This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

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5 List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I II III IV V VI Leijon, M., Waters, R., Rahm, M., Svensson, O., Boström, C., Strömstedt, E., Engström, J., Tyrberg, S., Savin, A., Gravråkmo, H., Bernhoff, H., Sundberg, J., Isberg, J., Agren, O., Danielsson, O., Eriksson, M., Lejerskog, E., Bolund, B., Gustafsson, S., Thorburn, K. "Catch the Wave to Electricity - The Conversion of Wave Motions to Electricity Using a Grid-Oriented Approach." IEEE Power and Energy Magazine, Vol. 7, Issue 1, pp.50-54, January-February Savin, A., Svensson, O., and Leijon, M." Azimuth-inclination angles and snatch load on a tight mooring system." Accepted for publication in Ocean Engineering, December 3, 2011, available online, December 22, 2011, Savin, A., Svensson, O., and Leijon, M. "Estimation of Stress in the Inner Framework Structure of a Single Heaving Buoy Wave Energy Converter." Conditionally accepted for publication in IEEE Journal of Oceanic Engineering, May 2011, revision submitted in June Savin, A., Svensson, O., Strömstedt, E., Boström, C., and Leijon, M. "Determining the service life of a steel wire under a working load in the Wave Energy Converter (WEC)." Proceedings of the 28th International Conference on Ocean, Offshore and Arctic Engineering, OMAE 2009, Honolulu, Hawaii, USA, OMAE , Vol. 4: Ocean Engineering, Ocean Renewable Energy, Ocean Space Utilization, Parts A and B, pp , Boström, C., Rahm, M., Svensson, O., Strömstedt, E., Savin, A., Waters, R., and Leijon, M. "Temperature measurements in a linear generator and marine substation for wave power." Journal of Offshore Mechanics and Arctic Engineering, Vol. 134, Issue 2, 6 pages, Gravråkmo, H., Leijon, M., Strömstedt, E., Engström, J., Tyrberg, S., Savin, A., Svensson, O., Waters, R. "Description of a torus shaped buoy for wave energy point absorber." Renewable Energy International Conference, 27 June-2 July, Pacifico Yokohama, Yokohama, Japan, (BEST PAPER AWARD RENEWABLE ENERGY 2010, Certificate of appreciation in Area VIII, Ocean Energy.)

6 VII VIII IX X XI Tyrberg, S., Stålberg, M., Boström, C., Waters, R., Svensson, O., Strömstedt, E., Savin, A., Engström, J., Langhamer, O., Gravråkmo, H., Haikonen, K., Tedelid, J., Sundberg, J., Leijon, M. "The Lysekil Wave Power Project: Status Update." World Renewable Energy Congress (WRECX), Glasgow, UK, pp , Svensson, O., Strömstedt, Savin, A., and Leijon, M. "Sensors and Measurements inside the Second and Third Wave Energy Converter at the Lysekil Research Site." Accepted for publication at the Ninth European Wave and Tidal Energy Conference (EWTEC 2011), Southampton, UK, 5-9 Sept., Lejerskog, E., Gravråkmo, H., Savin, A., Strömstedt, E., Haikonen, K., Tyrberg, S., Krishna, R., Boström, C., Ekström, C., Rahm, M., Svensson, O., Engström, J., Ekergård, B., Baudoin, A., Kurupath, V., Hai, L., Li, W., Sundberg, J., Waters, R. and Leijon, M. "Lysekil Research Site, Sweden: A Status Update." Accepted for publication at the Ninth European Wave and Tidal Energy Conference (EWTEC 2011), Southampton, UK, 5-9 Sept., Leijon, M., Rahm, M, Savin, A. "A wave power unit." International patent, International Publication Number: WO 2010/024741, International Publication Date: 4 March, Leijon, M., Savin, A., Leandersson. R., Waters, R., Rahm, M. "A wave power unit with guiding device." International patent, International Publication Number: WO 2011/149396, International Publication Date: 1 December, Other contributions of the author which are not included in the thesis. XII Boström, C., Lejerskog, E., Tyrberg, S., Svensson, O., Waters, R., Savin, A., Bolund, B., Eriksson, M. and Leijon, M." Experimental results from an offshore wave energy converter." Journal of Offshore Mechanics and Arctic Engineering, 132(4):5 pages, XIII Boström, C., Svensson, O., Rahm, M., Lejerskog, E., Savin, A., Strömstedt, E., Engström, J., Gravråkmo, H., Haikonen, K., Waters, R., Björklöf, D., Johansson, T., Sundberg, J., Leijon, M. "Design proposal of electrical system for linear generator wave power plants."proceedings of IECON 2009, 35th annual conference of the IEEE Industrial Electronics and Society, Porto, Portugal, PD : pp , 2009.

7 XIV XV Boström, C., Lejerskog, E., Tyrberg, S., Svensson, O., Waters, R., Savin, A., Bolund, B., Eriksson, M. and Leijon, M." Experimental results from an offshore wave energy converter." Proceedings of the ASME 27th International Conference on Ocean, Offshore and Arctic Engineering, OMAE 2008, Estoril, Portugal, OMAE , Boström, C., Rahm, M., Svensson, O., Strömstedt, E., Savin, A., Waters, R., and Leijon, M. "Temperature measurements in a linear generator and marine substation for wave power." Proceedings of the ASME 29th International Conference on Ocean, Offshore and Arctic Engineering, OMAE 2010, Shanghai, China, OMAE , Reprints were made with permission from the publishers.

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9 Contents 1 Introduction Wave energy conversion technologies Theory Equation of motion Straight strand cable Strain Strain gauge The principle of the Wheatstone bridge circuit The quarter bridge circuit The half bridge circuit The diagonal bridge circuit The full bridge circuit Shunt calibration Misalignment Effect Mechanical stress, force and bending moment Propagation of uncertainty The Wave Energy Converter (WEC) The WEC L The connection line Simulations Experiment Location of strain gauges on the capsule Location of strain gauges on the inner framework structure Estimation of stress in the inner framework structure Force and bending moments Magnetic attractive force Estimation of stress in the capsule Lateral force and azimuth angle Inclination angle between the generator and the floating buoy Position of the floating buoy on the water surface Results and discussion Estimation of stress in the inner framework structure Forces and bending moments Magnetic attractive force Estimation of stress in the capsule Lateral force and azimuth angle

10 8.6 Inclination angle between the generator and the floating buoy Snatch load Position of the floating buoy on the water surface Error estimation Concluding remarks Future Work Summary of Papers Svensk sammanfattning Acknowledgments Bibliography

11 Nomenclature and abbreviations A m 2 cross section area B δ T magnetic flux in the airgap C r r rolling resistance coefficient D m diameter of the funnel d m diameter of the steel wire E GPa Young s modulus F buoy N buoy force F em N electromagnetic force F es N end stop force F pt N pre-tension force F pc N pre-compression force F s N spring force F g N force due to the mass of the translator F f f N funnel friction force F r r N rolling resistance force F 0 N spring pre-tension F y N overall axial force g m/s 2 gravity k s,k es spring constants k strain gauge sensitivity or gauge factor L m length l m elastic prolongation l t m prolongation due to temperature changes M N m bending moment m kg mass P m pitch length R Ω resistance R p Ω parallel resistance (shunt) R h m helical curve radius 11

12 R w m wire radius R c m core radius S m 3 bending section modulus V b m 3 volume of the water displaced by the buoy V δ m 3 airgap volume W δ J airgap energy y m translator position ÿ m/s 2 acceleration of the translator α r ad lay angle ɛ µm/m strain µ 0 V s/am permeability ν Poisson s ratio ρ kg /m 3 water density θ y, y r ad/m twist angle per unit length θ d eg r ee inclination angle υ y, y µm/m axial strain υ a,a µm/m axial strain of the wire φ d eg r ee azimuth angle σ Pa stress in the material ϱ Ωm resistivity ϱ 0 Ωm resistivity at 20 C ς 1/ C temperature coefficient of resistivity α r ad variation of lay angle L m absolute change of length T C change in temperature 12

13 AW S Archimedes Wave Swing C RF Coefficient of Rolling Friction F E M Finite Element Method MW L Mean Wave Level N F P Normalized Funnel Pressure N d 2 Fe 14 B Neodymium-Iron-Boron OW C Oscillating Water Column PT O Power Take-Off Systems P M Permanent Magnet PC B Printed Circuit Board SSG Sea Slot Generator W EC Wave Energy Converter % MBL Percentage of minimum breaking load 13

