The Non-shallow Arch This inherently autoparametric system, shown in Fig. 4.6,
obeys coupled equations of motion having quadratic nonlinearities:
€ f þ 2b _
f þ ð1 À mx
2 uÞf ¼ 0;
€ u þ 2bx _
u þ x
2 u þ jðf € f þ _
f
2
Þ ¼
q
m
cosðXsÞ;
ð6:60Þ
where the variables f and u describe the amplitudes of antisymmetric and symmetric
vibrations, respectively. In Chap. 4 a local perturbation analysis was performed,
predicting large amplitudes of the autoparametrically excited antisymmetric mode
when X % x.
Fig. 6.23 Dynamics of the follower-loaded double pendulum with m = 2 and c = 0.1; regions of
similar type of motion in the plane of loading parameters (a, p). Black: chaotic response ( ^ k 1 [ 0);
gray: periodic response ( ^ k 1 % 0); White: static response ( ^ k 1 \0). (Thomsen 1995)
Fig. 6.24 Fluid-conveying pipe, clamped and spring-supported
Fig. 6.25 The fluttering panel (Dowell 1982)
6.6 Mechanical Systems and Chaos
373
obeys coupled equations of motion having quadratic nonlinearities:
€ f þ 2b _
f þ ð1 À mx
2 uÞf ¼ 0;
€ u þ 2bx _
u þ x
2 u þ jðf € f þ _
f
2
Þ ¼
q
m
cosðXsÞ;
ð6:60Þ
where the variables f and u describe the amplitudes of antisymmetric and symmetric
vibrations, respectively. In Chap. 4 a local perturbation analysis was performed,
predicting large amplitudes of the autoparametrically excited antisymmetric mode
when X % x.
Fig. 6.23 Dynamics of the follower-loaded double pendulum with m = 2 and c = 0.1; regions of
similar type of motion in the plane of loading parameters (a, p). Black: chaotic response ( ^ k 1 [ 0);
gray: periodic response ( ^ k 1 % 0); White: static response ( ^ k 1 \0). (Thomsen 1995)
Fig. 6.24 Fluid-conveying pipe, clamped and spring-supported
Fig. 6.25 The fluttering panel (Dowell 1982)
6.6 Mechanical Systems and Chaos
373