Langthjem (1995b) studied the dynamic stability of immersed fluid-conveying
tubes. Seemingly chaotic coupling between flutter and whirling modes was
observed for an experimental tube immersed in water.
2
Chaotic vibrations due to coupling between the symmetric and the antisymmetric
mode of an elastically buckled two-bar linkage have been studied by Sorokin and
Terentiev (1998).
These are just examples; Many more can be found in the monographs by Nayfeh
(1997) and Tondl et al. (2000).
Most systems with internal resonance may display chaotic motions, along with
the regular and predictable kinds of motions described above. Thus, we shall return
to internal resonant systems in Chap. 6 on chaos.
4.5 The Follower-Loaded Double Pendulum
In this section we examine possible periodic motions of a well-known archetype of
a flutter-prone structure (cf. Sect. 2.4.2), the double pendulum of Fig. 4.11. The
system differs from those already described in this chapter in that it is autonomous,
that is, the external load is not an explicit function of time. This case will provide us
with an opportunity to exercise the method of multiple scales on models written in
first-order matrix form. But first a brief account of the background.
Some structures are prone to flutter-instability due to the presence of
non-conservative follower-type forces. Examples are compressors, turbines, fluid
conveying pipes, aircraft wings, rockets, suspension bridges, and shafts with controlled speed of revolution. The load-limits of stability and the initial unstable
behavior of such systems may be established through linearization, whereas the
prediction of long-time behavior requires a nonlinear analysis. Traditionally,
long-time behavior has been characterized as either soft flutter (finite amplitude
periodic motion, possibly causing fatigue failure), explosive flutter (perhaps causing
ultimate failure), or escape to a different state of equilibrium. However, as pointed
out in Thomsen (1995), in some cases the final state may be none of the above but
rather stationary chaotic.
Theoretical interest in non-conservative problems of stability was stimulated by
a number of flutter-related structural failures, the most renowned undoubtedly being
the collapse of the Tacoma Narrows Bridge in 1940 (e.g., Billah and Scanlan 1991;
Koughan J 1996; Peterson 1990; Ross 1984). It has persevered ever since. Most
flutter research concentrates on certain archetypal flutter-prone models that are
generic, that is, they illustrate the phenomena in question in the simplest possible
setting. One such model is the Beck’s column – a fixed-free column subject to a
tangential load at the free end (e.g., Ziegler 1968; Langthjem and Sugiyama 2000;
2
Langthjem kindly demonstrated this experiment to the author, who got pretty wet in the attempt to
count the periods of the modes involved; they were nearly two-to-one.
228
4 Nonlinear Multiple-DOF Systems: Local Analysis
tubes. Seemingly chaotic coupling between flutter and whirling modes was
observed for an experimental tube immersed in water.
2
Chaotic vibrations due to coupling between the symmetric and the antisymmetric
mode of an elastically buckled two-bar linkage have been studied by Sorokin and
Terentiev (1998).
These are just examples; Many more can be found in the monographs by Nayfeh
(1997) and Tondl et al. (2000).
Most systems with internal resonance may display chaotic motions, along with
the regular and predictable kinds of motions described above. Thus, we shall return
to internal resonant systems in Chap. 6 on chaos.
4.5 The Follower-Loaded Double Pendulum
In this section we examine possible periodic motions of a well-known archetype of
a flutter-prone structure (cf. Sect. 2.4.2), the double pendulum of Fig. 4.11. The
system differs from those already described in this chapter in that it is autonomous,
that is, the external load is not an explicit function of time. This case will provide us
with an opportunity to exercise the method of multiple scales on models written in
first-order matrix form. But first a brief account of the background.
Some structures are prone to flutter-instability due to the presence of
non-conservative follower-type forces. Examples are compressors, turbines, fluid
conveying pipes, aircraft wings, rockets, suspension bridges, and shafts with controlled speed of revolution. The load-limits of stability and the initial unstable
behavior of such systems may be established through linearization, whereas the
prediction of long-time behavior requires a nonlinear analysis. Traditionally,
long-time behavior has been characterized as either soft flutter (finite amplitude
periodic motion, possibly causing fatigue failure), explosive flutter (perhaps causing
ultimate failure), or escape to a different state of equilibrium. However, as pointed
out in Thomsen (1995), in some cases the final state may be none of the above but
rather stationary chaotic.
Theoretical interest in non-conservative problems of stability was stimulated by
a number of flutter-related structural failures, the most renowned undoubtedly being
the collapse of the Tacoma Narrows Bridge in 1940 (e.g., Billah and Scanlan 1991;
Koughan J 1996; Peterson 1990; Ross 1984). It has persevered ever since. Most
flutter research concentrates on certain archetypal flutter-prone models that are
generic, that is, they illustrate the phenomena in question in the simplest possible
setting. One such model is the Beck’s column – a fixed-free column subject to a
tangential load at the free end (e.g., Ziegler 1968; Langthjem and Sugiyama 2000;
2
Langthjem kindly demonstrated this experiment to the author, who got pretty wet in the attempt to
count the periods of the modes involved; they were nearly two-to-one.
228
4 Nonlinear Multiple-DOF Systems: Local Analysis