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15 1. Introduction Water covers more than 70 % of the Earth s surface. Hence, the ocean is the biggest solar collector in the world. Everything on earth receive it s energy from the sun, and ocean waves are no exception. The nature of the ocean waves is directly related to the differences in pressure between warm and less warm areas. Ocean waves can be created in several ways, e.g. by an earthquake in the ocean or by tidal water. The waves created by an earthquake can travel long distances, up to 3,000 kilometers, without energy loss. In deep water they are almost invisible but once the bottom of the landscape is changing, they start to grow and near the coastline they transform into huge waves, e.g. tsunami waves. However, when we study the generation of energy from ocean waves, are the most important wind induced waves. The nature of these waves is directly related to solar energy. The sun heats the earth unevenly, which leads to a wind from a high pressure area to a low pressure area. The difference between day and night is a simple example of such a change of pressure. As the wind is blowing, a part of the kinetic energy is transferred by the process of friction between the wind and the ocean surface. The potential of ocean waves is huge due to the high density of energy contained in them. The most important differences in comparison to common electricity sources are no need for fuel and absence of pollution. Worldwide energy consumption will rise considerably over the coming decades. Moreover, it is obvious that the traditional methods of energy production are contributing to serious and fundamental environmental problems. The first attempts to use wave energy were dated back to 1799, and have evolved substantially since the 1970s. Nowadays, there are more than 1000 suggestions on how to convert wave energy and on how to design wave power plants. The main aim of many projects is the development of devices and systems that can withstand heavy loads, especially under extreme conditions. Research question The aim of this thesis is to present experimental and analytical results concerning the estimation and characterization of stress properties in the 15

16 outer structure and the inner framework structure of the wave energy converter that uses the principle of a power take-off system. The reason for this thesis is to answer the need for information about processes in the wave energy generator under working load. Measured results allow verifying and developing methods to predict stresses acting on the wave energy converter under real working conditions. This thesis introduces the method of offshore measurements, where strain gauge sensors instrumented on the outer structure and the inner framework structure were used. Simulations of the attractive force between the stator and the translator acting on the framework crossbar have been performed. A simulation of loading from the translator at the upper end stop structure has been made for maximal stress estimation in the inner framework structure. The stress in the inner framework structure is as important as the maximum stress. The measured results were compared with the predicted performance. A method for the evaluation of the lateral force acting on the capsule has been developed. The experimental data allow us to define the inclination and azimuth angle between the generator and the floating buoy. Moreover, the inclination and azimuth angle allow us to reproduce the position of the floating buoy on the water surface. 16

17 2. Wave energy conversion technologies Nowadays, there are a lot of different wave energy conversion technologies, varying in the principle of energy conversion. The devices are adjusted to different water depths and locations. There are three types of locations: offshore, near-shore and shoreline. The technologies are often divided into different groups depending on how they operate. Examples of different categories are: OWC (fixed and floating structures), oscillating bodies (submerged and floating) and overtopping (fixed and floating structures). Wave energy conversion devices using the OWC principal are mostly performed with an air turbine. Waves create oscillations on the surface of the water in a submerged hollow chamber. The oscillations continuously compress and decompress the air column. The difference in pressure rotates the turbine-driven generator, which converts rotational energy into electrical energy. For efficient energy extraction, a cross-sectional area OWC narrows as it approaches the turbine. This accelerates the air so the air reaches its maximum velocity. Such devices can be a fixed structure Limpet 1 [1] or as floating device Oceanlinx 2 [1] [2], see Fig The principle of oscillating bodies is constructed with a hydraulic motor, a hydraulic turbine or a linear electric generator. They can be introduced as floating structures or submerged structures. Some examples of oscillating body converters are Pelamis 3 [3], which is a floating structure and AWS 4, which is a submerged structure [4, 5], see Fig The Wave Energy Converter The WEC 5 that has been developed at the Swedish Centre for Renewable Electric Energy Conversion at Uppsala University is using the principle of an oscillating body and built as submerged structure connected to a floating buoy [6 10], see Fig The WEC consists of a linear direct drive generator placed on the sea floor. The translator, which is mounted with (N d 2 Fe 14 B) magnets, moves up and down and induces a current in the stator windings. 1 Limpet, Oceanlinx, Australia, Pelamis, UK, Archimedes wave swing, UK, Uppsala University s concept, Sweden,

18 (a) OWC (fixed structure) Limpet, Voith Hydro Wavegen Limited (b) OWC (floating) Oceanlinx Limited, Australia Figure 2.1: Wave energy conversion devices using the principal of an oscillating water column (OWC). (a) A fixed structure Limpet. (b) A floating device Oceanlinx. Oscillating bodies (floating) Pelamis Wave Power Ltd., UK Oscillating bodies (submerged) Archimedes Wave Swing Ltd., UK Figure 2.2: Wave energy conversion devices using the principle of oscillating bodies. (a) A floating structure Pelamis. (b) A submerged structure Archimedes wave swing. 18

19 Figure 2.3: The WEC developed at the Swedish Centre for Renewable Electric Energy Conversion at Uppsala University. A submerged structure using the principal of oscillating bodies. Overtopping (fixed structure), SSG-concept (Sea-wave Slot-cone Generator concept) Wave Energy AS, Norway Overtopping (floating structure) Wave Dragon Ltd., Denmark Figure 2.4: Wave energy conversion devices using the principle of overtopping devices. (a) A fixed structure in break water without concentration SSG-concept. (b) A floating device with concentration Wave Dragon. 19

20 The principle of overtopping devices is built with a low-head hydraulic turbine. There are two different types, i.e. a fixed structure in break water without concentration and a floating structure with concentration. The examples are the SSG-concept (sea-slot-cone Generator Concept) [11] for the first case and the Hydrodynamic System Wave Dragon [12] for the second case, see Fig The SSG 6 system consists of three reservoirs which are placed on top of each other. The reservoirs store the potential energy of the waves. The water rotates the turbines for energy conversion. The SSG-concept is built for using multiple reservoirs, which raises the efficiency significantly compared to the single overtopping system. The Wave Dragon 7 is a hydrodynamic system, one of the simplest concepts of overtopping devices. The bottom of the reservoir is floating above the mean wave level (MWL). The waves come to the reservoir and rotate a turbine, which is connected to a generator. This is the basic principle of overtopping for the Wave Dragon. The "wings" of the Wave Dragon catch the waves into the device. In all cases presented here, the energy of waves is converted by rotating or direct drive linear generators. 6 SSG-concept, Norway, Wave dragon, Denmark,

21 3. Theory The purpose of this chapter is to show an equation of motion for a linear generator system as well as describe a straight strand cable. A theory of strain gauge, the principle of the Wheatstone bridge circuit and the propagation of uncertainty are presented. 3.1 Equation of motion All working parts of the WEC contribute with forces which can be classified as either damping forces or energizing forces. When the buoy moves with the waves, the translator gets up-and-down motions. A force model of the system is shown in Fig The Cartesian coordinate system in two-dimensional space was employed for the equation of motion. The equation of motion can be expressed as: mÿ = F buoy + F g + F s + F es1 + F es2 + F em + F f f + F r r [N ], (3.1) where ÿ is the acceleration of the translator in the y direction, m indicates the mass of the translator and the buoy, if the buoy force is greater than zero, or the mass of the translator only, F buoy denotes the hydrodynamic forces acting on the buoy and is calculated by use of linear potential wave theory [13]. The force due to the mass of the translator can be written as F g = mg where g is the gravity constant. The spring force from the tension springs is given by F s = F pt k s y [N ], (3.2) where y is the translator displacement along the vertical axis from the equilibrium y 0 which corresponds to the position of the translator in the middle of the inner framework structure, opposite the stator, F pt denotes the pre-tension force in the equilibrium position of the translator, k s is the tension spring constant. 21

