NONLINEAR RESPONSES FOR A SALM BUOY
135
The compatible value of the drag coefficient in terms of the damping factor and
the other System constants is
y/aoh^Jocos2
For this study 5 percent damping was chosen, or ( = 0.05.
The last force component needed in équation (5.101) is that due to wave
excitation, or pi(t). The assumptions used to model this force are summarized.
The wave is a simple one based on linear theory with characteristics as described
in Table 3.1. The compatibility équation relating the wave frequency w, the wave
number k, and water depth d is given by équation (3.16). The wave loading on
the buoy is dominated by the inertia term q; in Morison’s équation, where the
ratio of buoy diameter to wave height falls in the range 0.5 < D/H < 1. The
inertia coefficient has an average value of Cm — 2. The horizontal wave particle
accélération û, evaluated at x = 0, is used to calculate the total horizontal wave
load on the buoy. When qj of équation (2.7) is integrated over the submerged
height of the buoy, this load is calculated as
Pi(t) = / qjdz=^-CMpD2 I ûdz = ——Cm pD2œ2 H sin ait (5.108)
J-d
4
J_d
8k
The location of this load from the sea floor is
hw — d + z
where its location from the still water line is
£ _ f°d zQi
1 — cosh kd
k sinh kd
(5.109)
(5.110)
It is noted that z is négative because it is below the still water line, and the
coordinate z is measured positive upward from the still water line.
When équations (5.102)-(5.104), (5.108), and (5.109) are substituted into the
original équation of motion (5.101), the resuit gives the final nonlinear model as
Joè + (a0h2 cos2 <ï>)0 - (aoh‘ c sin cos )010|
+(50/ic cos4
~ boh* sin cos3 1^1
+CDh2 dd = Çd+ z)^CMpD2a;2H sin wt
(5.111)
In this model, the absolute value sign was used on the even-order restoring force
terms to preserve the proper sign of that force, or to assure that the restoring
force is always antisymmetric about 0 = 0. For a fixed wave frequency and water
depth, k is calculated from équation (3.16), and Z is calculated from équation
135
The compatible value of the drag coefficient in terms of the damping factor and
the other System constants is
y/aoh^Jocos2
For this study 5 percent damping was chosen, or ( = 0.05.
The last force component needed in équation (5.101) is that due to wave
excitation, or pi(t). The assumptions used to model this force are summarized.
The wave is a simple one based on linear theory with characteristics as described
in Table 3.1. The compatibility équation relating the wave frequency w, the wave
number k, and water depth d is given by équation (3.16). The wave loading on
the buoy is dominated by the inertia term q; in Morison’s équation, where the
ratio of buoy diameter to wave height falls in the range 0.5 < D/H < 1. The
inertia coefficient has an average value of Cm — 2. The horizontal wave particle
accélération û, evaluated at x = 0, is used to calculate the total horizontal wave
load on the buoy. When qj of équation (2.7) is integrated over the submerged
height of the buoy, this load is calculated as
Pi(t) = / qjdz=^-CMpD2 I ûdz = ——Cm pD2œ2 H sin ait (5.108)
J-d
4
J_d
8k
The location of this load from the sea floor is
hw — d + z
where its location from the still water line is
£ _ f°d zQi
1 — cosh kd
k sinh kd
(5.109)
(5.110)
It is noted that z is négative because it is below the still water line, and the
coordinate z is measured positive upward from the still water line.
When équations (5.102)-(5.104), (5.108), and (5.109) are substituted into the
original équation of motion (5.101), the resuit gives the final nonlinear model as
Joè + (a0h2 cos2 <ï>)0 - (aoh‘ c sin
+(50/ic cos4
~ boh* sin cos3 1^1
+CDh2 dd = Çd+ z)^CMpD2a;2H sin wt
(5.111)
In this model, the absolute value sign was used on the even-order restoring force
terms to preserve the proper sign of that force, or to assure that the restoring
force is always antisymmetric about 0 = 0. For a fixed wave frequency and water
depth, k is calculated from équation (3.16), and Z is calculated from équation