Physical and Mathematical Modeling of a Wave Energy Converter Equipped with a Negative Spring Mechanism for Phase Control
Abstract
:1. Introduction
2. Experimental Setup
- The spring’s stiffness, k;
- The initial compression force of the spring, which is a function of its natural length , its actual length l, and the stiffness k, ;
- The angular displacement of arm () as a function of angular displacement of the arm axis (), cf. Figure 2;
- The change in distance as a function of angular displacement of the axis ().
3. Mathematical Model
3.1. Equation of Motion
3.2. Hydrodynamic Parameters
3.3. Optimal Load Resistance for Unconstrained Oscillation
3.4. Optimal Load Resistance for Constrained Oscillation
4. Experimental Testing Conditions
4.1. Model Variations
4.2. Wave Conditions
5. Results and Discussion
5.1. Uncertainty
5.2. Hydrostatic Stiffness
5.3. Friction
5.4. WaveSpring Torque
5.5. Actuator Reference Following
5.6. Regular Wave Results
5.6.1. Motion Response in Regular Waves
5.6.2. Power Absorption in Regular Waves
- Wave energy: Total energy taken from the wave field.
- Total absorbed energy: Total energy absorbed into the mechanical system from the water.
- Net absorbed energy: Absorbed energy as measured on the load cell between the actuator and the rotating arm that holds the buoy. Losses between total absorbed energy and net absorbed energy include friction losses in the shaft bearings and the WaveSpring mechanism.
- Converted energy: Derived quantity based on subtracting assumed losses between net absorbed energy and electrical energy output, notably losses in the electric generator and the pretension system. The generator losses are estimated based on the conversion efficiency for a typical electric generator. When the system works with pretension, some energy will be lost in the pretension mechanism. This loss is computed as a loss fraction on the pretension torque, .
5.6.3. Mathematical Model Validation
5.7. Irregular Wave Results
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. List of Tests Performed during the Experimental Campaign
Run ID | Setup | Wave Type | [Nm/rad] | [Nm s/rad] | [Nm] |
---|---|---|---|---|---|
A001 | Setup 1 | Rf1 | 0 | 3.64 | 0 |
A002 | Setup 1 | Rf2 | 0 | 3.53 | 0 |
A003 | Setup 1 | Rf3 | 0 | 7.31 | 0 |
A004 | Setup 1 | Rf4 | 0 | 11.30 | 0 |
A005 | Setup 1 | Rf5 | 0 | 15.12 | 0 |
A006 | Setup 1 | Rf6 | 0 | 18.78 | 0 |
A007 | Setup 1 | Rf7 | 0 | 22.31 | 0 |
A008 | Setup 1 | Rf8 | 0 | 25.73 | 0 |
A009 | Setup 2 | S2 | 0 | 11.56 | 8 |
A010 | Setup 2 | Rf2 | 0 | 6.11 | 8 |
A011 | Setup 2 | Rf4 | 0 | 13.12 | 8 |
A012 | Setup 2 | Rf6 | 0 | 19.95 | 8 |
A013 | Setup 2 | S2 | −59 | 6 | 8 |
A014 | Setup 2 | S2 | −40 | 4 | 8 |
A015 | Setup 2 | S2 | −59 | 4 | 8 |
A016 | Setup 3 | Rf1 | 0 | 1.00 | 7.4 |
A017 | Setup 3 | Rf1 | 0 | 2.00 | 7.4 |
A018 | Setup 3 | Rf1 | 0 | 4.00 | 7.4 |
A019 | Setup 3 | Rf1 | 0 | 6.00 | 7.4 |
A020 | Setup 3 | Rf2 | 0 | 1.00 | 7.4 |
