114
SINGLE DEGREE OF FREEDOM STRUCTURES
ship is displaced first in the surge direction by an amplitude of 3.0 ft from
its equilibrium State and then is released from rest. Then repeat the saine
calculation for the sway State. The numerical values of
and fc3 for each
motion are given as the coefficients of équations (2.72) and (2.73). The results
for surge motion are
Wq
20,300
3.52 x 105
= 0.240 rad/sec
i
. 0.2402 + 3(40Q)32
= 0.255 rad/sec
-
y u
4(3.52 x 105)
7
The results for sway motion are
12,700
ni..
um - <------------ r = 0.144rad/ sec
\ 6.12 x 105
'
These results show that, although the surge frequency is increased by only
about 6 percent by the nonlinear restoring force term, the sway frequency is
increased by a significant amount, 22.9 percent. The calculation of the displacement responses using équation (5.56) is straightforward.
Example Problem 5.6. Investigate the general behavior in free oscillations
of the nonlinear System modeled by équation (5.43). Consider two classes of
restraints: the hardening restraint for which the slope of the restoring force
q(v) = fciv + fc3u3 increases with v and fc3 > 0; and the softening restraint for
which the slope of q(v) decreases with v and fc3 < 0. Softening restraint occurs
in a mooring line with clumped weights attached to it at the sea floor, where
these weights lift off the sea floor when the side displacement of the tethered
structure is sufficiently large (Wilson and Orgill, 1984). In particular, choose
several fixed ratios fc3/fci ,both positive and négative, and plot the free vibration
amplitude A vs. the frequency ratio û/ojq. Use ratios of fc3/fci in the range
given for the moored LST in Example Problem 2.10 where fc3/fci ranges from
0.02 ft"2 to 0.1 ft"2.
For ease of plotting and discussion, the governing équation (5.55) that relates
frequency to amplitude is recast in the following form:
y 3 /c3
(5.57)
The latter équation shows that for a hard restraint with k^/k\ = 0.1 and 0.02,
then .4 is a real number only for û/wo
1. These results are shown in the right
SINGLE DEGREE OF FREEDOM STRUCTURES
ship is displaced first in the surge direction by an amplitude of 3.0 ft from
its equilibrium State and then is released from rest. Then repeat the saine
calculation for the sway State. The numerical values of
and fc3 for each
motion are given as the coefficients of équations (2.72) and (2.73). The results
for surge motion are
Wq
20,300
3.52 x 105
= 0.240 rad/sec
i
. 0.2402 + 3(40Q)32
= 0.255 rad/sec
-
y u
4(3.52 x 105)
7
The results for sway motion are
12,700
ni..
um - <------------ r = 0.144rad/ sec
\ 6.12 x 105
'
These results show that, although the surge frequency is increased by only
about 6 percent by the nonlinear restoring force term, the sway frequency is
increased by a significant amount, 22.9 percent. The calculation of the displacement responses using équation (5.56) is straightforward.
Example Problem 5.6. Investigate the general behavior in free oscillations
of the nonlinear System modeled by équation (5.43). Consider two classes of
restraints: the hardening restraint for which the slope of the restoring force
q(v) = fciv + fc3u3 increases with v and fc3 > 0; and the softening restraint for
which the slope of q(v) decreases with v and fc3 < 0. Softening restraint occurs
in a mooring line with clumped weights attached to it at the sea floor, where
these weights lift off the sea floor when the side displacement of the tethered
structure is sufficiently large (Wilson and Orgill, 1984). In particular, choose
several fixed ratios fc3/fci ,both positive and négative, and plot the free vibration
amplitude A vs. the frequency ratio û/ojq. Use ratios of fc3/fci in the range
given for the moored LST in Example Problem 2.10 where fc3/fci ranges from
0.02 ft"2 to 0.1 ft"2.
For ease of plotting and discussion, the governing équation (5.55) that relates
frequency to amplitude is recast in the following form:
y 3 /c3
(5.57)
The latter équation shows that for a hard restraint with k^/k\ = 0.1 and 0.02,
then .4 is a real number only for û/wo
1. These results are shown in the right