where, to obtain autonomous equations, a new phase variable w has been substituted for u:
w ¼ rT 1 À u:
ð3:220Þ
Stationary oscillations correspond to singular points of the modulation equations, that is, to those (a, u) for which a′ = w′ = 0. Letting a′ = w′ = 0 in (3.219)
we obtain, by squaring and adding the two equations, the frequency response
equation for the superharmonic case:
bx
2
0
À
Á 2 þ rx 0 À 3cðQ
2
þ
1
8
a
2
Þ
2
!
a
2
¼ cQ
3
À
Á 2 ;
ð3:221Þ
with corresponding phase w given by (divide the two equations):
tan w ¼
Àbx
2
0
rx 0 À 3cðQ 2 þ
1
8 a 2 Þ
:
ð3:222Þ
The frequency response equation (3.221) is not readily solved for a or for X
(which is hidden in Q, cf. (3.212)). However, numerical solutions are rather easily
obtained.
We may now write down the response for the superharmonic case. To a first
approximation it becomes, by (3.208), (3.211), (3.220) and (3.217):
uðtÞ ¼ u 0 þ OðeÞ ¼ AðT 1 Þe
ix 0 T 0 þ Qe
iXT 0 þ cc þ OðeÞ ¼ Á Á Á
¼ a cosð3Xt À wÞ þ
q
x 2
0 À X
2
cos Xt þ OðeÞ:
ð3:223Þ
For stationary oscillations the constant amplitude a and phase w is given by
(3.221)–(3.222), whereas for nonstationary (transient) motions the time-varying
amplitudes a(t) and phases w(t) are given by the modulation equation (3.219).
As appears from (3.223) the superharmonic response consists of oscillations at
three times the excitation frequency, overlaying the linear response at frequency X.
Note that the frequency 3X approximately equals x 0 for this case.
Typical superharmonic frequency responses are shown in Fig. 3.21 for different
values of excitation level q, and in Fig. 3.22 for different levels of damping b. The
response curves were obtained by solving (3.221) numerically, with (3.217) and
(3.212) substituted for r and Q. The backbone curves are obtained by letting Q =
b = 0 in (3.221, using (3.217) to substitute r, and solving for a, which gives:
a ¼ x 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
8
3c
3
X
x 0
À 1
s
ðon backbone):
ð3:224Þ
3.7 Externally Excited Duffing Systems
163
w ¼ rT 1 À u:
ð3:220Þ
Stationary oscillations correspond to singular points of the modulation equations, that is, to those (a, u) for which a′ = w′ = 0. Letting a′ = w′ = 0 in (3.219)
we obtain, by squaring and adding the two equations, the frequency response
equation for the superharmonic case:
bx
2
0
À
Á 2 þ rx 0 À 3cðQ
2
þ
1
8
a
2
Þ
2
!
a
2
¼ cQ
3
À
Á 2 ;
ð3:221Þ
with corresponding phase w given by (divide the two equations):
tan w ¼
Àbx
2
0
rx 0 À 3cðQ 2 þ
1
8 a 2 Þ
:
ð3:222Þ
The frequency response equation (3.221) is not readily solved for a or for X
(which is hidden in Q, cf. (3.212)). However, numerical solutions are rather easily
obtained.
We may now write down the response for the superharmonic case. To a first
approximation it becomes, by (3.208), (3.211), (3.220) and (3.217):
uðtÞ ¼ u 0 þ OðeÞ ¼ AðT 1 Þe
ix 0 T 0 þ Qe
iXT 0 þ cc þ OðeÞ ¼ Á Á Á
¼ a cosð3Xt À wÞ þ
q
x 2
0 À X
2
cos Xt þ OðeÞ:
ð3:223Þ
For stationary oscillations the constant amplitude a and phase w is given by
(3.221)–(3.222), whereas for nonstationary (transient) motions the time-varying
amplitudes a(t) and phases w(t) are given by the modulation equation (3.219).
As appears from (3.223) the superharmonic response consists of oscillations at
three times the excitation frequency, overlaying the linear response at frequency X.
Note that the frequency 3X approximately equals x 0 for this case.
Typical superharmonic frequency responses are shown in Fig. 3.21 for different
values of excitation level q, and in Fig. 3.22 for different levels of damping b. The
response curves were obtained by solving (3.221) numerically, with (3.217) and
(3.212) substituted for r and Q. The backbone curves are obtained by letting Q =
b = 0 in (3.221, using (3.217) to substitute r, and solving for a, which gives:
a ¼ x 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
8
3c
3
X
x 0
À 1
s
ðon backbone):
ð3:224Þ
3.7 Externally Excited Duffing Systems
163