both
programmes will not be running at the same time and will therefore not interact. This
means that the inertial forces exerted by the mechanism on the buoy frame do not affect
the buoy in the Dyna simulation in any way. This induces a small error because the buoy
movement is simulated as if the mechanism was completely rigid, which it is not. This
error is completely negligible because of the small ratio of the mechanism’s weight to
the buoys. Any force the mechanism might exert on the buoy will not change its
movement significantly. The extremely slight gain of accuracy a true coupled simulation
might provide does not justify the extra difficulties. The actual procedure is pictured in
figure 117.
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Figure 117: Flow of data and relation between programmes
From the output of the Dyna simulations for each buoy, representing the movement of a
specific node in both translation and rotation, is exported and converted to a text file
format. After the exporting process is completed, the part of LS-Dyna in this simulation
is completed. The definition of the frame movement in relation to UM’s inertial
coordinate system is effectuated using the text files. UM simulates the mechanism’s
movement just like it has before. Only now the frame translates and rotates the exact
same way as the buoy moves, instead of just in a sinusoidal fashion. The text files contain
a sampled record of the position of a node (X, Y, Z position) and its rotation (rotation
around X, Y, Z axes). Because the buoy model is rigid, this means that the position of all
the buoy’s points can be determined. UM interpolates the points in the position and
rotation datavectors using spline functions in order to be able to calculate the first and
second derivative. Please note that a total of six files/variables was needed.
K-file Dyna model of wavetank with buoy.
LS Dyna calculation
(Dyna output) Buoy position X-Y-Z
(Dyna output) Buoy rotation X-Y-Z
(Text file) Buoy position X-Y-Z
(Text file) Buoy rotation X-Y-Z
Conversion
Universal mechanism simulation
Frame movement definition
(Plot & data) Lidar rotation X-Y
Stabilization Performance analysis This section
Buoy movement analysis See REF
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The Lidar performance was measured earlier using the Lidar inclination around the Z –
axis. This was sufficient because performance numbers were only relevant in the
relative sense. It was all about getting the inclination as low as possible and checking if
the mechanism reacts positively or negatively to certain changes. For the coupled and
realistic simulations, the absolute numbers have true meaning. That is why the
mechanisms performance will be measured using the absolute value of the total rotation
vector, which is the algebraic sum of all three rotation vectors. This inclination is
directly correlated to the quality of the Lidar’s measurements.
5.1 3E prototype
The 3E prototype was tested first. The movement of the buoy itself was visualised in
Figure 118. The simulations are repeated for different flywheel speeds: 0, 3000 and
6000 rpm. This will show how much effect the flywheel has in a realistic situation and
will reveal the extent of the benefit to increase speed to 6000 rpm. Please remember
that the inclination is now an absolute value and is always positive. The total duration is
28 seconds.
Figure 118: Simulation result for Lidar inclination on the 3E prototype buoy (absolute value of the rotation vector). No
flywheel curve is the highest.
It looks like it is definitely worthwhile to install the flywheel.
No flywheel 3000 rpm 6000 rpm Inclination [Rad] Time [s]
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Figure 119: Detail of Figure 118
Flywheel
speed
Average inclination Relative Maximum inclination Relative
0 0.09449 rad 5.41° 0.23623 rad 13.54°
3000 0.01160 rad 0.66° -87.72% 0.02549 rad 1.46° -89.21%
6000 0.01029 rad 0.59° -89.11% 0.02315 rad 1.33° -90.20%
Table 35: Results of coupled simulation of 3E prototype buoy
It seems the performance increase related to a faster spinning flywheel that looked
promising during the kinematic study is not so relevant with more realistic boundary
conditions. Compared to the simulation with the flywheel present, but inactive, the
maximum inclination is about 89.91% lower for 3000 rpm and just 90.20% for 6000
rpm. It can be concluded that the flywheel increases performance dramatically, but it is
not very beneficial to increase speed to 6000 rpm.
3000 rpm 6000 rpm Inclination [Rad] Time [s]
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5.2 PEM 58
Next up is the PEM 58. Only 18 seconds were simulated because of limitations in
computing power and time. Similar tests to the ones of the 3E buoy will be executed:
Figure 120: Simulation result for Lidar inclination on the PEM 58 buoy (absolute value of the rotation vector). No
flywheel curve is the highest.
