BEAM RESPONSES
265
For n
1, then (ai,/^)
(2.92; 1.46). This coordinate set lies in the cross
hatched région of Figure 10.8, and thus the transverse chain displacements (and
tension forces) are bounded. This same conclusion is reached for (ân;3„) based
on n = 2,3, ... . However, if w = 1.1 rad/sec, which is at the high end of the
driving frequency for this mooring chain, then an ~ n2 and 0n ~ 0.5n2. For
n = 1, then
in the absence of damping, but stable motion in the presence of small, realistic
damping. For n — 2, 3,..., the motion is stable even without damping.
The conclusion of the preceeding calculations is that the peak responses in
Figure 10.11 are not due to either transverse or parametric résonance of this
particular bow chain. How, then, can these two major response peaks recorded
in the month of Match be reconciled with the single peak of wave excitation in
Figure 10.10? The answer can be found by studying the motion of the whole
ship.
Referring again to Example Problems 2.10 and 5.5, the ship motion v = v(t)
in either surge or sway can be modeled by
mv + CiU + ktv + k3v3 — posinût
(10.68)
where the coefficients on the left side are given by équation (2.72) or (2.73),
where po is the wave force in line with v, and where w ~ 0.65 rad/sec. The
results of Chapter 5 showed that peak responses of such a nonlinear System
under harmonie excitation occur not only at the excitation frequency _> = 0.65
rad/sec but also for w/3, which is near wq- Since u>o = 0.144 rad/sec for sway
(Example Problem 5.5), then the peak at w ~ 0.2 rad/sec can be explained as
a subharmonic ship response of order one-third. The existence of the one-third
subharmonic for équation (10.68) is shown in Section 5.5, where the amplitude
parameter is shown in Figure 5.11. Thus, the lower frequency (or longer period)
ship response refiected in mooring chain No. 2 arises from the group behavior of
ail the mooring lines and cornes about because of the nonlinear restoring force
constant À , of équation (10.68), during sway motion of the ship. Whether drift
currents at a frequency of about 0.22 rad/sec existed during these sea tests and
also contributed to the lower résonance peak of Figure 10.11 is not known.
10.3
BEAM RESPONSES
As for cables, submerged beams and pipelines are subjected to three main typts
of excitation: transverse loading due to vortex shedding, transverse end motion,
and parametric excitation. Considered first in this section are deterministi
responses of uniform beams with common types of end supports and with an
arbitrary transverse load per unit length q — q(x,t). Following this is a stalnlitt
study of a simply supported beam subjected to parametric excitation. I hix
section concludes with a calculation of the statistical responses of beams to Mi
tionary, ergodic excitation. Modal analysis is employed throughout and dosed
form solutions are sought.
265
For n
1, then (ai,/^)
(2.92; 1.46). This coordinate set lies in the cross
hatched région of Figure 10.8, and thus the transverse chain displacements (and
tension forces) are bounded. This same conclusion is reached for (ân;3„) based
on n = 2,3, ... . However, if w = 1.1 rad/sec, which is at the high end of the
driving frequency for this mooring chain, then an ~ n2 and 0n ~ 0.5n2. For
n = 1, then
in the absence of damping, but stable motion in the presence of small, realistic
damping. For n — 2, 3,..., the motion is stable even without damping.
The conclusion of the preceeding calculations is that the peak responses in
Figure 10.11 are not due to either transverse or parametric résonance of this
particular bow chain. How, then, can these two major response peaks recorded
in the month of Match be reconciled with the single peak of wave excitation in
Figure 10.10? The answer can be found by studying the motion of the whole
ship.
Referring again to Example Problems 2.10 and 5.5, the ship motion v = v(t)
in either surge or sway can be modeled by
mv + CiU + ktv + k3v3 — posinût
(10.68)
where the coefficients on the left side are given by équation (2.72) or (2.73),
where po is the wave force in line with v, and where w ~ 0.65 rad/sec. The
results of Chapter 5 showed that peak responses of such a nonlinear System
under harmonie excitation occur not only at the excitation frequency _> = 0.65
rad/sec but also for w/3, which is near wq- Since u>o = 0.144 rad/sec for sway
(Example Problem 5.5), then the peak at w ~ 0.2 rad/sec can be explained as
a subharmonic ship response of order one-third. The existence of the one-third
subharmonic for équation (10.68) is shown in Section 5.5, where the amplitude
parameter is shown in Figure 5.11. Thus, the lower frequency (or longer period)
ship response refiected in mooring chain No. 2 arises from the group behavior of
ail the mooring lines and cornes about because of the nonlinear restoring force
constant À , of équation (10.68), during sway motion of the ship. Whether drift
currents at a frequency of about 0.22 rad/sec existed during these sea tests and
also contributed to the lower résonance peak of Figure 10.11 is not known.
10.3
BEAM RESPONSES
As for cables, submerged beams and pipelines are subjected to three main typts
of excitation: transverse loading due to vortex shedding, transverse end motion,
and parametric excitation. Considered first in this section are deterministi
responses of uniform beams with common types of end supports and with an
arbitrary transverse load per unit length q — q(x,t). Following this is a stalnlitt
study of a simply supported beam subjected to parametric excitation. I hix
section concludes with a calculation of the statistical responses of beams to Mi
tionary, ergodic excitation. Modal analysis is employed throughout and dosed
form solutions are sought.