For the system whose output pitch displacement contour lines are horizontal, Figure 8.14-(a) and the deep CoM values in Figures 8.13-(a), the peak frequency of the PDTF occurs at very high frequencies where the input PM spectrum amplitudes are essentially zero. The amplitude of the PDTFs for the different CoM values are then equal in the lower frequency regions where the input PM spectrum is significant, therefore the output pitch displacements for these systems do not depend on the CoM, resulting in the horizontally banded contour lines.
Figure 8.15: PDTF for a 20kg buoy with 0.10m and 0.20m radius, for various CoM and MoI values.
the maximum output power for the two cases is equivalent. At a radius value of 0.14m, whereωnb= 3.9 rad/s andωng= 5.9 rad/s, the constrained translator mass case reaches its largest maximum power output of about 2W and the two cases begin diverging.
0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21
1 1.5 2 2.5
Buoy radius (m) Power (W) m<MTotal
m<0.5M
Total
Figure 8.16: Maximum output power for 40kg bouy for varying radius values.
The output power contour plot for the 0.14m radius 40kg WEC, as a function of generator damping and translator mass (constrained to less that 50% of the the total mass), is shown in Figure 8.17. Here it can be seen that the maximum output power of 2W occurs for generator damping values of 20 - 60 Ns/m combined with translator masses 35 - 50% of the total WEC mass. Figure8.17also shows that the output power exceeds 1W for a very large range of generator damping and translator mass values, spanning from generator damping values of 3Ns/m combined with translator mass values over 5%
of the total WEC mass up to generator damping values of 3,000Ns/m combined with translator mass values 50% of the total WEC mass.
0.2 0.4
0.6 0.8 1 1.2 1.4
1.6
Generator damping (Ns/m) Ratio of translator mass to total mass
100 101 102 103 104
0.1 0.2 0.3 0.4 0.5
Power(W)
0.5 1 1.5 2
Figure 8.17: Output power as a function of generator damping and mass for a 40kg buoy with 0.14m radius (ωnb 3.9 rad/s,ωng 5.9 rad/s).
8.3.2 Stroke
Figure8.18displays a contour plot of the RMS stroke displacement for the 0.14m radius 40kg CIPMLG WEC. Like the 0.10m radius 20kg CIPMLG WEC in Figure 8.8, the maximum RMS stroke displacement occurs when the generator is very lightly damped, but for the present 40kg WEC the maximum stroke displacement is slightly less with a value of 0.20m. Once again, the RMS stroke displacement is seen to decrease with
increasing generator damping, with a value of about 0.05m at 30 Ns/m, where the maximum output power occurs in Figure8.17.
The maximum allowable stroke length for the 0.14m radius 40kg buoy with a draught of 0.64m, equals 0.32m. Therefore the RMS stroke displacements in Figure 8.18 do not exceed the maximum allowable stroke length. Similar to the time domain analysis results for the 0.10m radius 20kg CIPMLG WEC, the stroke displacement in the time domain for the 0.14m radius 40kg WEC exceeds the RMS value on average 33% of the time and the maximum stroke lengths are on average 2.9 times larger than the RMS values.
Therefore, the stroke displacements for all the systems in Figure 8.18 with generator damping values larger than 7Ns/m will remain less than the maximum allowable stroke length.
0.02
Generator damping (Ns/m) Ratio of translator mass tot toal mass
100 101 102 103 104
0.1 0.2 0.3 0.4 0.5
RMS Stroke (m)
0.05 0.1 0.15
Figure 8.18: RMS stroke displacement as a function of generator damping and trans-lator mass for a an 40kg buoy with 0.14m radius.