22 Figure 3.1: Forces acting on the generator during operation. The upper end stop force, modeled as a stiff spring to prevent the translator from leaving the generator can be expressed as { kes1 (y es1 y), y > y es1 F es1 = 0, el se, [N ] (3.3) where k es1 is the upper spring constant, y es1 denotes the length from the upper end stop spring to the translator equilibrium. The lower end stop force, modeled as a stiff spring to prevent the translator to collide with the bottom of the generator can be written as { kes2 (y es2 + y), y > y es2 F es2 = 0, el se, [N ] (3.4) where k es2 is the lower spring constant, y es2 denotes the length from the lower end stop spring to the translator equilibrium. The electromagnetic force is given by F em = si g n(ẏ) dw δ d y d B 2 δ = si g n(ẏ) V δ [N ], (3.5) d y 2µ 0 where W δ is the energy in the air-gap, B δ is the magnetic flux in the air gap, µ 0 is the permeability and V δ is the air gap volume. 22

23 The funnel friction force can be expressed as F f f = si g n(ẏ)µθf buoy [N ], (3.6) where µ is a friction coefficient and θ is the inclination angle between the generator and the floating buoy, given in radians. The rolling resistance force from the translator is given by F r r = si g n(ẏ)c r r F pc [N ], (3.7) where F pc is the pre-compression of the disk spring and C r r is the dimensionless rolling resistance coefficient or the coefficient of rolling friction (CRF). The buoy force is the force extracted from the waves. The electromagnetic force is calculated from the electromagnetic field in the air gap. An end stop force appears due to compression of the spring mounted on the top and the lower part of the generator. The funnel friction force appears due to the angle between the generator and the floating buoy. The spring force changes depending on the position of the translator inside the mechanical structure. The spring constant is a major parameter for the spring force. The rolling resistance force can be calculated by the pre-compression force from the disk spring and the coefficient of rolling friction (CRF). 3.2 Straight strand cable Wire ropes are widely used in various applications, such as hoist installations, elevators, bridging applications, mooring lines for offshore oil platforms as well as the connection line for wave energy converters. A wire rope is a mechanical structure. It is made of layers of strands wrapped helically around a central core, see Fig The geometry of such a wire is characterized by the lay angle α corresponding to the angle between the cable y-axis and the helical wire, the wire radius R w, the core radius R c and the pitch length of wire P. The centerline of such a wire is a helical curve with radius R h, Eq R h = R w + R c [m] (3.8) The pitch length can be calculated according to Eq P = 2πR h tanα [m] (3.9) For a simple straight strand, it is assumed that change in the angle between the cable y-axis and the helical wire is small. The twist angle per unit 23

24 Figure 3.2: 6+1 straight strand cable made of six helical wires wrapped around a straight core.r w is the wire radius, R c is the core radius, R p is the pitch length of wire, α is the angle between the cable y-axis and the helical wire. length θ y, y, the axial strain υ y, y and the axial strain of the wire υ a,a in each wire are linearized with respect to α. The relationship for each wire can be expressed as υ y, y = υ a,a + αtanα [m/m] (3.10) The rotation of one end of the strand with respect to the other would be R h θ y, y = υ a,a tanα α + νtanα R cυ y, y + R w υ a,a R h [r ad] (3.11) 3.3 Strain The strain of a structure is always caused by an external force or an internal effect, e.g. thermal expansion. Any mechanism or structure, which is subjected to loading, requires determination of the stresses in the material, as well as the places where these stresses are highest. External force applied to an elastic material produces an absolute change of length, see Fig Hence, strain is defined as the displacement or deformation that occurs, Eq ɛ = L L 0 L 0 = L L 0 [m/m], (3.12) where ɛ is the strain, L is the absolute change of length, L 0 is the original length 24

25 Figure 3.3: Changes of length caused by tension or compression forces. L 0 is the original length, L is the section length, L is the absolute change of length Figure 3.4: Various resistive strain gauges designs. There is tensile and compressive strain, distinguished by a positive or negative sign. The word "strain" is a technical expression for the elongation process from the international standard ISO Strain gauge The strain gauge is one of the most important measurement sensors for the electrical measurements of mechanical quantities. A strain gauge is a sensor constructed by bonding a measuring grid whose resistance varies with the applied force, see Fig It converts force, pressure, tension, weight, etc., into a change in electrical resistance that can then be measured. Basically, there are tree types of bonded resistive strain gauges: 1) wire gauge 2) foil gauge 3) semiconductor gauge There are new developments, which are mainly regarded as alternatives to conventional strain gauges. For example, for use at high temperatures beyond the limit of a metal strain gauge, capacitive strain gauge can be used. A piezoelectric strain gauge is an active device. It provides an electrical charge on its surface which is proportional to the strain. The electrical charge can be measured with a charge amplifier. 25

26 Figure 3.5: Resistive strain gauge. A vapour-deposited (thin-film) strain gauge uses a technique where the measuring element is deposited directly onto the measurement place. A photoelastic strain gauge uses a princip which is based on the displacement of the isochromatic field as a result of the strain. Mechanical strain gauges have a long tradition. The principle of a resistive strain gauge is based on the strain/resistance relationship of the electrical conductors. Electrical conductor is the wire in the strain gauge. Mechanical force applied to the conductor material results in micro structural changes due to the resistivity change of the material. This results in an electrical resistance change. The resistance is given by R = ϱ L A = ϱ 4L πd 2 [Ω], (3.13) where R is the electrical resistance, ϱ is the resistivity, L is the length of the conductor, D is the diameter of the conductor, A is the cross sectional area of the conductor. The resistivity depends on the temperature. The amount of that change can be calculated using the temperature coefficient of resistivity (ς). ϱ = ϱ 0 (1 + ς T ) [Ωm], (3.14) where ϱ 0 is the resistivity at 20 C, ς is the temperature coefficient of resistivity, T is the change in temperature As the wire stretches it becomes longer and thinner because the volume of the wire stays approximately the same. Hence, the resistance of the wire is increased by stretching. When a wire is compressed it becomes shorter and thicker. This reduces the resistance, see Fig The material structure changes slightly during stretching and compression which produces small resistivity changes The principle of the Wheatstone bridge circuit Resistance strain gauges must be connected to an electrical circuit in order to measure strain in material. The Wheatstone bridge is a usual circuit 26

27 Figure 3.6: The Wheatstone bridge circuit. used to measure electrical resistance. The Wheatstone bridge circuit is adjusted for temperature compensation with very good accuracy. The number of active strain gauges, which must be connected to the circuit, varies depending on the application. If all arms of the Wheatstone bridge circuit are active, the temperature compensation is automatic. If two active arms of the Wheatstone bridge circuit are orthogonal, the temperature compensation is also automatic. For temperature compensation, one of the arms can be also a dummy strain gauge. If a supply voltage V s is applied to the two points A and C, the two halves of the bridge R1, R2 and R4, R3 divide it up as a ratio of the corresponding bridge resistances. In other words, each half of the bridge forms a voltage divider, see Fig For a given supply voltage V s, the current flowing through ABC and ADC depends on the resistances, Eq V s = V AB C = V ADC = I AB C (R 1 + R 2 ) = I ADC (R 4 + R 3 ) [V ], (3.15) where V AB C, V ADC and I AB C, I ADC correspond to voltage and current flowing trough ABC, ADC respectively The partial voltage V AB can be calculated as: The partial voltage V AD : V AB = I AB C R 1 = V AD = I ADC R 4 = V s R 1 + R 2 R 1 (3.16) V s R 4 + R 3 R 4 (3.17) The bridge output voltage V o is the difference between two partial voltages: V o = V AB V AD = V s R 1 + R 2 R 1 V s R 4 + R 3 R 4 = R 1 R 3 R 2 R 4 (R 1 + R 2 )(R 4 + R 3 ) V s (3.18) 27