The flywheel proves itself yet again.
Figure 121: Detail of Figure 120
No flywheel 3000 rpm 6000 rpm Inclination [Rad] Time [s]
3000 rpm 6000 rpm Inclination [Rad] Time [s]
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Flywheel
speed
Average inclination Relative Maximum inclination Relative
0 0.10953 rad 6.28° 0.25000 rad 14.32°
3000 0.01244 rad 0.71° -88.64% 0.02542 rad 1.46° -89.83%
6000 0.01095 rad 0.63° -90.01% 0.02311 rad 1.32° -90.76%
Table 36: Results of coupled simulation of PEM buoy
Similar observations can be made to the 3E prototype. The comparison will be most
revealing.
Please remember that the SPAR buoy was not simulated (see 3.9.2.3) because of
limitations in computation power.
5.3 Comparison
All results have been put together. 3E prototype simulations were longer than the PEM
simulations respectively 28 and 17 seconds:
Figure 122: Simulation result for Lidar inclination of both buoys (absolute value of the rotation vector).
3E No flywheel 3E 3000 rpm 3E 6000 rpm PEM No flywheel PEM 3000 rpm PEM 6000 rpm Z inclination [rad] Time [s]
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Figure 123: Detail of Figure 122
3E 3000 rpm 3E 6000 rpm PEM 3000 rpm PEM 6000 rpm Z inclination [rad] Time [s]
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6 Conclusion
It is obvious that, despite the lesser dynamic performance of the PEM compared to the
3E prototype as a wave filter, the results are very similar. The stabilisation mechanism
takes the more violent movement of the PEM 58 in its stride. So if a choice had to be
made between the two buoys, cost would be the deciding factor and not performance.
This comparison would likely favour the PEM 58 (see 2.3.3).
Graph 25: Average inclination values of both buoy types compared
Graph 26: Maximum inclination values of both buoy types compared
0.00000
0.02000
0.04000
0.06000
0.08000
0.10000
0.12000
No flywheel 3000 rpm 6000 rpm
Lid
ar in
clin
atio
n [
rad
]
Average
3E
PEM
0.00000
0.05000
0.10000
0.15000
0.20000
0.25000
0.30000
No flywheel 3000 rpm 6000 rpm
Lid
ar in
clin
atio
n [
rad
]
Maximum
3E
PEM
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The graphs showcase the flywheel performance once more. System performance
remains almost identical despite the different behavior of each buoy. The choice of buoy
should really be based on cost effectiveness rather than dynamic stability because the
stabilizing mechanism irons out most of the bad behavior the chosen buoy might have.
Just as suggested in the beginning (part 2.3.3) the PEM 58 is, according to this research,
more suited for the job than 3E’s own prototype because it more practical. As long as the
Lidar is suspended in a similar construction as the reference mechanism with a flywheel,
the Lidar accuracy can be superior to that achieved by the support mechanism currently
used by 3E.
On the practical side, lots of improvements could be made. These can be seen in figure
124. Ideally, only the laser emitting part of the Lidar should be stabilised. As a
consequence, the stabilisation mechanism could be constructed much smaller yielding
similar, of even better, accuracy with only a fraction of the costs.
Figure 124: Concept sketch with added practical improvements
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Master thesis – Mechanical Engineering –References Page 156
7 References
[1] V. nieuwsdienst, “Deredactie.be,” 3 Mei 2012. [Online]. Available:
http://www.deredactie.be/cm/vrtnieuws/regio/westvlaanderen/1.1150084.
[Accessed 3 Mei 2012].
[2] Resinex, Resinex, Maart 2012. [Online]. Available: http://www.resinextrad.com/.
[3] M. B. L. G. R. Liu, Smoothed particle hydrodynamics, World Scientific Publishing Co.
Pte. Ltd., 2003.
[4] D. L. H. S. W. A. J. Swegle, “J. Computational physics,” 1995, pp. 116-123.
[5] L. S. T. Corporation, “LS Dyna,” LSTC, 2012. [Online]. Available:
http://www.lstc.com/.