A021 | Setup 3 | Rf2 | 0 | 2.00 | 7.4 |
A022 | Setup 3 | Rf2 | 0 | 4.00 | 7.4 |
A023 | Setup 3 | Rf2 | 0 | 6.00 | 7.4 |
A024 | Setup 3 | Rf3 | 0 | 1.00 | 7.4 |
A025 | Setup 3 | Rf3 | 0 | 2.00 | 7.4 |
A026 | Setup 3 | Rf3 | 0 | 4.00 | 7.4 |
A027 | Setup 3 | Rf3 | 0 | 6.00 | 7.4 |
A028 | Setup 3 | Rf4 | 0 | 1.00 | 7.4 |
A029 | Setup 3 | Rf4 | 0 | 2.00 | 7.4 |
A030 | Setup 3 | Rf4 | 0 | 4.00 | 7.4 |
A031 | Setup 3 | Rf4 | 0 | 6.00 | 7.4 |
A032 | Setup 3 | Rf5 | 0 | 1.00 | 7.4 |
A033 | Setup 3 | Rf5 | 0 | 2.00 | 7.4 |
A034 | Setup 3 | Rf5 | 0 | 4.00 | 7.4 |
A035 | Setup 3 | Rf5 | 0 | 6.00 | 7.4 |
A036 | Setup 3 | Rf6 | 0 | 1.00 | 7.4 |
A037 | Setup 3 | Rf6 | 0 | 2.00 | 7.4 |
A038 | Setup 3 | Rf6 | 0 | 4.00 | 7.4 |
A039 | Setup 3 | Rf6 | 0 | 6.00 | 7.4 |
A040 | Setup 3 | Rf7 | 0 | 1.00 | 7.4 |
A041 | Setup 3 | Rf7 | 0 | 2.00 | 7.4 |
A042 | Setup 3 | Rf7 | 0 | 4.00 | 7.4 |
A043 | Setup 3 | Rf7 | 0 | 6.00 | 7.4 |
A044 | Setup 3 | Rf8 | 0 | 1.00 | 7.4 |
A045 | Setup 3 | Rf8 | 0 | 2.00 | 7.4 |
A046 | Setup 3 | Rf8 | 0 | 4.00 | 7.4 |
A047 | Setup 3 | Rf8 | 0 | 6.00 | 7.4 |
A048 | Setup 3 | S1 | 0 | 6.00 | 7.4 |
A049 | Setup 3 | S2 | 0 | 3.37 | 7.4 |
A050 | Setup 3 | S3 | 0 | 3.00 | 7.4 |
A051 | Setup 3 | S4 | 0 | 10.00 | 7.6 |
A052 | Setup 3 | Sweep | 0 | 3.5 | 7.6 |
A053 | Setup 4 | S1234 | 0 | 4 | 7.6 |
A054 | Setup 4 | Sweep1min | 0 | 4 | 7.6 |
A055 | Setup 4 | Rf1 | 0 | 4 | 7.6 |
A056 | Setup 4 | Rf2 | 0 | 4 | 7.6 |
A057 | Setup 4 | Rf3 | 0 | 4 | 7.6 |
A058 | Setup 4 | Rf4 | 0 | 4 | 7.6 |
A059 | Setup 4 | Rf5 | 0 | 4 | 7.6 |
A060 | Setup 4 | Rf6 | 0 | 4 | 7.6 |
A061 | Setup 4 | Rf7 | 0 | 4 | 7.6 |
A062 | Setup 4 | Rf8 | 0 | 4 | 7.6 |
A063 | Setup 4 | S1 | 0 | 4 | 7.6 |
A064 | Setup 4 | S2 | 0 | 4 | 7.6 |
A065 | Setup 4 | S3 | 0 | 4 | 7.6 |
A066 | Setup 5 | S2 | 0 | 4 | −0.6 |
A067 | Setup 5 | Rf2 | 0 | 6.90 | −0.6 |
A068 | Setup 5 | Rf4 | 0 | 2.45 | −0.6 |
A069 | Setup 5 | Rf6 | 0 | 1.72 | −0.6 |
A070 | Setup 5 | S1234 | 0 | 4 | −0.6 |
A071 | Setup 5 | Sweep1min | 0 | 4 | −0.6 |
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Quantity | Unit | Setup 2/3 | Setup 4 | Setup 1/5 |
---|---|---|---|---|
Mass of buoy—lever and attachments | kg | 2.564 | 2.564 | 4.210 |
Mass of WaveSpring bracket | kg | 0.027 | 0.027 | 0.027 |
Estimated moment of inertia | kg m | 0.430 | 0.430 | 0.900 |
Buoy diameter | m | 0.263 | 0.263 | 0.263 |
Buoy total volume | dm | 7.28 | 7.28 | 7.28 |
Buoy submerged volume (hemisphere) | dm | 3.45 | 3.45 | 3.45 |
Waterline beam | m | 0.256 | 0.256 | 0.256 |
Height of cylindrical part of bouy | m | 0.050 | 0.050 | 0.050 |
Height of point A above MWL | m | 0.285 | 0.285 | 0.285 |
Distance : | m | 0.510 | 0.510 | 0.510 |
Distance | m | 0.470 | 0.470 | 0.470 |
Distance | m | 0.400 | 0.405 | 0.405 |
Distance | m | 0.320 | 0.320 | 0.320 |
Distance | m | 0.090 | 0.090 | 0.090 |
Distance : | m | 0.306 | 0.306 | 0.400 |
Angle of line with horizontal | deg | −34.0 | −34.0 | −34.0 |