28 If resistors change their resistances by some amount R, the strain gauges circuit becomes unbalanced: V o = (R 1 + R 1 )(R 3 + R 3 ) (R 2 + R 2 )(R 4 + R 4 ) (R 1 + R 1 + R 2 + R 2 )(R 4 + R 4 + R 3 + R 3 ) V s (3.19) If R 1 = R 2 = R 3 = R 4 = R: R( R 1 + R 3 R 2 R 4 ) + R 1 R 3 R 2 R 4 V o = 4R 2 + 2R( R 1 + R 2 + R 3 + R 4 ) + ( R 1 + R 2 )( R 3 + R 4 ) V s (3.20) If the bridge is initially balanced, the output voltage V o should be zero. V o = It is possible at two conditions: 1) all resistors in the bridge have equal value, R 1 R 3 R 2 R 4 (R 1 + R 2 )(R 4 + R 3 ) V s = 0 (3.21) R 1 = R 2 = R 3 = R 4 (3.22) or 2) the ratios of the resistors in the two halves of the bridge are the same, R 1 R 2 = R 4 R 3 (3.23) It is common to use the approximation below because the resistances changing in the strain gauges are very small. [14] V 0 V s = 1 4 ( R 1 R 1 R 2 R 2 + R 3 R 3 R 4 R 4 ) (3.24) Each strain gauge has a specific ratio between strain and change of resistance, known as strain gauge sensitivity or gauge factor k. A gauge factor is expressed as: k = R R ɛ, (3.25) Due to: R R = kɛ, (3.26) the Eq can be rewritten as: 28

29 V o V s = k 4 (ɛ 1 ɛ 2 + ɛ 3 ɛ 4 ) (3.27) Eq and Eq show that the strain, which contributes to the resistance changes, unbalances the bridge and hence the output voltage The quarter bridge circuit. In the quarter bridge circuit there is only one sensor and the other three are fixed resistances. If R 1 = R and R 2 = R 3 = R 4 =0, then Eq appears in the form: V o = R R 4R 2 + 2R R V R s = 4R + 2 R V s (3.28) Using Eq. 3.26, and if it is assumed that R << R, Eq becomes: The half bridge circuit. V 0 = R 4R V s = 1 4 kɛv s (3.29) In the half bridge circuit there is two sensors in the same leg and the other two are fixed resistances. If R 1 = R, R 2 = R, R 3 = R 4 =0, and using Eq. 3.26, Eq becomes: V o = R2 R 4R 2 V s = R 2R V s = 1 2 kɛv s (3.30) Any changes in temperature affect both gauges in the same manner. Because of this, the temperature changes are identical in the two gauges and the effects of the temperature change are compensated The diagonal bridge circuit. In the diagonal bridge circuit there is two sensors and two fixed resistances. If R 1 = R 3 = R and R 2 = R 4 =0, then Eq appears in the form: V o = R2 R + R 2 4R 2 + 2R2 R + R 2 V R(2R + R) s = (2R + R) 2 V s = R 2R + R V s (3.31) 29

30 Figure 3.7: Principle of shunt calibration. Using Eq. 3.26, and if it is assumed that R << R, Eq becomes: The full bridge circuit. V o = R 2R V s = 1 2 kɛv s (3.32) In the full bridge circuit we have four active sensors. If R 1 = R 3 = R and R 2 = R 4 = R, and using Eq. 3.26, Eq becomes: V o = R4 R 4R 2 V s = R R V s = kɛv s (3.33) Any changes in temperature affect all gauges in the same manner. Because of this, the temperature changes are identical in all gauges and the effects of the temperature change are compensated Shunt calibration. If the measurement equipment can not be calibrated with good accuracy directly, i.e. by applied force, an alternative method can be used. The principe is based on using the device which is producing the defined effect on the measurement equipment. For example, the strain gauges circuit can be unbalanced with a parallel shunt resistance, see Fig It is simulating the strain by changing the resistance of an arm in the circuit by known quantity R p, see Eq The output of the circuit is then compared with the calculated amount. The result is used to form the ratio between the calculated calibrated value and the indicated measurement which is called the correction factor. 30 R p where R p is a parallel resistance (shunt) V o = 1 4 ( 1)V s (3.34) R 4 + R p

31 3.3.8 Misalignment Effect The measured strain that is misaligned by an angle ψ from the principal strain direction can be calculated according to the relationship: ɛ ψ = 1 2 ((ɛ 1 + ɛ 2 ) + (ɛ 1 ɛ 2 )cos2ψ) [m/m] (3.35) If ɛ 2 = νɛ 1 under the uniaxial stress condition, ɛ ψ = 1 2 ɛ 1((1 ν) + (1 + ν)cos2ψ) [m/m], (3.36) where ɛ ψ corresponds to the misaligned strain, ɛ 1 is the principal strain or the strain in the active direction of the force, ɛ 2 is the strain perpendicular to ɛ 1 and ν is the Poisson ratio. 3.4 Mechanical stress, force and bending moment Mechanical stresses σ can not be measured by direct measurement except by X-ray technology where the stress in the material can be determined in the microscopic range. Stresses can be calculated either according to the theory of strength of materials or from strain gauges measurements. In this paper, stresses were calculated by measured strain in material, which were caused by tensile, compressive and bending forces. The calculations of the force and bending moment were based on Hooke s Law, see Eq The force, F, was calculated by the tension or compression, ɛ, with a given cross section, A, as well as Young s modulus, E, see Eq σ = Eɛ [Pa] or L = F E A L = σ L [m] (3.37) E F = σa = (ɛ 1 + ɛ 2 ) E A [N ] (3.38) 2 The bending moment, M, was calculated by the tensile and compressive strains, ɛ, with a given bending section modulus, S, as well as Young s modulus, E, see Eq M = σs = (ɛ 1 ɛ 2 ) ES [N m] (3.39) 2 31

32 3.5 Propagation of uncertainty The uncertainty of measurements can usually be defined by the accuracy of the measurement equipment. Calculated variables from experimental measurements have an error due to measurement limitations that propagate to the combination of variables in the function. The uncertainty is usually defined by the absolute error [15] [16]. When the strain ɛ in the structure is calculated from the measured voltage, the absolute error of strain calculations can be expressed as: ɛ = ɛ( V, V s, k, ν, r ), (3.40) where ɛ is the error for strain, V o is the error for output voltage, V s is the error for bridge excitation voltage, k is the error for gauge factor, ν is the error for Poissons s ratio, r is the error for amplification. We can calculate the error propagation ɛ for the strain using partial derivatives to propagate error according to: ɛ = ɛ V V o + ɛ o V V s + ɛ s k k + ɛ ν ν + ɛ r r (3.41) The error propagation σ can be calculated according to the following formula: σ = σ ɛ ɛ + σ E E, (3.42) where σ the error for stress, E error for Young s modulus. 32

33 4. The Wave Energy Converter (WEC) Research on wave energy converters has been going on for decades. A variety of proposals have been developed to transform the energy of ocean waves into electricity and some pilot installations have been built worldwide [4, 5, 17, 18]. A lot of wave energy converters have been introduced as large and expensive facilities, often located on the water surface and including a lot of complex and sensitive equipments. Because of their size and complex system, they can have some problems to handle the harsh wave climate they were intended for. At the Swedish Centre for Renewable Electric Energy Conversion, Ångström Laboratory, Uppsala University a concept for wave energy conversion has been developed and the first full-scale generator was launched offshore outside the Swedish west coast in March 2006 [6 10]. The wave energy concept that has been tested in the Lysekil project differs in many aspects from earlier attempts. The WEC consists of a linear direct drive generator placed on the sea floor as shown in Fig Instead of adapting rotating standard generators to the waves rolling motion, a completely new type of generator have been developed which is specifically adapted to the standard or usual Figure 4.1: A concept of wave energy conversion by the linear generator. waves. The device can be represented as a direct drive wave energy converter with a linear generator. The main differences between this generator and a rotating one are that it does not have such a complex system of transmission of motion and that the number of moving parts is reduced. 33

34 Figure 4.2: In a) floating buoys for L1, L2 and L3 from the left to the right, b) L1, c) L2, d) L3, e) L4, L7 and L8 that were built by Seabased Industry AB, f) L9. The translator, which is mounted with permanent Neodymium Iron Boron (N d 2 Fe 14 B) magnets, moves up and down and induces a current in the stator windings according to Faraday s law. At a translator speed of 0.7 m/s and connected to a load of 4 Ohm, the generator produces 10 kw. The efficiency of the generator is in this case 86 % [9]. Four full-scale generators and marine substation were installed offshore at Lysekil off the Swedish west coast between Four more fullscale generators were installed during Many components were modified in terms of weight reduction and more effective energy conversion. Floating buoys with different shapes were connected to determine the difference in the power absorption. Pictures of different wave energy converters and floating buoys can be seen in Fig The WEC L2 Since the author was responsible for the WEC L2, see Fig. 4.3, it should be mentioned that development and construction of the WEC has not been a trivial process. As a new device, the WEC has needed improvements in the design and modeling as well as careful inspection after each building step. 34