[6] R. A. D. Robert G. Dean, Water wave mechanics, World scientific publishing Co. Pte.
Ltd., 1992.
[7] J. Pelfrene, Study of the SPH method for simulation of regular and breaking waves,
Gent: Universiteit Gent, 2011.
[8] P. Chapman, “Ocean currents,” Advameg, Inc, [Online]. Available:
http://www.waterencyclopedia.com/Mi-Oc/Ocean-Currents.html. [Accessed 24
April 2012].
[9] V. Hydrografie, “Golftheorie,” Vlaamse Hydrografie, 2012. [Online]. Available:
http://www.vlaamsehydrografie.be/hm_atlas_cd/www/theorie/golftheorie/golfth
eorie.htm. [Accessed 19 April 2012].
[10] M. Chicago, “Tsunami wave tank,” MSI Chicago, 2012. [Online]. Available:
http://www.msichicago.org/whats-here/exhibits/science-storms/the-
exhibit/tsunami/tsunami-wave-tank/. [Accessed 19 April 2012].
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8 Figures, graphs and tables
8.1 List of figures
Figure 1: Lidar measuring equipment ..................................................................................................... 1
Figure 2: Schematic overview of measuring device operating conditions ................................ 2
Figure 3: Location of metacentre .............................................................................................................. 5
Figure 4: Conventional catamaran versus SWATH type catamaran ............................................ 6
Figure 5: New SWATH type catamaran pilot ship for the port of Zeebrugge [1].................... 6
Figure 6: Small catamaran research vessel ........................................................................................... 7
Figure 7: Stable floating plastic modular buoy for mooring applications [2] .......................... 8
Figure 8: Resinex PEM 43 Catamaran buoy. Diameter approximately 4.3m ........................ 11
Figure 9: Resinex PEM 58 catamaran buoy. Diameter approximately 5.8 m ........................ 11
Figure 10: Schematic of a typical spar buoy design. ....................................................................... 13
Figure 11: Support domain ....................................................................................................................... 16
Figure 12: 2D representation of support and influence domains for illustrating possible
problems when using different smoothing lengths. ....................................................................... 17
Figure 13: Crash test [5] ............................................................................................................................ 18
Figure 14: From geometry to useable LS-Dyna code by meshing ............................................. 19
Figure 15: Hierarchical structure used in the LS-Dyna code for a rigid shell part. ............. 20
Figure 16: Extract of k-file......................................................................................................................... 21
Figure 17: Paddle .......................................................................................................................................... 22
Figure 18: Spacing between the parts and particles ....................................................................... 22
Figure 19: Wave characteristics [6] ...................................................................................................... 24
Figure 20: Wave Tank CSI Chicago [10] ............................................................................................... 29
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Figure 21: Flap type wave maker ........................................................................................................... 30
Figure 22: Wave tank design with sponge layer from [7] ............................................................. 33
Figure 23: Model with 3E prototype, cable and sponge layer ..................................................... 33
Figure 24: Travelling wave in Dyna model ......................................................................................... 34
Figure 25: Boundaries and dimensions ............................................................................................... 35
Figure 26: Scheme of undercurrent duct ............................................................................................ 36
Figure 27: Undercurrent circuit model ................................................................................................ 37
Figure 28: Lumped plate friction model .............................................................................................. 38
Figure 29: Friction plate model ............................................................................................................... 38
Figure 30: Buoy model with mooring cable and two forces: downwards and sideways . 39
Figure 31: Initial grid of the SPH-particles ......................................................................................... 40
Figure 32: Failing contact .......................................................................................................................... 42
Figure 33: Wave tanks with 25kg (upper) and 1000kg(lower) particles............................... 43
Figure 34: Sea level for 1000 kg(upper) and 25 kg(lower) particles ....................................... 44
Figure 35: Wave period; Z-displacement (in meters) versus time (in seconds) ................. 47
Figure 36: Wave height; Z-displacement (in metres) versus the time (in seconds) .......... 47
Figure 37: Measured wavelength ........................................................................................................... 48
Figure 38: Buoyancy check ....................................................................................................................... 49
Figure 39: Screenshots of breaking waves ......................................................................................... 49
Figure 40:- Z-coordinate of water level for LS-Dyna Dam Break test (left) and
