126
SINGLE DEGREE OF FREEDOM STRUCTURES
There are at least two equal real roots if
1
-,
9
9
,
^(u>
-;,'=Î6^,V'
(593)
There is only one real root if
To explain the ship’s behavior, two reasonable assumptions are made. First,
the wave excitation parameters p0 and w remained constant, or nearly so, from
before to after the mooring lines were adjusted. Since the ship’s mass m remained constant, then the right sides of équations (5.92)-(5.94) also remained
constant. Second, the inequality given by équation (5.92) was true before the
line adjustment since multiple real roots â lead to erratic motion, which is explained in more detail in the next section. This inequality (5.92) implies that
w > wq. The equality condition of équation (5.93) can be eliminated because
such exactness is rarely possible in the physical world.
After tightening the mooring lines, Aq definitely increased; and ks decreased
somewhat because the static restraint stiffness curve q(v) vs. v would tend to
straighten out as the initial slope of this curve increased. For a small increase in
ki, the u)q term increased, causing the term in brackets on the left of équation
(5.92) to decrease. Since this latter term is to the power three, its decrease
more than offset the increase of l/k^. The inequality of équation (5.92) with
its multiple amplitudes â thus reverted to the inequality of équation (5.94), the
condition of a single amplitude i. with regular oscillations.
Jumps
The erratic changes or jumps in the response amplitudes for the nonlinear
model described by équation (5.81) can be predicted from the |A| vs. Q behavior
of équation (5.88). This behavior is shown in Figure 5.9 for two values of K.
Several observations can be made about these curves. First, as FC is decreased at
constant fl, as from the solid curves to the dashed curves, the response behavior
approaches that of the undamped, linear System response |H(lü)| of Figure 5.4.
Second, as K is increased at constant Q, the response curves lean more to the
right. Third, multiple amplitudes exist for values of Q > Qm where Q,„ is
the value of Q at the knee of each curve. Fourth, if linear damping had been
mcluded in the nonlinear model, équation (5.81), then that effect would hâve
shown up as the amplitude-limiting, small dashed line shown as the upper knee
for K = 0.001, around the point labeled 2' (Stoker, 1963).
Consider an example based on Figure 5.9 in which a particular set of conditions can lead to a jump in response amplitude. For a wave load of constant
magnitude pg and for K = 0.001, suppose that the wave excitation frequency
initially given by w = 1.2 u>0 is decreased very slowly. The response amplitude
the path from point 1 to point 2 on the lower knee of the solid curve,
ic point the amplitude must almost double in magnitude by jumping to
SINGLE DEGREE OF FREEDOM STRUCTURES
There are at least two equal real roots if
1
-,
9
9
,
^(u>
-;,'=Î6^,V'
(593)
There is only one real root if
To explain the ship’s behavior, two reasonable assumptions are made. First,
the wave excitation parameters p0 and w remained constant, or nearly so, from
before to after the mooring lines were adjusted. Since the ship’s mass m remained constant, then the right sides of équations (5.92)-(5.94) also remained
constant. Second, the inequality given by équation (5.92) was true before the
line adjustment since multiple real roots â lead to erratic motion, which is explained in more detail in the next section. This inequality (5.92) implies that
w > wq. The equality condition of équation (5.93) can be eliminated because
such exactness is rarely possible in the physical world.
After tightening the mooring lines, Aq definitely increased; and ks decreased
somewhat because the static restraint stiffness curve q(v) vs. v would tend to
straighten out as the initial slope of this curve increased. For a small increase in
ki, the u)q term increased, causing the term in brackets on the left of équation
(5.92) to decrease. Since this latter term is to the power three, its decrease
more than offset the increase of l/k^. The inequality of équation (5.92) with
its multiple amplitudes â thus reverted to the inequality of équation (5.94), the
condition of a single amplitude i. with regular oscillations.
Jumps
The erratic changes or jumps in the response amplitudes for the nonlinear
model described by équation (5.81) can be predicted from the |A| vs. Q behavior
of équation (5.88). This behavior is shown in Figure 5.9 for two values of K.
Several observations can be made about these curves. First, as FC is decreased at
constant fl, as from the solid curves to the dashed curves, the response behavior
approaches that of the undamped, linear System response |H(lü)| of Figure 5.4.
Second, as K is increased at constant Q, the response curves lean more to the
right. Third, multiple amplitudes exist for values of Q > Qm where Q,„ is
the value of Q at the knee of each curve. Fourth, if linear damping had been
mcluded in the nonlinear model, équation (5.81), then that effect would hâve
shown up as the amplitude-limiting, small dashed line shown as the upper knee
for K = 0.001, around the point labeled 2' (Stoker, 1963).
Consider an example based on Figure 5.9 in which a particular set of conditions can lead to a jump in response amplitude. For a wave load of constant
magnitude pg and for K = 0.001, suppose that the wave excitation frequency
initially given by w = 1.2 u>0 is decreased very slowly. The response amplitude
the path from point 1 to point 2 on the lower knee of the solid curve,
ic point the amplitude must almost double in magnitude by jumping to