35 (a) The generator inside the watertight capsule (b) The WEC L2 in the harbor of Lysekil before deployment Figure 4.3: The WEC L2. The funnel on top of the WEC, see Fig. 4.4, was designed as two separate parts to allow easy installation. The parameters of the funnel were modified depending on the parameters of the connection line. The purpose of the funnel was to guide the connection line inside the guiding system. The author was deeply involved in the design of the funnel. The inner framework structure consists of four corner pillars that are connected by twelve framework crossbars. Four stator packages are mounted on the corner pillars by sixteen stator beams, see Fig The upper end stop structure has an upper end stop spring inside the cylinder in terms of protecting the translator leaving the generator, see Fig The lower end stops structure has a lower end stop spring to avoid hitting between the translator and the bottom plate. Eight springs mounted at the bottom of the generator pull the translator down. The generator is protected against the water by the outer structure. A rotation resistant hoist rope Powerplast made of compacted strands was used as the connection line between the floating buoy and the generator, (Paper II). The water depth at the designated site is 25 metres. The height of the generator is 8 m, the diameter of the outer cylindrical structure is 1.6 m, and the bending radius of the funnel is 600 mm. The parameters for the WEC L2 are given in Table 4.1. The installation of such a device requires a study of many details related to offshore installation as well as knowledge of equipment for pumping pressure at launching of the WEC. 35

36 (a) part (b) assembly Figure 4.4: The funnel was designed as two separate parts in terms of easy installation. Fig show assembly, transportation and deployment of the WEC L2. 36

37 Figure 4.5: Mechanical structure. Generator length with foundation 9506 mm Generator width with foundation 5400 mm Generator diameter 1566 mm Stator length 1264 mm Translator length 1867 mm Translator width 401 mm Nominal power at 0.7 m/s 10 kw Main voltage 200 V Air gab between stator and translator 3 mm Translator weight 1300 kg Bending radius of the funnel 600 mm Connection line: type Powerplast length mm diameter 28 mm tensile strength of wire 1960 N/mm 2 minimum breaking load kn Table 4.1: Parameters for the generator and the connection line. 37

38 Figure 4.6: Springs. Figure 4.7: Final assembly. 38

39 Figure 4.8: Transportation and deployment of the WEC L2. Figure 4.9: Picture shows the deployment of the WEC L2 Energi världen, nummer 2, 2010 (Permission for publication). 39

40

41 5. The connection line This chapter is concerned with the design and analysis of the steel wire that was used as the connection line between the floating buoy and the generator, see Fig Figure 5.1: The floating buoy conected to the WEC L2 and the WEC L2 under deployment. The connection line has a shorter service life than the other parts of the WEC. Therefore, the service life of the line appears to be a very important characteristic. In the operation of the steel wire, its cable wires are exposed to tensile stress, bending, twisting, and contact loading. The role of these stresses on the wear process of the wire is different. Significant influence on the mechanical strength, the life of the wire, have an effect on the material, on the process of laying, the lay angle, and a number of other technical factors. Experimental work on large diameter wires needs large and expensive testing devices. At the Lysekil research site an approach was made for determining the service live of the connection line under a working load in the WEC. The choice of the connection line was not trivial. There are different ropes and wires, which could be suitable as the connection lines for the WEC. The MegaTwin Vectran 36 mm with a minimum breaking load of 485 kn was chosen as the first connection line, see Fig It consists of a Vectran core, covered with Dynema Sling, made of a braided Vectran cores and over braided with white Dynema. For the second experiment, a rotation resistant steel wire Powerplast 28 mm (with a tensile strength of wire at 1960 N/mm 2 ) made of compacted 41

42 Figure 5.2: MegaTwin Vectran 36 mm with a minimum breaking load of 485 kn. strands with a minimum breaking load at kn was chosen as the connection line, see Fig. 5.3 The steel wire was fully lubricated, which prevented metal-to-metal contact between the outer and inner strands, and also excluded water and abrasive elements that could penetrate conventional ropes. The steel wire was also housed in a black high density (HD) jacketing compound to prevent metal-to-metal contact between the steel wire and the funnel. Figure 5.3: A steel wire Powerplast 28 mm with a minimum breaking load of kn, housed in a black high density (HD) jacketing compound. The steel wire Powerplast was chosen because of its rough surface, constructed by the outer strands of the wire, which makes a better contact between the wire and the jacketing compound and prevents sliding. Moreover, the jacket was an effective method to prevent wire corrosion completely. The mechanical properties of the jacketing material were the following: flexural modulus of 850 MPa, tensile strain at break of 700 %, tensile strength of 25 MPa. The scratch resistances as well as a good abrasion and heat deformation resistance made this rope suitable to use in the WEC where the service life of the steel wire is a very important characteristic. The surface-floating buoy transfers both a normal force and torque from waves to the steel wire. It can lead to the untwisting of the steel wire. To prevent this, the steel wire Powerplast has a steel core which is an independent wire, closed in the opposite direction to the outer strands [19]. Under load, the core tries to twist the wire in one direction and the outer strands try to twist it in the opposite direction. The geometrical design of 42

43 a rotation-resistant Powerplast steel wire is such that the moments in the core and the outer strands compensate for each other. Even at great lifting heights no wire twist occurs. At the same time, it is necessary to take into account a prolongation in the wire, which occurs when the wire is subjected to loading. Fundamental reasons to prolongation can be classified in three phases: 1) initial prolongation 2) elastic prolongation (strain) 3) permanent prolongation (change of the temperature, wear, corrosion) Phase 1 occurs among the strands and wires when a wire is subjected to the axial tension load. Strands and wires press together and fill a space between individual wires. This gives some reduction in the wire diameter and some prolongation of the wire. Phase 2 occurs at further load. An elastic prolongation can be formulated as: l = W L E A [m], (5.1) where l is an elastic prolongation, W is the load, L is the length, E is the modulus of elasticity, and A represents area. If the loading gets the yield strength, phase 3 occurs. Temperature changes also make some contribution to a prolongation: l t = klt [m], (5.2) where l t is a prolongation due to temperature changes, k is the coefficient, linear depending in the change of the temperature in the material, and t is the change of the temperature. Determining the service life of the steel wire includes failure modes, which can be grouped under five headings: bending at the funnel, tension-tension, torsion, free bending, and snatch load. Bending fatigue at the funnel correlates fatigue life (cycles to failure) to "normalized funnel pressure" N F P = % MBL, (5.3) D d where N F P is a normalized funnel pressure, % MBL is a load in the steel wire expressed as a percentage of its minimum breaking load, D is a diameter of the funnel and d is a diameter of the wire. In a taut-mooring system tension-tension fatigue correlates to fluctuations due to motions induced by waves. This category of fatigue is the simplest, based on stress changing due to axial loading from the floating buoy. 43

44 Tension-tension fatigue in the connection line can be maximized by a 10:1 margin on the peak load. Torsional oscillations of the wire from the floating buoy contribute to fatigue, for which the dominant parameter seems to be the twist amplitude. Free bending fatigue is of importance as a degradation mechanism with bending deformation, which does not include contact with another subject. Snatch load has an unpredictable influence on the service life of the steel wire and is of course, of special interest to wave power community. However, the dominant source of failure in the steel wire was linked to another degradation mechanism, namely fretting wear by friction between the funnel and the steel wire. Repeated back-and forth cycling over one spot on the wire created a rapid heat build-up in a (HD) compound and an unexpected damage mechanism. A black high density (HD) jacketing compound on the steel wire was cracked at the contact area with the funnel after 2 months, see Fig Figure 5.4: Underwater pictures of the funnel and the connection line with a cracked black high density (HD) jacketing compound. This resulted in a significant increase of the friction coefficient between the funnel and the wire. A fretting wear notch on the funnel appeared. This resulted in crack initiation, propagation and fracture of the outer strands of the steel wire. The period until fracture of the outer strands was about 3 months from the splitting of the jacketing compound. After that, the wire was working for about 3 weeks until the wire fractured. The first experimental service life of the steel wire under a working load in the WEC was more than 5 months. Moreover, a new approach with another compound covering a (HD) compound has been recently tested. In this case, the steel wire was housed in 2 layers of jacketing compounds. Firstly, the steel wire was housed in a black high density (HD) jacketing compound. Secondly, the wire was 44

