Now, what is the maximum velocity of the beam when oscillating in a potential
well? Rewriting (6.31) in terms of a new coordinate u = x − 1, centered at the well
at x = 1, one obtains:
€ u þ b _
u þ u þ
3
2
u
2
þ
1
2
u
3
¼ p cos Xt; x ¼ u þ 1:
ð6:35Þ
Performing a local perturbation analysis for small u and p Moon finds that, to
first order:
uðtÞ ¼ AðXÞ cosðXtÞ þ ðhigher order termsÞ;
ð6:36Þ
where the stationary amplitude A(X) is given by the frequency response equation
A
2
1 À X
2
À
3
2
A
2
2 þ b
2
X
2
!
¼ p
2
:
ð6:37Þ
By (6.36) the maximum velocity is
_
x max ¼ _
uðtÞ
j
j max ¼ ÀXAðXÞ sin Xt
j
j max ¼ XAðXÞ;
ð6:38Þ
which takes on the critical value v c ¼ av 0 ¼
1
2 a when
XAðXÞ ¼
1
2
a:
ð6:39Þ
Substituting this into (6.37), the critical velocity v c is found to correspond to a
critical amplitude of excitation p c :
p c ¼
a
2X
1 À X
2
À
3
8
a
2
X
2
2
þ b
2
X
2
"
# 1=2
; a % 1:
ð6:40Þ
Using p > p c as a criterion for chaos and a = 0.86, Moon finds a fair agreement
with experimental observations when b ( 1 and X is within ±50% of the natural
frequency of the beam.
Fig. 6.19 compares, for the system (6.31), the multiwell potential criterion (6.40)
(solid line) to the Melnikov criterion (6.27) (dashed lined). Thresholds of chaos as
observed by numerical simulation are also depicted (as circles). Seemingly, for this
case the multiwell potential criterion provides a closer fit to the numerical results
than does the Melnikov criterion.
The multiwell criterion has been successfully applied to other multiwell potential
problems (Moon 1987).
Other Criteria based on Perturbation Analysis. In a number of papers
Szemplinska-Stupnicka and co-workers employ local perturbation techniques for
the study of the Duffing-system:
360
6 Chaotic Vibrations