الفهرس 
SADDLE-NODE BIFURCATION CONTROL FOR ANOnly 14 pages are availabe for public view
 |  | from | 151 |  |  |
| | | from | 151 | | |
Abstract
On the behavior studying of certain dynamical systems, there exist some unwanted properties from the application view. For instance, the jump phenomena, the bistability region, the unstable solutions, the vibrations and the resonance due to the existing of bifurcations at some fixed points. The objective of this thesis is how to control the behavior of these dynamical systems for removing the unwanted properties or at least delaying its occurrence for along time, so performance the system work. These dynamical systems are mathematically represented by a weakly nonlinear ordinary differential equation of second order or by two coupled weakly nonlinear differential equations. The study is performed by applying the perturbation technique. This thesis consists of four chapters as following: Chapter one is concerned with the study of controlling saddle-node bifurcations which may occur in the frequency-response curves in the case of primary resonance of a forced single-degree-of-freedom (SDOF) weakly nonlinear system. This system represents the cable of the hanging bridge system of gas turbine. The appearance of this bifurcation may lead to jump and hysteresis phenomena. A feedback control law is designed to remove or delay the occurrence of jump and hysteresis phenomena. The structure of feedback control law is determined by analyzing the eigenvalues of the modulation equations. It is shown that three types of feedback linear, nonlinear or a combination of linear and nonlinear are adequate for the bifurcation control. Also, it is shown by illustrative examples that the proposed feedback control law has a positive effect. In chapter two, a saddle-node bifurcation was found in case of superharmonic resonance of order five for the dynamical system that mentioned in the first chapter. A combination of feedback controllers was designed. Bifurcation control equations were obtained. It is found that the linear feedback gain can delay the occurrence of saddle node bifurcations which occur in the uncontrolled system, while the nonlinear feedback gains can eliminate saddle node
bifurcations. Then a feasible way of further research of saddle-node bifurcation
was provided. Finally, we show that an appropriate nonlinear feedback control
gain can suppress the amplitude of steady-state response.
Chapter three is devoted to control the primary, subharmonic of order onethird
and superharmonic of order five resonances for a weakly non-linear
dynamical system with two distinct time-delays which playing as a control law.
This system represents the simplest model for many practical controlled systems
such as active vehicle suspension systems when the non-linearity in the tires is
taken into account. The effect of the feedback gains and time-delays is discussed
and it is found that: an appropriate feedback can enhance the control
performance. A suitable choice of the feedback gains and time-delays can
enlarge the critical force amplitude and reduce the peak amplitude of the
response for the case of primary resonance. Also, an appropriate choice of the
feedback gains and time-delays can reduce the peak amplitude of the free
oscillation term for the case superharmonic resonance. Furthermore, a proper
feedback can eliminate saddle-node bifurcation, thereby eliminating jump and
hysteresis pheomena taking place in the corresponding uncontrolled system. For
subharmonic resonance, an adequate feedback can reduce the regions of
subharmonic resonance response.
Chapter four is devoted to suppress the vibrations of the first bending modes
of the structural dynamic model of an F-15 twin tails assembly when subjected
to parametric excitations. The dynamics of the first flexural modes of the twin
tails were modeled by two second-order nonlinear coupled ordinary differential
equations. Then, bifurcation analysis was conducted to examine the stability of
-iiithe
system and to investigate the performance of the control law. The vibrations
of the first bending modes of the structural dynamic model were reduced by
using the control laws. To compare the performance of linear and nonlinear
control techniques, power requirements for a simple system were calculated.