45 covered with a new compound. This new solution showed a significant increase in the service life of the connection line. 45

46

47 6. Simulations Simulations of the mechanical structure of the generator has been performed in Pro/ENGINEER Mechanica. Simulation of loading from the translator at the upper end stop structure for maximum stress estimation in the inner framework structure is shown in Fig Stress in the inner framework structure is as important as the maximum stress. The load value of 80 kn was chosen based on real measurements of the line force from the floating buoy at a sea state of 3 m waves. The yellow color shows the most likely places for crack propagation to occur. Figure 6.1: FEM simulation of stress (MPa) in the corner pillar when the translator hits an end stop structure. Simulations of the attractive force between the stator and the translator acting at the framework crossbar can be seen in Fig. 6.2(a,b). The attractive force was previously investigated in [20]. This simulation is based on recent results of the attractive force performed by the Ansoft program. The attractive force corresponded to N for each side of the generator. 47

48 (a) Uniaxial stress (MPa) (b) Bending stress (MPa) Figure 6.2: FEM simulations of the stress in the framework crossbar occurring due to the attractive force between the stator and the translator. 16 stator beams were loaded with the force (yellow arrows) that corresponded to the magnetic attractive force between the stator and the translator ( kn). (a) shows the uniaxial stress in the framework crossbar, (b) represents the bending stress in the framework crossbar. 48

49 7. Experiment In this chapter, the offshore experiment is presented. The first two sections describe the location of the strain gauge circuits on the capsule and the inner framework structure. This is followed by calculation of the stresses in the structure, the lateral force, the azimuth angle and the inclination angles. The position of the floating buoy on the water surface is calculated. 7.1 Location of strain gauges on the capsule The capsule was instrumented with two separate 2-active-gauge systems for bending strain measurement (Strain gauge circuits nr.1 and nr.2) mounted inside the capsule with an angle of 90 degree to each other, see Fig. 7.1(a),(b). Gauge length 5 mm Resistance 350±1.8 Ω Gauge factor 1.97±2% (a) Table 7.1: Properties of Kyowa KFN C9-16 Every circuit had two resistive strain gauges (KFN C9-16), see Table 7.1, that were mounted in an angle of 180 degree to each other. The strain gauges were glued with (KYOWA CC-33A, strain gauge (b) cement). Each strain gauge circuit was amplified inside the WEC to Figure 7.1: (a),(b) Position of strain get a sufficient S/N (signal to noise) gauges inside the capsule. ratio to transfer the signal through a twisted pair of 70 metres long shielded cables to the data acquisition sys- 49

50 tem where the signals were digitized with a sampling frequency of 256 Hz. For this purpose a strain measurement system was applied. The strain gauge circuit was followed by an amplifier with reinforcement of times ± 5.3% and an analog digital counter (ADC NI 9205). 7.2 Location of strain gauges on the inner framework structure (a) Upper part of the inner framework structure Different parts of the inner framework structure were provided with different strain gauge circuits both for bending strain measurements and for uniaxial strain measurements occurring in uniform tension/compression conditions, see Fig. 7.2(a),(b),(c). When the WEC is under working load, the translator moves up and down and stress occurs in the inner framework structure. To evaluate the maximum stress, (b) Middle part of the inner framework structure two strain gauge circuits of an active-dummy 2-gauge system (Ch. 3 and Ch. 6) were mounted on the corner pillars close to the upper end stop structure, see Fig. 7.2(a). One of the corner pillars was also provided with two circuits of an active-dummy 2-gauge system (Ch. 2 and Ch. 11) applied to mirror-imaged cross-sectional shapes, see Fig. 7.2(b). Uniaxial stress and bending stress were calculated on the basis of these two (c) Lower part of the inner framework structure separate measurements. Figure 7.2: Position of strain gauge circuits. The framework crossbars were instrumented with an orthogonal 4-active-gauge system (Ch. 9) for uniaxial strain measurement as well as with a 2-active-gauge system (Ch. 4) for 50

51 bending strain measurement, see Fig. 7.2(b). The lower part of the inner framework structure was provided with an orthogonal 4-active-gauge system (Ch. 10) for uniaxial strain measurement, see Fig. 7.2(c). 7.3 Estimation of stress in the inner framework structure The stress on a structure was caused by a force from the floating buoy, the reciprocating translator and the electromagnetic force. The stress was calculated by measured strain. As the inner framework structure is symmetric on all four sides, a Cartesian coordinate system in two-dimensional space can be employed for the description of the directions of loading. The X dimension corresponds to the horizontal direction. The Y dimension corresponds to the vertical direction. Calculations of stress in the corner pillar were made according to Eq. 7.1 for channels 2, 3, 6 and 11 and Eq. 7.2 for channel 10. Differences in calculations were due to the fact that different strain gauge circuits were used for strain measurements. For channels 2, 3, 6 and 11, active-dummy 2-gauge systems were used and for channel 10, an orthogonal 4-active-gauge system was used. σ i y = ɛ i E = 4V o (i ) V s kr i E [Pa] (7.1) σ i y is the uniaxial stress in the Y direction for each channel, ɛ i is the strain for each channel, V o (i ) is the output voltage for each channel, V s is the bridge excitation voltage, E is Young s modulus, k is the strain gauge s sensitivity or gauge factor, r i is the amplification for each channel σ 10 y = ɛ 10 E = 2V o (10) (1 + ν)v s kr 10 E [Pa] (7.2) ɛ 10 is the strain from the strain gauge circuit Ch. 10, V o (10) is the output voltage for Ch. 10, r 10 is the amplification for Ch. 10 Calculations of stress in the framework crossbar were made according to Eq. 7.3 for Ch. 4 and Eq. 7.4 for Ch. 9. For Ch. 4 a 2-active-gauge system (for bending strain measurements) was used and for Ch. 9 an orthogonal 4-active-gauge system was used. σ 4x = ɛ 4 E = 2V o (4) V s kr 4 E [Pa] (7.3) σ 4x is the bending stress in the X direction for Ch. 4, ɛ 4 is the strain from 51

52 the strain gauge circuit Ch. 4, V o (4) is the output voltage for Ch. 4, r 4 is the amplification for Ch. 4 σ 9x = ɛ 9 E = 2V o (9) (1 + ν)v s kr 9 E [Pa] (7.4) σ 9x is the uniaxial stress in the X direction for Ch. 9, ɛ 9 is the strain from the strain gauge circuit Ch. 9, V o (9) is the output voltage for Ch. 9, ν is the Poisson ratio, r 9 is the amplification for Ch Force and bending moments To obtain the force and the bending moment that were acting on the inner framework structure in the Y direction, one of the corner pillars was provided with 2 strain gauge circuits (Ch. 2 and Ch. 11) applied to mirror imaged cross-sectional shapes. Two separate strain measurements were made. Force and bending moments were calculated based on these two separate measurements according to Eq. 7.5 and 7.6. F y = (ɛ 2 + ɛ 11 ) E A p 2 = 2E A ( p V o (2) + V ) o (11) [N ], (7.5) V s k r ( 2) r (11) where F y is the force in the Y direction, ɛ 2 and ɛ 11 are the strains in the Y direction from strain gauge circuits Ch. 2 and Ch. 11, A p is the crosssectional area of the corner pillar. M y = (ɛ 2 ɛ 11 ) ES p 2 = 2E A ( p V o (2) V ) o (11) [N m], (7.6) V s k r ( 2) r (11) where M y is a bending moment that was acting on the corner pillar, S p is a section modulus of the corner pillar. To obtain the bending moment in the X direction, a 2-active-gauge system (Ch. 4) was mounted on the framework crossbar. Eq. 7.7 was used for the calculation of the bending moment. M x = ɛ 4 ES g [N m], (7.7) where M x is a bending moment that was acting on the framework crossbar, S g is a section modulus of the framework crossbar. 52