experimental and Joren’s data (right). Right picture from [7] .................................................... 50
Figure 41: Series of screenshots of the executed dam break test .............................................. 51
Figure 42: Detailed CAD drawing (left) and simplified mesh (right) for the PEM58 buoy
.............................................................................................................................................................................. 52
Figure 43: Platform drawn with CAD ................................................................................................... 53
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Figure 44: Detailed 3E buoy Dyna model ............................................................................................ 53
Figure 45: Detailed PEM58 buoy Dyna model .................................................................................. 54
Figure 46: Detailed spar buoy Dyna model ........................................................................................ 54
Figure 47: 3E working prototype ........................................................................................................... 55
Figure 48: Buoyancy check ....................................................................................................................... 55
Figure 49: Two layer model ...................................................................................................................... 57
Figure 50: Dyna two layer model: Two layers, two parts ............................................................. 58
Figure 51: Dyna two layer model: Two layers, one part ............................................................... 59
Figure 52:Wave tank (left), wave tank and buoy (centre), wave tank and moored buoy
(right) ................................................................................................................................................................ 61
Figure 53: Displacement in x- (red), y- (green) and z-direction (blue) for the 3E buoy . 62
Figure 54: rotation around x- (red), y- (green) and z-axis (blue) for the 3E buoy ............. 63
Figure 55: side view of 3-buoy model at 13.33 seconds showing the maximum
inclination ........................................................................................................................................................ 64
Figure 56: rotational speed around y-axis for the 3E buoy ......................................................... 64
Figure 57: rotation around the y-axis for the PEM58..................................................................... 65
Figure 58: side view of PEM58 model at 16 seconds showing the maximum inclination
.............................................................................................................................................................................. 66
Figure 59: rotational speed around y-axis for the PEM58 buoy ................................................ 66
Figure 60: Barely submerged, but floating SPAR buoy .................................................................. 67
Figure 61: sketch showing vortices when placing the spar buoy .............................................. 69
Figure 62: rotation around y-axis. Red for the PEM58 and green for the 3E buoy ............. 70
Figure 63: rotational speed [rad/s] versus time [s] for the 3E and the PEM58 buoy........ 70
Figure 64: CAD drawing of linkage system to make the Lidar move independently ......... 73
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Figure 65: UM animation window: representation of Lidar stabilizing mechanism model
.............................................................................................................................................................................. 74
Figure 66: Offset parameters of mechanism model in UM and reference points. ............... 75
Figure 67: Relative movement and restoring moment. ................................................................. 78
Figure 68: Result of 120 sec simulation of the reference mechanism. X and Z inclination
is shown with max. inclination value of 0.07 degrees. Speed: 6000 rpm ............................... 80
Figure 69: Sine function of angular movement definition of frame. ......................................... 81
Figure 70: Inclination signal component definition ........................................................................ 82
Figure 71: 600 sec simulation of reference mechanism highlighting the slow decline in
precession ........................................................................................................................................................ 82
Figure 72: Simulation of Lidar inclination for varying flywheel speeds. Lower amplitude
curves represent higher speeds and vice versa. ............................................................................... 83
Figure 73: 20 minute simulations of Lidar inclination for varying flywheel speeds. Lower
amplitude curves represent higher speeds and vice versa. ......................................................... 85
Figure 74: Simulation of reference mechanism with changing amplitudes Smaller
amplitudes yield lower maxima. ............................................................................................................. 87
Figure 75: Scheme of flywheel dimensions and offset. .................................................................. 88
Figure 76: Reference mechanism. Observe the thin flywheel with a large diameter. ....... 89
Figure 77: Simulation of Lidar inclination for changing flywheel radius with constant
height. Speed is 6000rpm. ......................................................................................................................... 90
Figure 78: Simulation of Lidar inclinations of Z axis for varying flywheel thickness.
