NONLINEAR RESPONSES FOR A SALM BUOY
131
Such results are shown in Figure 5.11. For accuracy and consistency with firstorder perturbation theory, neither Ka nor Q3 should exceed the values shown.
The knees in these curves indicate that for each Ka there is a minimum frequency below which the subharmonic cannot exist. This minimum, derived
from équation (5.97) using dOajdAa = 0, yields the following équation from
which the critical points (roots)
= <),„ can be computed:
y
-A', -0
(5.98)
O
With these roots, the critical amplitudes Aa = Am are computed from
Am = 1612^
An approximate formula for the critical frequencies at the knee of each curve
of constant Ka can be derived by substituting expanding the frequency terms, and ignoring terms in e2, e3,... . The resuit is
7
~ 1 + — Ka
(5.100)
This is an important resuit because it shows that, for hardening restraints where
ks > 0, the minimum frequency for a one-third subharmonic to exist is just a
little higher than luq.
Other important results by Cunningham (1964) are summarized. First, the
subharmonic motion is stable only in the portions of the curves of Figure 5.11
that hâve positive slopes. No sustained subharmonic motion exists in the région
below the broken line shown in this figure. Second, the effect of including linear
damping of the form Civ in the nonlinear model, équation (5.81), is to decrease
As at constant Ka and Qs, and to eliminate subharmonic motion at some cutoff
value of Qm > 1. The value of Qc decreases as the damping
increases. Third, for a subharmonic of order 1/n to exist for équation (5.81)
where the excitation frequency is nu>o, the highest power of v in the restraining
force polynominal must be at least of order n. Thus, a one-fifth subharmonic
response cannot exist for équation (5.81), but a one-half subharmonic response
can.
In summary, it is clear from these classical results for a nonlinear System that
a résonance response amplitude As of frequency
(near to but a little greater
than wo) exists for excitation frequencies w = 3a>„. Experimental evidence for
moored ship responses that support this analysis was presented by O Brien
and Muga (1964) and by Wilson and Awadalla (1973). This evidence will be
discussed in Chapter 10.
5.6
NONLINEAR RESPONSES FOR A SALM BUOY
In practice, the responses to wave loading of offshore structures with nonlinear
restraints are generally calculated numerically from the governing differential
131
Such results are shown in Figure 5.11. For accuracy and consistency with firstorder perturbation theory, neither Ka nor Q3 should exceed the values shown.
The knees in these curves indicate that for each Ka there is a minimum frequency below which the subharmonic cannot exist. This minimum, derived
from équation (5.97) using dOajdAa = 0, yields the following équation from
which the critical points (roots)
= <),„ can be computed:
y
-A', -0
(5.98)
O
With these roots, the critical amplitudes Aa = Am are computed from
Am = 1612^
An approximate formula for the critical frequencies at the knee of each curve
of constant Ka can be derived by substituting expanding the frequency terms, and ignoring terms in e2, e3,... . The resuit is
7
~ 1 + — Ka
(5.100)
This is an important resuit because it shows that, for hardening restraints where
ks > 0, the minimum frequency for a one-third subharmonic to exist is just a
little higher than luq.
Other important results by Cunningham (1964) are summarized. First, the
subharmonic motion is stable only in the portions of the curves of Figure 5.11
that hâve positive slopes. No sustained subharmonic motion exists in the région
below the broken line shown in this figure. Second, the effect of including linear
damping of the form Civ in the nonlinear model, équation (5.81), is to decrease
As at constant Ka and Qs, and to eliminate subharmonic motion at some cutoff
value of Qm > 1. The value of Qc decreases as the damping
increases. Third, for a subharmonic of order 1/n to exist for équation (5.81)
where the excitation frequency is nu>o, the highest power of v in the restraining
force polynominal must be at least of order n. Thus, a one-fifth subharmonic
response cannot exist for équation (5.81), but a one-half subharmonic response
can.
In summary, it is clear from these classical results for a nonlinear System that
a résonance response amplitude As of frequency
(near to but a little greater
than wo) exists for excitation frequencies w = 3a>„. Experimental evidence for
moored ship responses that support this analysis was presented by O Brien
and Muga (1964) and by Wilson and Awadalla (1973). This evidence will be
discussed in Chapter 10.
5.6
NONLINEAR RESPONSES FOR A SALM BUOY
In practice, the responses to wave loading of offshore structures with nonlinear
restraints are generally calculated numerically from the governing differential