53 7.5 Magnetic attractive force Little attention seems to have been devoted in publications on how to estimate the attractive forces in wave energy systems. Stress in the framework crossbar was mostly caused by the magnetic attractive force between the stator and the translator. To obtain the attractive force that was acting on the framework crossbar, one of the framework crossbars was provided with an orthogonal 4-activegauge system for uniaxial strain measurement, see Fig (a) Orthogonal 4-active-gauge system for uniaxial strain measurement (b) Position of strain gauges Figure 7.3: Estimation of the attractive force. Eq. 7.8 was used for calculation of the attractive force. F a = ɛ 9 E A g [N ], (7.8) where F a is the attractive force, A g is the cross-sectional area of the framework crossbar. 7.6 Estimation of stress in the capsule The outer structure of the WEC consists of the funnel, the guiding system, the capsule, and the foundation, see Fig The outer structure protects the inner framework structure, with the linear generator inside, from the water. It has to be strong enough to withstand axial and lateral forces from the floating buoy. Therefore, estimation of stress is of importance. The funnel The guiding system The capsule The foundation Figure 7.4: The outer structure of the WEC. 53

54 When the buoy moves with the waves, the angle between the wire and the generator changes. The outer structure obtains a lateral force from the wire. The stress in the capsule, caused by this lateral force from the buoy s movements, can be obtained through strain measurements in the capsule. Strain gauge sensors are a common data unit to measure strain in this case. Bending stress in the capsule was calculated according to Eq σ b = σ 2 S g c 1 + σ 2 S g c 2 = E ɛ 2 S g c 1 + ɛ 2 S g c 2 = 2E ) ( V 2 ( ) o (S g c 1) V 2 o (S g c 2) + [Pa] (7.9) V s k r S g c 1 r S g c 2 σ b is the bending stress in the capsule, σ S g c 1 is the bending stress from strain gauge circuit 1, σ S g c 2 is the bending stress from strain gauge circuit 2, ɛ S g c 1 is the strain from strain gauge circuit 1, ɛ S g c 2 is the strain from strain gauge circuit Lateral force and azimuth angle As was mentioned earlier, the outer structure obtains a lateral force from the floating buoy, see Fig The lateral force as well as the azimuth angle between the generator and the floating buoy are of primary interest. The lateral force can be calculated by the stress in the capsule. Lateral force calculations were performed according to Fq F l = M b L = σ D bπ 4 d 4 32d L [N ], (7.10) where M b is a bending moment, σ b is a bending stress, S is a section modulus of the capsule, D,d are the outer and inner diameters of the cylindrical capsule, and L is the length from the strain gauge circuits to the upper flange of the cylindrical outer structure. 7.8 Inclination angle between the generator and the floating buoy That the angle between the generator and the floating buoy will vary depending on the sea state is evident. To physically measure the angle, i.e. by using measuring equipment, is a very difficult experiment. The angle is constantly changing. 54

55 A method for the estimation of inclination angle based on strain measurements in the capsule was developed, see Paper II. The inclination angle was calculated by the lateral force and the line force from the floating buoy. The lateral force was calculated according to Eq The line force from the floating buoy was measured with the force transducer HBM U2B 200 kn, see Fig The calculated lateral force on the funnel and the line force allow calculation of the inclination angle (θ) between the buoy and the linear generator, see Fig The inclination angle was calculated according to Eg ( ) Fl θ = ar csi n [deg r ee], F (7.11) where F l is the lateral force and F is the line force from the floating buoy. Figure 7.5: Force transducer HBM U2B 200 kn. 7.9 Position of the floating buoy on the water surface The calculated azimuth (φ) and inclination (θ) angles between the generator and the floating buoy synchronized with the measured vertical position of the buoy (r ) allow reproduction of the position of the floating buoy on the water surface, see Paper II. A spherical coordinate system can be used to plot the buoy s position. x = r sinθ cosφ y = r sinθ sinφ z = r cosφ, where r is a radial distance, θ is an inclination angle, φ is an azimuth angle, see Fig Figure 7.6: F is the line force, F l is the lateral force, θ is the inclination angle. 55

56 The position of a point is specified by three parameters: radial distance of the point from an origin, inclination angle, and azimuth angle. Figure 7.7: Spherical coordinates. 56

57 8. Results and discussion This chapter represents the measurement results as well as discussion. 8.1 Estimation of stress in the inner framework structure Strain measurements in the inner framework structure were synchronized with measurement of the position of the translator in the linear generator so as to obtain a maximum information about the stress. The translator position was measured by a standard wire sensor which was mounted on the upper end stop structure and attached to the top of the translator. The translator is adjusted to be located in the middle of the inner framework structure, opposite the stator at a sea state of 0 m waves. It corresponds to 0 on the measurement scale. Asymmetrical movements around 0 were depending on a tide water level. As the inner framework structure is symmetric on all four sides, a Cartesian coordinate system in two-dimensional space can be employed for description of the directions of loading. The X dimension corresponds to the horizontal direction. The Y dimension corresponds to the vertical direction. Fig. 8.1 and 8.2 present the stress along the Y axis where the translator moves up and down. Two different sea states are represented. The stress obtains the maximum value in the upper part of the inner framework structure where strain gauge circuits Ch. 3 and Ch. 6. were instrumented. Ch. 10 was mounted near the base of the inner framework structure. Such positioning of the strain gauge circuits explains the need for information about stress if the translator hits a lower end stop structure. Fig. 8.2 shows that at a sea state of 3.2 m, similar cogging curves have appeared. The most probable explanation for similar cogging curves is that the translator was pressing against the end stop spring. Stress values at those periods of time were corresponding to MPa. At normal working load, stress values corresponded to 4.5 MPa in both sea states. Fig. 8.3 and 8.4 show the stress in the framework crossbars. Under working load, the stress has varied depending on the position of the translator regarding to the stator. 57

58 Figure 8.1: Uniaxial stress in the corner pillars at a sea state of 1.32 m waves. σ y is the stress in the Y direction, L y is the position of the translator inside the generator. 0 corresponds to the middle position of the translator regarding to the stator structure. Channel 3 and 6 correspond to stress in the upper part of the structure, see Fig. 7.2(a). Ch. 2, 10 and 11 are from the middle and lower part of the structure, see Fig. 7.2(b)(c). (Results from Paper III). Figure 8.2: Uniaxial stress in the corner pillars at a sea state of 3.2 m waves. σ y is the stress in the Y direction, L y is the position of the translator inside the generator. It can be seen that similar cogging curves have appeared. (Results from Paper III). 58

59 Figure 8.3: Stress in the framework crossbar at a sea state of 1.32 m waves. σ x is the stress in the X direction, L y is the position of the translator inside the generator. 0 corresponds to the middle position of the translator regarding to the stator structure. Ch. 4 corresponds to the bending stress, Ch. 9 corresponds to the uniaxial stress. (Results from Paper III). Figure 8.4: Stress in the framework crossbar at a sea state of 3.2 m waves. σ x is the stress in the X direction. (Results from Paper III). 59

60 Moreover, the direction of the stress has varied. The stress obtained by Ch. 9 at a sea state of 1.32 m waves was compressive, see Fig The stress has varied in the range of -2 MPa, 0 MPa for Ch. 4 and in the range of -1.5 MPa, 0 MPa for Ch. 9. The compressive and tensile stress was obtained by Ch. 9 at a sea state of 3.2 m waves. Sudden jumps of the curve were registered when the translator passed 0 positions, see Fig The stress has varied in the range of - 2 MPa, + 1 MPa for Ch. 4 and in the range of -1.5 MPa, +1 MPa for Ch. 9. Simulated stress and measured stress were compared. Measured uniaxial stress matched the simulation. Measured bending stress matched the simulation. The uniaxial stress had a maximum value in the upper part of the corner pillars. 8.2 Forces and bending moments Fig. 8.5 shows the force and bending moments acting on the inner framework structure during system operation. The force varies only slightly at different sea states while bending moments acting on the corner pillar varies significantly. 8.3 Magnetic attractive force Fig. 8.6 shows the attractive force acting on the framework crossbar during system operation at two different sea states. The objective of the measurements included evaluation of how the lateral direction of the attractive force has been changing during the linear generator system operation at different sea states. The force obtained at a sea state of 1.32 m waves was compressive. The force has varied in the range of N, 0 N. The compressive and tensile force occurred at a sea state of 3.2 m waves. The force has varied in the range of N, N. 8.4 Estimation of stress in the capsule Fig. 8.7 presents time history of stress at sea states of 1.32 m and 3.2 m waves. The stresses are not of critical value even at a sea state of 3.2 m waves which can be defined as an abnormal condition because the amplitude of the translator under operation is 2 meters. 60