Lower radii yield higher maxima. .......................................................................................................... 92
Figure 79: 90 second simulation of reference mechanism with a flywheel made from
varying materials. Heavier materials yield lower maximum inclinations. ............................ 95
Figure 80: Simulation of Lidar inclinaton for varying flywheel offset. Smaller amplitudes
represent smaller offset and vice versa. .............................................................................................. 97
Figure 81: Mechanism configurations respectively with the Lidar moved downwards and
upwards. ........................................................................................................................................................... 99
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Figure 82: Comparison of the reference mechanism and a mechanism with the Lidar
positioned lower. .......................................................................................................................................... 99
Figure 83: Illustration of Lidar subsystem swaying. ................................................................... 100
Figure 84: Comparison of the reference mechanism and a mechanism with the Lidar
positioned higher. The reference position yields higher excitation amplitude. ............... 101
Figure 85: Screenshot of ‘Super mechanism’ in UM. Would be difficult to build. Physical
axes not drawn. Frame and Lidar module drawn in wireframe.............................................. 102
Figure 86: ‘Super mechanism’. From below. The flywheel is a ring around the Lidar itself
with a hub at the bottom......................................................................................................................... 103
Figure 87: Side view of ‘super mechanism’. Look at the Lidar module’s orientation which
is, as far as the eye can see, perfectly upright while the mechanism is moving. ............... 103
Figure 88: Flywheel of the ‘super mechanism’. A ring instead of a disc. A central hub was
added to make it possible to attach a motor. .................................................................................. 104
Figure 89: Simulations of Lidar inclination comparing the super mechanism to the
reference mechanism ............................................................................................................................... 105
Figure 90: Simulations of Lidar inclination comparing the super mechanism to the
reference mechanism. Simulation time is 1200 seconds or 20 minutes.............................. 105
Figure 91: Simulation of Lidar inclination for changing damping. Higher amplitudes
mean higher damping values. Flywheel speed is 3000 rpm. .................................................... 107
Figure 92: Mechanism with misaligned start-up position. Frame is fixed. ......................... 108
Figure 93: Simulation results for verying damping with misaligned startup of 0.1 rad and
stationary frame. 1200 sec simulation. 3000 rpm ........................................................................ 109
Figure 94: Simulation results for changing damping with misaligned startup of 0.1 rad
and stationary frame. ............................................................................................................................... 110
Figure 95: Flywheel angular speed function with parameters composed of constants and
cosines ........................................................................................................................................................... 112
Figure 96: Detail of flywheel angular speed for the parameters described above for the
‘disastrous’ speed dips............................................................................................................................. 114
Figure 97: Simulation of Lidar inclination for large speed dips of 200, 400 and 800 rpm.
Before t equals 120 the situation is the reference situation. Total duration is 240 sec. 114
Stability of LIDAR measuring buoy for offshore wind profile registration Master Thesis
Master thesis – Mechanical Engineering –Figures, graphs and tables Page 162
Figure 98: Simulation of Lidar inclination with a dip in the speed at 120 seconds. Varying
speed dip duration. ................................................................................................................................... 116
Figure 99: Flywheel angular speed and acceleration for said simulations. ........................ 117
Figure 100: Flywheel angular acceleration for speed dip simulations for constant dip
time. ................................................................................................................................................................ 118
Figure 101: Simulation of Lidar inclination with very fast but small speed dip at t equals
120 seconds. ................................................................................................................................................ 120
Figure 102: Flywheel speed for experiment of gradual variation in flywheel speed. .... 121
Figure 103: Simulation result for Lidar inclination with gradually changing flywheel
speed. Transition from 3000 to 6000 rpm. Transition time is 120 seconds. ..................... 122
Figure 104: Excitation function with envelope and its parameters....................................... 123
Figure 105: Plots of frame movement functions for amplitudes 0.05, 0.10, 0.15, 0.20 . 124
Figure 106: Simulation results for Lidar inclination for a disturbance amplitude of 0.05 ,
0.1, 0.15, 0.2 rad. ........................................................................................................................................ 125
Figure 107: Direction of translational motion (1,4,1) ................................................................. 126
Figure 108: Simulations of Lidar inclination. Sudden translation at t equals 120 seconds.
........................................................................................................................................................................... 127
Figure 109: Simulation of Lidar inclination for gradually changing amplitude from 0.1 to
0.2 rad. ........................................................................................................................................................... 128
Figure 110: Simulations of Lidar inclination at start-up for varying start-up durations.
