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Wave Energy Chapter | 5 131

WECs, in general, are underdamped. In the absence of external wave forces,
a WEC will oscillate and gradually stop due to frictional damping. The solution
of an underdamped system can be found using Eq. (5.54). In order to better
formulate the solution, we define the damping ratio (ζ)as
c d c d c d
ζ = * = = √ → ζ< 1 ≡ Δ< 0 (5.56)
c 2mω N 2 mk
d
Using Eq. (5.54) and the previous definitions for damping ratio and natural
frequency, it can easily be shown that

√
ξ =−ζω N ± −1 1 − ζ 2 ω N (5.57)
and the solution of an underdamped system becomes

ξt
2
z = z o e = z o e −ζω N t cos 1 − ζ ω N t + g 1 (5.58)
where z o and g 1 are constants, and depend on the initial conditions. When the
damping ratio is zero, the previous solution reduces to an undamped system
(Eq. 5.55). When the damping ratio is between 0 and 1 (i.e. underdamped
system), the term e −ζω N t leads to an exponential decay of the amplitude as can
be seen in Fig. 5.12. Finally, when the damping ratio is 1 (i.e. critically damped),
the solution is z o cos(g 1 )e −ζω N t , which is an exponential decay line.

Analytical Solution of Forced Vibration
In the presence of an external force, the solution of the differential equation
is a combination of the solution for free vibration and a particular solution. In
the theory of ordinary differential equations, the solution of the free vibration
(RHS = 0) is also called homogeneous solution, and the forced vibration is called
nonhomogeneous solution (RHS = 0).
In a similar way to free vibration, the solution of the forced vibration case
ξt
(Eq. 5.51) can also be found by replacing z = z 1 e = z 1 e iω F t or z = z 1 sin(ω F t+
g F ). Note that here the solution frequency is ω F , which is the frequency of the
external force. We do not present the steps for this case, as it is very similar to
what we saw in the free vibration case.
F a
z = z 1 sin(ω F t + g F ) where z 1 = and
2
2
2 2
m (ω − ω ) + (c d ω F ) 2
N F
c d ω F
tan g F =− (5.59)
2
2
m(ω − ω )
N F
In the previous equation, z 1 is the amplitude of the vibration or response of the
system to the external forcing. As we can see, this amplitude is dependent on
the frequency of the external force compared with the natural frequency of the
system, and also the damping coefficient. For instance, we can see that as ω F