Bifurcations in dynamical systems with parametric excitation

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. 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.