61 (a) (b) Figure 8.5: Force and bending moments that occurred due to the translator movements in the inner framework structure of the WEC under working load. F y corresponds to the force in the Y direction, L y is the position of the translator inside the generator, M y is a bending moment that was acting on the corner pillar, M x is a bending moment that was acting on the framework crossbar. a) corresponds to a sea state of 1.32 m waves, b) corresponds to a sea state of 3.2 m waves. (Results from Paper III). 61

62 (a) (b) Figure 8.6: The attractive forces F a acting on the framework crossbar during system operation at sea states of 1.32 m and 3.2 m waves. L y corresponds to the translator position inside the system during operation. a) corresponds to a sea state of 1.32 m waves, b) corresponds to a sea state of 3.2 m waves. 62

63 (a) Sea state of 1.32 m waves (b) Sea state of 3.2 m waves Figure 8.7: Figures show the stresses in the capsule under generator operation. Two different sea states are presented. 63

64 8.5 Lateral force and azimuth angle Fig. 8.8 represents the lateral force and the azimuth angle for two operational sea states. Fig. 8.9 shows the direction and magnitude of the loading on the funnel from the connection line for a period of 54 sec. The maximum lateral force acting on the wave energy converter was 3300 N at a sea state of 1.32 m waves and N at a sea state of 3.2 m waves. The experimental results allow evaluation of the most loaded part of the funnel. Consequently, it gives us very valuable information about the waves direction and the magnitude of the loading which are key factors for the design of the WEC. Moreover, an overturning moment acting on the generator under working load can be calculated through the lateral force from the floating buoy. 8.6 Inclination angle between the generator and the floating buoy The inclination angle between the generator and the floating buoy is shown in Fig Fig shows also the line force from the floating buoy and the position of the translator inside the inner framework structure for two different sea states for a period of 54 sec. The translator position was measured by a standard wire sensor which was mounted on the upper end stop structure and attached to the top of the translator. Fig focuses on the estimation of the inclination angle, but for better visualization of the results, the author has included the position of the translator and the line force. It is very clear that the inclination angle becomes very large when the translator is on the way down and the line force is dropped to 0. An explanation for this can be found in the next section. However, the inclination angle, when the line force was larger than the force caused by the translator weight, was not significant and did not exceed 10 degrees even for a sea state of 2 m waves. The inclination angle is one of the general parameters for the design of the outer structure as well as for the design of the concrete foundation. 8.7 Snatch load The connection line of the WEC suffers from snatch loads following any slackening. Slackening occurred when the translator was on the way down and the line force dropped to 0. 64

65 (a) Sea state of 1.32 m waves. The lateral force on the funnel is largest at an azimuth angle of 92 degrees, see also Fig. 8.9 (b) Sea state of 3.2 m waves. The lateral force on the funnel is largest atan azimuth angle of 290 degrees, see also Fig. 8.9 Figure 8.8: (F l ) is the lateral force on the funnel from the floating buoy. (φ) is the azimuth angle between the floating buoy and the generator. (Results from Paper II). 65

66 (a) (b) Figure 8.9: Direction and magnitude of the loading on the funnel from the connection line. Two sea states are represented: (a) 1.32 m waves (b) 3.2 m waves. (Results from Paper II). 66

67 (a) Sea state of 1.32 m waves. (b) Sea state of 2 m waves. Figure 8.10: Experimental results from the research site. F corresponds to the line force from the floating buoy, Position corresponds to the position of the translator inside the inner framework structure, and θ corresponds to the inclination angle between the generator and the floating buoy. (Results from Paper II). 67

68 The experiments were conducted under different sea states corresponding to 1.32 m waves and 2 m waves. The results from the previous section demonstrate the sudden jump of the inclination angle, see Fig Fig shows a detailed investigation of a sudden jump of the inclination angle. Three cases are represented. Case 1 corresponds to the inclination angle when the line force from the floating buoy is lower than the force caused by the line weight; case 2: the line force is between the force caused by the line weight and the force caused by the translator weight; case 3: the line force is higher than the force caused by translator weight. Fig. 8.11(a) shows an interesting observation. The translator has already passed the equilibrium position between the translator and the stator when an sudden jump of the inclination angle occurred. The equilibrium position corresponds to the zero scale on the diagram for the position of the translator. It can be assumed that the sudden jump of the inclination angle occurred due to the connection line hanging slack. This could happen if the translator moved down more slowly than the buoy followed the wave motion downwards. Electromagnetic damping could be one of the major factors that affected the translator motion. The connection line hung slack for a moment until the force caused by the mass of the translator and the spring forces pulled on the wire. Fig. 8.11(b) shows clearly that the sudden jump of the inclination angle occurred when the line force from the floating buoy had fallen appreciably. Fig. 8.11(c) shows that the signal from strain gauge sensors mounted on the capsule had a pronounced tendency to oscillate which is typical for a force ripple (cogging) on the translator due to electromagnetic damping. Fig represents the generated power, position and inclination angle. It can be seen clearly that the sudden jump of the inclination angle occurred when the generated power reached a value of 20 kw, which is 2 times higher then the nominal value of the generated power. It can be assumed that a snatch load appeared during the generator s operation. Hence, the jump of the inclination angle can be assumed to be a snatch load angle between the connection line and the funnel. 8.8 Position of the floating buoy on the water surface Fig shows a reproduction of the position of the floating buoy on the water surface at a sea state of 1.32 m and 2 m waves for a period of 54 seconds. R x and R y correspond to the motion of the buoy on the water surface in the x and y directions, see Fig

69 0 angle (case 1) angle (case 2) angle (case 3) position Angle (degree) Position (mm) Time (s) -400 (a) angle (case 1) angle (case 2) angle (case 3) axial force Angle (degree) Line force (N) Time (s) (b) angle (case 1) angle (case 2) angle (case 3) lateral force Angle (degree) Lateral force (N) Time (s) (c) Figure 8.11: Inclination angle between the buoy and the generator with: (a) the position of the translator inside the generator, (b) the line force from the floating buoy and (c) the lateral force on the funnel. Case 1: the inclination angle when the line force from the floating buoy is lower than the force caused by the line weight; case 2: the line force is between the force caused by the line weight and the force caused by the translator weight; case 3: the line force is larger than the force caused by the translator weight. The zero scale for the position of the translator corresponds to the middle position of the translator with regards to the stator. (Results from Paper II). 69

70 (a) The sudden jump of the inclination angle (b) Narrower time span Figure 8.12: Experimental results from the research site. Power corresponds to the generated power, Position corresponds to the position of the translator inside the inner framework structure, and θ corresponds to the inclination angles between the generator and the floating buoy. 70

71 Reciprocating movements in the z direction were taken from the difference in the position of the translator inside the generator plus the length of the connection line outside the funnel. (a) A sea state of 1.32 m waves (b) A sea state of 2 m waves Figure 8.13: Reproduction of the position of the floating buoy on the water surface at two different sea states for a period of 54 sec. Case 3 for the inclination angle was applied, see Fig R x and R y are corresponding to the x and y directions according to a spherical coordinate system, see Fig (Results from Paper II). The majority of the data points for the position of the floating buoy on the water surface are located within a distance of 1.8 meter in the x direction and within a distance of 2 meter in the y direction for a sea state of 1.32 m waves and within a distance of 1.2 meter in the x direction and within a distance of 4.5 meter in y the direction for a sea state of 2 m waves. 71

72 8.9 Error estimation Error estimation of the stresses was made according to Eq For example, Fig shows the results of error estimation for Ch. 6 at two different sea states. It gives an idea of how accurate the measurements are. Fig shows the result of the error estimation for Ch. 11 under 2.5 seconds for better visualization of the error. (a) (b) Figure 8.14: The figures show the accuracy of the measurements. For example, channel 6 is presented for two different sea states. (a) 1.32 m waves. (b) 3.2 m waves. The figures indicate the possible maximum and minimum values of the measured variable. (Results from Paper III). 72

73 Figure 8.15: Accuracy of the measurement for Ch. 11 at a sea state of 3.2 m waves under 2.5 seconds. The green lines are corresponding to (+), (-) errors. (Results from Paper III). 73