Angle of line with horizontal | deg | −28.5 | −28.5 | −25.9 |
Coil spring natural length | m | 0.165 | 0.165 | 0.165 |
Shortest spring length | m | 0.130 | 0.135 | 0.135 |
Coil spring number of turns | m | 58 | 58 | 58 |
Coil spring thread diameter | mm | 1.50 | 1.50 | 1.50 |
Inner coil diameter | mm | 7.0 | 7.0 | 7.0 |
Coil spring fabricant | —– Lesjøfors 3992 —– | |||
Approximate spring stiffness | kN/m | 1.32 | 1.32 | 1.32 |
Spring stiffness estimated from measurements, | kN/m | 1.50 | 1.50 | 1.50 |
Acronym | Description | Mass of the Buoy [kg] | Shortest Spring Length [m] |
---|---|---|---|
Setup 1 | No WaveSpring, large mass | 4.210 | - |
Setup 2 | No WaveSpring, small mass | 2.564 | - |
Setup 3 | With WaveSpring, small mass | 2.564 | 0.130 |
Setup 4 | Softer WaveSpring, small mass | 2.564 | 0.135 |
Setup 5 | Softer WaveSpring, large mass | 4.210 | 0.135 |
Name | Wave Period | Wave Height | ||
---|---|---|---|---|
Target | Measured | Reference | ||
Rf1 | 0.7 | 0.0300 | 0.0293 | 0.0250 |
Rf2 | 0.9 | 0.0300 | 0.0270 | 0.0250 |
Rf3 | 1.1 | 0.0300 | 0.0250 | 0.0250 |
Rf4 | 1.3 | 0.0300 | 0.0250 | 0.0250 |
Rf5 | 1.5 | 0.0300 | 0.0230 | 0.0250 |
Rf6 | 1.7 | 0.0300 | 0.0236 | 0.0250 |
Rf7 | 1.9 | 0.0300 | 0.0236 | 0.0250 |
Rf8 | 2.1 | 0.0300 | 0.0225 | 0.0250 |
Seastate | |||
---|---|---|---|
S1 | 0.0193 | 0.778 | 0.869 |
S2 | 0.0556 | 1.029 | 1.205 |
S3 | 0.0895 | 1.334 | 1.596 |
Measurement | Unit | Uncertainty |
---|---|---|
Nm | 0.05 | |
v | rad/s | 0.03 |
rad/s | 0.02 |
Configuration | Stiffness [Nm/rad] | Resonance Period [s] | ||||
---|---|---|---|---|---|---|
Mean | Peak | at | ||||
Setup 1 | 88.5 | 95.6 | 91.5 | 0.807 | 0.770 | 0.791 |
Setup 2 | 85.4 | 93.8 | 89.3 | 0.637 | 0.599 | 0.619 |
Setup 3 | 33.4 | 9.13 | 21.4 | 1.15 | 2.31 | 1.47 |
Setup 4 | 42.3 | 23.2 | 29.0 | 1.00 | 1.41 | 1.25 |
Setup 5 | 45.2 | 26.3 | 32.8 | 1.18 | 1.58 | 1.40 |
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Têtu, A.; Ferri, F.; Kramer, M.B.; Todalshaug, J.H. Physical and Mathematical Modeling of a Wave Energy Converter Equipped with a Negative Spring Mechanism for Phase Control. Energies 2018, 11, 2362. https://doi.org/10.3390/en11092362
Têtu A, Ferri F, Kramer MB, Todalshaug JH. Physical and Mathematical Modeling of a Wave Energy Converter Equipped with a Negative Spring Mechanism for Phase Control. Energies. 2018; 11(9):2362. https://doi.org/10.3390/en11092362
Chicago/Turabian StyleTêtu, Amélie, Francesco Ferri, Morten Bech Kramer, and Jørgen Hals Todalshaug. 2018. "Physical and Mathematical Modeling of a Wave Energy Converter Equipped with a Negative Spring Mechanism for Phase Control" Energies 11, no. 9: 2362. https://doi.org/10.3390/en11092362
APA StyleTêtu, A., Ferri, F., Kramer, M. B., & Todalshaug, J. H. (2018). Physical and Mathematical Modeling of a Wave Energy Converter Equipped with a Negative Spring Mechanism for Phase Control. Energies, 11(9), 2362. https://doi.org/10.3390/en11092362