RESPONSE OF NONLINEAR STRUCTURES
129
real roots of â for 0.7 < w/o?o < 1-3. As in Figure 5.9, the response amplitude is
négative or opposite in sign to po to the right of the po — 0 line and is positive to
the left of this line. The two branches of each po/fcj curve were connected by the
broken lines to show the approximate effects of light damping, or ( = 0.05. The
estimated peak amplitude of each was chosen as ten times the respective static
response, the same amplification calculated for its linear counterpart, Figure
5.4. Damping is discussed in detail by Stoker (1963).
With the damping shown and with a value of w/w0 — 1.15 for the lower
load ratio of po/k^ = 0.4 ft, the System oscillâtes initially with an amplitude
of â — 1.3 ft at point 1. For this lower load ratio and with System damping,
the response amplitude is single-valued at this frequency. As the load ratio is
increased to po/ky = 0.6 ft, the response amplitude increases to â = 2.85 at the
knee of the solid curve, or point 2. However, a response amplitude of â — 5.4
ft could also exist on the upper branch of the solid curve. It follows that there
can be an upward jump in amplitude response between points 2 and 3 as the
excitation load ratio is increased. Furthermore, there can be a downward jump
from points 3 to 2 and then a smooth transition to point 1 if the excitation load
ratio were gradually decreased to its initially lower value. There is of course
the possibility of jump behavior at load levels between those given, not only for
w/lüq = 1.15 but for larger values of the forcing frequency as well. The two load
level ratios chosen, however, sufficiently illustrate the combination of System
parameters that can lead to erratic jump responses in this nonlinear System.
Subharmonics
The preceding studies for the nonlinear structure modeled by équation (5.81)
showed high amplitude responses or résonance for excitation frequencies w near
wq. This is not too surprising since résonance for w near u>q is well-known in its
linear System counterpart (k-j = 0) and was shown in Figure 5.4. In nonlinear
Systems, however, résonance near wq may sometimes occur when w is not near
ojq. For instance, if high amplitude responses also exist near wq = w/n where
n = 2,3,... , then the response is defined as a subharmonic of order 1/n. Such
responses do not occur in linear Systems, but exist under spécial conditions in
nonlinear Systems such as cable-stayed offshore structures in seas with regular,
harmonie wave components.
The subharmonic response most commonly observed is for n — 3, or the
one-third subharmonic (Wilson and Awadalla, 1973). There are several alternative, classical methods that can be used to show the existence of this and other
subharmonics (Stoker, 1963). For instance for u> =
Cunningham (1964)
applied first-order perturbation theory to équation (5.81 ) in the same, straightforward manner illustrated previously in this chapter. Cunningham’s results are
summarized as follows. The necessary relationship between the subharmonic response of amplitude as and its frequency of oscillation w, (close to wq), when
the System of équation (5.81) is excited by the force pi(t) = pocosSo^f, is
K
, a
3fc3
4ma„w2 s , 32a’mM\
«
UT — üJnüj = ----------- Ô Pn
1------------------------• ---------- 9-------s
0 s
128m3'0 V
po
Po
J
129
real roots of â for 0.7 < w/o?o < 1-3. As in Figure 5.9, the response amplitude is
négative or opposite in sign to po to the right of the po — 0 line and is positive to
the left of this line. The two branches of each po/fcj curve were connected by the
broken lines to show the approximate effects of light damping, or ( = 0.05. The
estimated peak amplitude of each was chosen as ten times the respective static
response, the same amplification calculated for its linear counterpart, Figure
5.4. Damping is discussed in detail by Stoker (1963).
With the damping shown and with a value of w/w0 — 1.15 for the lower
load ratio of po/k^ = 0.4 ft, the System oscillâtes initially with an amplitude
of â — 1.3 ft at point 1. For this lower load ratio and with System damping,
the response amplitude is single-valued at this frequency. As the load ratio is
increased to po/ky = 0.6 ft, the response amplitude increases to â = 2.85 at the
knee of the solid curve, or point 2. However, a response amplitude of â — 5.4
ft could also exist on the upper branch of the solid curve. It follows that there
can be an upward jump in amplitude response between points 2 and 3 as the
excitation load ratio is increased. Furthermore, there can be a downward jump
from points 3 to 2 and then a smooth transition to point 1 if the excitation load
ratio were gradually decreased to its initially lower value. There is of course
the possibility of jump behavior at load levels between those given, not only for
w/lüq = 1.15 but for larger values of the forcing frequency as well. The two load
level ratios chosen, however, sufficiently illustrate the combination of System
parameters that can lead to erratic jump responses in this nonlinear System.
Subharmonics
The preceding studies for the nonlinear structure modeled by équation (5.81)
showed high amplitude responses or résonance for excitation frequencies w near
wq. This is not too surprising since résonance for w near u>q is well-known in its
linear System counterpart (k-j = 0) and was shown in Figure 5.4. In nonlinear
Systems, however, résonance near wq may sometimes occur when w is not near
ojq. For instance, if high amplitude responses also exist near wq = w/n where
n = 2,3,... , then the response is defined as a subharmonic of order 1/n. Such
responses do not occur in linear Systems, but exist under spécial conditions in
nonlinear Systems such as cable-stayed offshore structures in seas with regular,
harmonie wave components.
The subharmonic response most commonly observed is for n — 3, or the
one-third subharmonic (Wilson and Awadalla, 1973). There are several alternative, classical methods that can be used to show the existence of this and other
subharmonics (Stoker, 1963). For instance for u> =
Cunningham (1964)
applied first-order perturbation theory to équation (5.81 ) in the same, straightforward manner illustrated previously in this chapter. Cunningham’s results are
summarized as follows. The necessary relationship between the subharmonic response of amplitude as and its frequency of oscillation w, (close to wq), when
the System of équation (5.81) is excited by the force pi(t) = pocosSo^f, is
K
, a
3fc3
4ma„w2 s , 32a’mM\
«
UT — üJnüj = ----------- Ô Pn
1------------------------• ---------- 9-------s
0 s
128m3'0 V
po
Po
J