........................................................................................................................................................................... 129
Figure 111: Simulation of Lidar Z axis inclination for varying friction coefficient.
Simulation time is 90 seconds. ............................................................................................................. 131
Figure 112: Simulation of the reference mechanism excited with 25 ms of excitation
period or 40Hz of excitation frequency. Excitation amplitude is about 0.5°. Duration: 18
seconds. ......................................................................................................................................................... 138
Figure 113: Reaction torque in Lidar-flywheel joint in UM model of reference mechanism
and the reference mechanism with doubled flywheel offset. Time window is 10 to 20
seconds in the simulation. ...................................................................................................................... 141
Stability of LIDAR measuring buoy for offshore wind profile registration Master Thesis
Master thesis – Mechanical Engineering –Figures, graphs and tables Page 163
Figure 114: Function visualisation of motor start-up model ................................................... 144
Figure 115: Maximum torque for a final speed of 3000 rpm for varying acceleration time.
........................................................................................................................................................................... 145
Figure 116: Maximum torque for a final speed of 3000 rpm for varying acceleration time.
........................................................................................................................................................................... 146
Figure 117: Flow of data and relation between programmes .................................................. 148
Figure 118: Simulation result for Lidar inclination on the 3E prototype buoy (absolute
value of the rotation vector). No flywheel curve is the highest. .............................................. 149
Figure 119: Detail of Figure 118 .......................................................................................................... 150
Figure 120: Simulation result for Lidar inclination on the PEM 58 buoy (absolute value
of the rotation vector). No flywheel curve is the highest. .......................................................... 151
Figure 121: Detail of Figure 120 .......................................................................................................... 151
Figure 122: Simulation result for Lidar inclination of both buoys (absolute value of the
rotation vector). ......................................................................................................................................... 152
Figure 123: Detail of Figure 122 .......................................................................................................... 153
Figure 124: Concept sketch with added practical improvements .......................................... 155
8.2 List of Graphs
Graph 1: Maximum Lidar inclination versus flywheel speed and the interpolating power
function. ............................................................................................................................................................ 84
Graph 2: Lidar inclination components versus flywheel speed (primary vertical axis) and
precession period versus flywheel speed (secondary vertical axis). ....................................... 85
Graph 3: Lidar inclination versus excitation amplitude. ............................................................... 87
Graph 4: Lidar inclinations (primary axis) and flywheel weight (secondary axis) versus
changing radius ............................................................................................................................................. 91
Graph 5: Maximum Lidar inclination versus flywheel thickness for a constant radius. ... 93
Graph 6: XY-plot of the product of inertias and the maximum Lidar inclination. ............... 93
Stability of LIDAR measuring buoy for offshore wind profile registration Master Thesis
Master thesis – Mechanical Engineering –Figures, graphs and tables Page 164
Graph 7: Lidar inclination versus flywheel radius. Product of inertias is constant at the
reference value. ............................................................................................................................................. 94
Graph 8: Inclination for varying flywheel density ........................................................................... 96
Graph 9: Maximum Lidar inclination versus flywheel offset for the reference mechanism.
.............................................................................................................................................................................. 98
Graph 10: Comparison of the reference mechanism to the hypothetical ‘super
mechanism’. ................................................................................................................................................. 106
Graph 11: Lidar inclination versus damping for the reference mechanism. ...................... 107
Graph 12: Half-life of precession motion. ........................................................................................ 109
Graph 13: Results of 6000 rpm simulation series compared to 3000 rpm......................... 110
Graph 14: Lidar inclination for simulation with different speed dips. ................................. 115
Graph 15: Lidar inclination for a constant spring back and changing dip duration. ....... 117
Graph 16: Lidar inclination versus dip duration for simulation series with constant
speed dip time. ............................................................................................................................................ 119
Graph 17: Results for simulation of mechanism reaction to sudden rotational jerking
with the set of parameters described above. The depicted values are the amplitude of the
precession motion in steady state and not of the violent transition maxima. The same
experiment was carried out of 6000rpm. ........................................................................................ 125
Graph 18: Maximum Lidar inclination in steady state after flywheel acceleration from 0
to 6000 rpm. ................................................................................................................................................ 129
Graph 19: Graphs of friction coefficient research for both Lidar axes. ................................. 132
Graph 20: Comparison of the results for both versions of the reference mechanism for
both amplitudes. This confirms the proposition that a change of amplitudes doesn’t
cause a shift in the frequency behavior. ........................................................................................... 135
Graph 21: Same as the previous graphs but rearranged to show the different behaviour
caused by the extra stiffness. ................................................................................................................ 136
Graph 22: Simulation of Lidar z inclination for varying vibration frequencies. ............... 139
Graph 23: Simulation of Lidar x inclination for varying vibration frequencies. ............... 139
Stability of LIDAR measuring buoy for offshore wind profile registration Master Thesis
Master thesis – Mechanical Engineering –Figures, graphs and tables Page 165
Graph 24: Reaction torque in Lidar-flywheel joint in UM model of reference mechanism.
