Page 143 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

134 Fundamentals of Ocean Renewable Energy

absorber from its equilibrium position. If no external force is applied to the
point absorber, it will gradually lose its energy and stop after a few oscillation
cycles. During oscillation, waves are radiated from the object. These radiated
waves are taking energy off the point absorber. According to linear wave theory,
the radiating force can be approximated as follows
2
d z dz
F rad =−(m add a z + b hd u z ) =− m add 2 + c hd (5.62)
dt dt
in which the force consists of two parts: added mass inertial force and
hydrodynamic damping force. The added mass m add and b hd are dependent on
the shape of the point absorber. They are determined by experiment or numerical
simulation.
To formulate the restoring force, assume that there is no wave (calm sea) and
the point absorber is in equilibrium. The downward force is the weight of the
point absorber, and the upward force is the hydrostatic pressure or Archimedes
force. When the point absorber is displaced from its equilibrium position, the
difference of the Archimedes force and weight tends to bring the system back
to equilibrium, and acts as a spring. If the displacement from the equilibrium
position is denoted by z, the restoring force would be equal to the mass of
the displaced water according to Archimedes law. If the cross-sectional area
of the point absorber is A a (waterline area), the weight of the displaced water, or
the restoring force, would be

F s =−mg = ρV d g =−ρA a zg = k s z and k s = ρA a g (5.63)
The power take-off force, which is used to generate electricity, can be
linearly formulated similar to a damping force as follows:
dz
F power take-off =−c pto (5.64)
dt
To have more control on the vibration of the point absorber, and to maximize
the energy output, we can use tuning or an additional added mass. By using this
additional mass, we can change the natural frequency of the system and control
the phase lag between the vibration of the point absorber and waves. The inertia
force of this additional mass is
dz 2
F tuning = m tun 2 (5.65)
dt
Replacing all forces into Eq. (5.61), leads to
2
dz 2 d z dz dz
dz 2
m = F w sin(ω F t + g F ) − k s z − m add + c hd + c pto − m tun
dt 2 dt 2 dt dt dt 2
(5.66)
which after rearranging becomes
dz 2 dz
{m + m add + m tun } + b hd + b pto + k s z = F w sin(ω F t + g F ) (5.67)
dt 2 dt