Page 138 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Wave Energy Chapter | 5 129

Assuming just one degree of freedom for motion (e.g. a heaving point
absorber), the equation of motion can be written as

F = ma z =−k s z + F damp + F ext (t) (5.50)
where k s is the spring constant, m is the mass, and a z is acceleration, and F damp
and F ext are damping and external forces, respectively. The damping force is
dz
usually proportional to the velocity and is expressed as c d u z = c d , where c d
dt
is velocity in the z direction. The spring force is the restoring force, which is a
combination of Archimedes/Buoyancy force and gravity, which tends to return
the system to its equilibrium position. Therefore, the single degree of freedom
(SDOF), mass-spring-damper equation can be written as
2
d z dz
m + c d + k s z = F a sin(ω F t + φ) (5.51)
dt 2 dt
Note that in the earlier equation, we assumed that the external force is a simple
harmonic force with an amplitude of F a and angular frequency of ω F .This
makes sense if we assume that the force is generated by a harmonic wave.
Eq. (5.51) is a linear ordinary differential equation, and so can be solved
ξt
analytically. The solution can be found by replacing z = z o e into the earlier
equation, where z o is a constant.

Analytical Solution of Free Vibration
First, consider a case where the external force is zero. We refer to that case
as free vibration. This will be equivalent to a WEC in a calm sea, which is
displaced from its equilibrium position. It will gradually be brought back to its
equilibrium position after a few oscillations. The governing equation for free
vibration reduces to
2
d z dz
m + c d + kz = 0 (5.52)
dt 2 dt
ξt
Replacing z o e into the previous equation, leads to
2 ξt
mξ + c d ξ + k s z o e = 0 (5.53)
2
ξt
Because z o e is not zero, mξ + c d ξ + k s should be zero. Using the quadratic
formula, the solution becomes

2
−c d ± c − 4mk s
d 2
ξ = and Δ = c − 4mk s (5.54)
d
2m
Consider a special case, where there is no damping (undamped system: c d = 0).
For an undamped system,
√
0 ± 0 − 4mk s k s
ξ = = −
2m m