8.3.3 Pitch
A similar method is used to analyse the pitch for the 0.14m radius 40kg CIPMLG WEC, as used for the 0.10m radius 20kg CIPMLG WEC in Section 8.2.3. Here, the range of MoI values are chosen considering a 40kg ball of steel, I = 0.16kgm2 and for a point mass of 40kg located the draught length of 0.64m away from the CoM, I = 16kgm2. From these two extreme cases, the analysis selects MoI values of 0.16, 1.6 and 16 kgm2 for the 0.14m radius 40kg CIPMLG WEC, displayed in Figures 8.19 -(a), (b) and (c), respectively.
Using time domain analysis, the maximum pitch displacement for the 0.14m radius 40kg WEC was found to be on average up to 3.6 times larger than the RMS pitch displacement, similar to the results shown in Figure 8.12 for the 0.10m radius 20kg CIPMLG WEC.
Therefore, once again the RMS pitch displacement should remain below 11 degrees to satisfy the pitch constraint. Except for the systems with very large MoI values and CoM near the centre of the buoy, shown in Figure 8.19-(c), the majority of the systems in Figure 8.19would not be able to satisfy the pitch constraint.
1.4
1.61.4 22.21.8 2.4 2.6
(a) I=0.16kgm2
5 10 15
1.5 2 2.5
11.21.4
1.6 1.6
1.8
2
2
2.2
2.4
(b) I=1.6kgm2
5 10 15
log 10(RMS pitch displacement) (deg) 1 1.5 2
0
0.5 0.5
1 1
1.5 1.5
2 2
2
2.5
2.5 3
Ratio of CoM to draught
Wind speed (m/s)
(c) I=16kgm2
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
5 10 15
0 1 2 3
Figure 8.19: RMS pitch displacement, for a 40kg CIPMLG WEC with 0.14m radius, as a function of CoM depth and input PM spectrum sea state parameterised by the
wind speed.
8.3.3.1 Increasing the radius
Here the radius of the 40kg CIPMLG WEC is increased to 0.20m in Figure8.20 and to 0.25m in Figure8.21. Similar to the situation in Section8.2.3.1, increasing the radius is seen to decrease the pitch displacement of the CIPMLG WEC. However, increasing the radius also decreases the output power, with a maximum power outputs for the 0.20m and 0.25m radius CIPMLG WECs of 1.24W and 0.66W respectively.
1 1
1.05 1.05
1.1 1.1
1.1
1.15 1.15
1.15 1.2 1.25 1.3 1.3
5 1.4
(a) I=0.16kgm2
5 10 15
1 1.1 1.2 1.3 1.4
1.5 1.4 1.6
1.7 1.8 1.9
2
2
2.1
2.1
(b) I=1.6kgm2
5 10 15
log 10(RMS pitch displacement) (deg) 1.4 1.6 1.8 2
0
0.5 0.5
1 1
1
1.5 1.5
1.5
2 2
2 2
Ratio of CoM to draught
Wind speed (m/s)
(c) I=16kgm2
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
5 10 15
0 0.5 1 1.5 2
Figure 8.20: RMS pitch displacement, for a 40kg buoy with 0.20m radius, as a function of CoM and PM spectrum sea state parameterised by the wind speed.
0.8 0.8
0.8 0.85 0.85 0.85 0.85
0.85 0.9 0.9 0.9 0.9
0.9 0.95 0.95 0.95 0.95
0.95
1 1
1 1 1
1.05
(a) I=0.16kgm2
5 10 15
0.8 0.9 1
1.3 1.4
1.6 1.5 1.7
1.8 1.9
(b) I=1.6kgm2
5 10 15
log 10(RMS pitch displacement) (deg) 1.4 1.6 1.8
00.5 0.5
1 1 1
1.5 1.5 1.5
2 2
2
2
2.5 2.5
2.5
3
3
2 2
Ratio of CoM to draught
Wind speed (m/s)
(c) I=16kgm2
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
5 10 15
0 1 2 3
Figure 8.21: RMS pitch displacement, for a 40kg buoy with 0.25m radius, as a function of CoM and PM spectrum sea state parameterised by the wind speed.