Time window is 10 to 20 seconds in the simulation.................................................................... 142
Graph 25: Average inclination values of both buoy types compared.................................... 154
Graph 26: Maximum inclination values of both buoy types compared ................................ 154
8.3 List of tables
Table 1: Specification of Resinex Buoys .................................................................................................. 9
Table 2: Material and section card parameters for SPH ................................................................ 23
Table 3: Sea state date on the Thornton bank ................................................................................... 26
Table 4: Summary the design parameters for basic wave tank .................................................. 29
Table 5: Table of wave measurements in Dyna models for 1000 kg particles ..................... 43
Table 6: wave measurements in Dyna models for 1000 kg and 100kg particles ................ 45
Table 7: Experimental wave data by GeoSea ..................................................................................... 46
Table 8: Original dimensions (deep sea) ............................................................................................. 57
Table 9: Summary of the simulation results for 3E and PEM58 buoy ..................................... 71
Table 10: Parameters of Frame movement. ....................................................................................... 76
Table 11: Lidar offset parameter ............................................................................................................ 77
Table 12: Flywheel geometry and movement parameters .......................................................... 77
Table 13: Joint restoring force parameters ........................................................................................ 78
Table 14: Summation of reference mechanism model parameters .......................................... 79
Table 15: Results of simulation series for varying speeds ........................................................... 84
Table 16: Results and calculated values of simulation with changing radius and constant
thickness .......................................................................................................................................................... 90
Table 17: Results and calculated values of simulations with changing thickness and
constant radius .............................................................................................................................................. 92
Stability of LIDAR measuring buoy for offshore wind profile registration Master Thesis
Master thesis – Mechanical Engineering –Figures, graphs and tables Page 166
Table 18: Results and calculated values of simulations with constant product of inertias.
.............................................................................................................................................................................. 94
Table 19: Results of simulation series with different flywheel materials. ............................. 96
Table 20: Results and calculated values of simulations with changing offset ...................... 97
Table 21: Results and calculated values of simulations with changing offset. .................. 108
Table 22: Parameters of flywheel speed function ........................................................................ 113
Table 23: Table with parameters for testing of detrimental speed dip: changing speed
difference ...................................................................................................................................................... 113
Table 24: Table with parameters for testing of detrimental speed dip: changing dip time.
........................................................................................................................................................................... 116
Table 25:Table with parameters for testing of detrimental speed dip: changing dip time
with constant total dip period. ............................................................................................................. 118
Table 26: Parameterset for fast speed dip simulation ................................................................ 120
Table 27: Parameterset for gradual speed change. ...................................................................... 121
Table 28: Parameters of amplitude function of sudden change model ................................ 123
Table 29: Rotational jerking simulation parameters: amplitude of impulse ..................... 124
Table 30: Translational jerking simulation parameters ............................................................. 126
Table 31: Parameters for friction research ..................................................................................... 131
Table 32: Friction simulation results. Absolute numbers. ........................................................ 133
Table 33: Statistics of torque shown in Graph 24 ......................................................................... 142
Table 34: Statistics of evaluated functions for flywheel start-up model. ............................ 144
Table 35: Results of coupled simulation of 3E prototype buoy ............................................... 150
Table 36: Results of coupled simulation of PEM buoy ................................................................ 152