Abstract

This thesis is a collection of studies on coupled nonconservative oscillator
systems which contain an oscillator with parametric excitation. The emphasis
this study will, on the one hand, be on the bifurcations of the simple
solutions such as fixed points and periodic orbits, and on the other hand on
identifying more complicated dynamics, such
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as chaotic solutions.
We study an autoparametric system, that is a vibrating system which
consists of at least two subsystems: the oscillator and the excited subsystem.
This system is governed by di®erential equations where the equations
representing the oscillator are coupled to those representing the excited subsystem
in a nonlinear way and such that the excited subsystem can be at
rest while the oscillator is vibrating. We call this solution the semi-trivial
solution. When this semi-trivial solution becomes unstable, non-trivial solutions
can be initiated. In this study we consider the oscillator and the
subsystem are in 1 : 1 internal resonance. The excited subsystem is in 1 : 2
parametric resonance with the external forcing. Using the method of averaging
and numerical bifurcation continuation, we study the dynamics of this
system. In particular, we consider the stability of the semi-trivial solutions,
where the oscillator is at rest and the excited subsystem performs a periodic
motion. We find various types of bifurcations, leading to non-trivial
periodic or quasi-periodic solutions. We also find numerically sequences of
period-doublings, leading to chaotic solutions. Finally, we mention that in
the averaged system we encounter a codimension 2 bifurcation.
In the separated chapter we analytically study aspects of local dynamics
and global dynamics of the system. The method of averaging is again used
to yield a set of autonomous equation of the approximation to the response
of the system. We use two di®erent methods to study this averaged system.
First, the center manifold theory is used to derive a codimension two
bifurcation equation. The results we found in this equation are related to
local dynamics in full system. Second, we use a global perturbation technique
developed by Kova ci c and Wiggins to analyze the parameter range
for which a Silnikov type homoclinic orbit exists. This orbit gives rise to a
well-described chaotic dynamics. We finally combine these results and draw
conclusions for the full averaged system.
There is also a study on coupled oscillator systems with self excitation
which is generated by flow-induced vibrations. In application the flowinduced
model can describe, for instance, the fluid flow around structures
that can cause destructive vibrations. These vibrations have become increasingly
important in recent years because designers are using materials
to their limits, causing structures to become progressively lighter and more
flexible.
Suppressing flow-induced vibrations by using a conventional spring-mass
absorber system has often been investigated and applied in practice. It is
also well-known that self-excited vibrations can be suppressed by using different
kinds of damping. However, only little attention has been paid to
vibration suppression by using interaction of di®erent types of excitation.
In the monograph by Tondl, some results on the investigation of synchronization
phenomena by means of parametric resonances have lead to the
idea to apply a parametric excitation for suppressing self-excited vibrations.
The conditions for full vibration suppression (also called quenching) were
formulated.
In this thesis we discuss the possibility of suppressing self-excited vibrations
of mechanical systems using parametric excitation in two degrees of
freedom. We consider a two-mass system of which the main mass is excited
by a flow-induced, self excited force. A single mass which acts as a dynamic
absorber is attached to the main mass and, by varying the sti®ness between
the main mass and the absorber mass, represents a parametric excitation.
It turns out that for certain parameter ranges full vibration cancellation is
possible. Using the averaging method the fully non-linear system is investigated
producing as non-trivial solutions stable periodic solutions and tori.
In the case of a small absorber mass we have to carry out a second-order
calculation.
We provide open problems of models with three degrees of freedom. These
models also contain an interaction between self-excitation and parametric
excitation. There is a basic stability analysis for a linear case, although far
from simple. We leave the analysis of the nonlinear case for further study.
Corresponding with results obtained in two degrees of freedom, the outcome
of the analysis will be interesting phenomena, such as the appearance of tori
or chaotic behavior.
This thesis is a collection of studies on coupled nonconservative oscillator
systems which contain an oscillator with parametric excitation. The emphasis
this study will, on the one hand, be on the bifurcations of the simple
solutions such as fixed points and periodic orbits, and on the other hand on
identifying more complicated dynamics, such as chaotic solutions.
We study an autoparametric system, that is a vibrating system which
consists of at least two subsystems: the oscillator and the excited subsystem.
This system is governed by di®erential equations where the equations
representing the oscillator are coupled to those representing the excited subsystem
in a nonlinear way and such that the excited subsystem can be at
rest while the oscillator is vibrating. We call this solution the semi-trivial
solution. When this semi-trivial solution becomes unstable, non-trivial solutions
can be initiated. In this study we consider the oscillator and the
subsystem are in 1 : 1 internal resonance. The excited subsystem is in 1 : 2
parametric resonance with the external forcing. Using the method of averaging
and numerical bifurcation continuation, we study the dynamics of this
system. In particular, we consider the stability of the semi-trivial solutions,
where the oscillator is at rest and the excited subsystem performs a periodic
motion. We find various types of bifurcations, leading to non-trivial
periodic or quasi-periodic solutions. We also find numerically sequences of
period-doublings, leading to chaotic solutions. Finally, we mention that in
the averaged system we encounter a codimension 2 bifurcation.
In the separated chapter we analytically study aspects of local dynamics
and global dynamics of the system. The method of averaging is again used
to yield a set of autonomous equation of the approximation to the response
of the system. We use two di®erent methods to study this averaged system.
First, the center manifold theory is used to derive a codimension two
bifurcation equation. The results we found in this equation are related to
local dynamics in full system. Second, we use a global perturbation technique
developed by Kova ci c and Wiggins to analyze the parameter range
for which a Silnikov type homoclinic orbit exists. This orbit gives rise to a
well-described chaotic dynamics. We finally combine these results and draw
conclusions for the full averaged system.
There is also a study on coupled oscillator systems with self excitation
which is generated by flow-induced vibrations. In application the flowinduced
model can describe, for instance, the fluid flow around structures
that can cause destructive vibrations. These vibrations have become increasingly
important in recent years because designers are using materials
to their limits, causing structures to become progressively lighter and more
flexible.
Suppressing flow-induced vibrations by using a conventional spring-mass
absorber system has often been investigated and applied in practice. It is
also well-known that self-excited vibrations can be suppressed by using different
kinds of damping. However, only little attention has been paid to
vibration suppression by using interaction of di®erent types of excitation.
In the monograph by Tondl, some results on the investigation of synchronization
phenomena by means of parametric resonances have lead to the
idea to apply a parametric excitation for suppressing self-excited vibrations.
The conditions for full vibration suppression (also called quenching) were
formulated.
In this thesis we discuss the possibility of suppressing self-excited vibrations
of mechanical systems using parametric excitation in two degrees of
freedom. We consider a two-mass system of which the main mass is excited
by a flow-induced, self excited force. A single mass which acts as a dynamic
absorber is attached to the main mass and, by varying the sti®ness between
the main mass and the absorber mass, represents a parametric excitation.
It turns out that for certain parameter ranges full vibration cancellation is
possible. Using the averaging method the fully non-linear system is investigated
producing as non-trivial solutions stable periodic solutions and tori.
In the case of a small absorber mass we have to carry out a second-order
calculation.
We provide open problems of models with three degrees of freedom. These
models also contain an interaction between self-excitation and parametric
excitation. There is a basic stability analysis for a linear case, although far
from simple. We leave the analysis of the nonlinear case for further study.
Corresponding with results obtained in two degrees of freedom, the outcome
of the analysis will be interesting phenomena, such as the appearance of tori
or chaotic behavior.
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