Singularity Method For Nonlinear Dynamical Analysis Of Systems With Two Parameters And Its Application In Engineering

Bifurcation theory and methods of the dynamical systems are the important parts of nonlinear dynamics and widely applied in the eigneering fields. For the systems with parameters, when the parameters change, the dynamical behavior may be aroused change—bifurcation. In theses years, many methods for bifurcation analysis of the dynamical systems have been proposed. Among theses methods, singularity theory is of much imporatance and has been widely applied as a quanlative analysis method. Singularity theory is an effective method to study the reduced eqations of the dynamical systems, which can solve the bifurcation problems uniformly and definitely. But up to now, singularity theory is mainly applied in the dynamical systems with one bifurcation parameter and one state variable. As the development of the science technology, there are more and more dynamical systems with multiple bifurcation parameters and multiple state variables. Therefore, the bifurcation analysis of such systems is challenged.In this dissertation, we pay our attention to the bifurcations of the dynamical systems with two bifurcation parameters and the ones with two state variables. The mechanics of the galloping and optimal control of the transmission line are analyzed, and the bifurcation of a class of biochemical reaction model is studied.1. For the actual systems, there are many structural parameters. Which parameter can be considered as bifurcation parameter and which parameter will arouse the change of the solution structure of the system are two important issues. In this dissertation, a method to find the main bifurcation parameter of the dynamical systems is given. As known that when the parameter is subject to some small perturbations the solution structure maybe changes and this change can be reflected by the eigenvalues of the Frechet derivatives matrix of the system. Therefore, expanding the eigenvalues of the Frechet derivatives matrix near the critical value, the effects of the parameters can be discussed. For the cases of simple eigenvalue and semi-simple eigenvalue, this method is easy to operate. For the case of defective eigenvalue, although this method has some complexity, it is applicable as well. Furhermore, this method can be extended to the dynamical systems with periodic coefficients. 2. For the system with multiple DOFs, there maybe exists internal resonance. Usually the bifurcation equations can be reduced to the one with one state variable by the elimination method, the proportion method or combine of these two methods. After study, it can be found that some bifurcation properties are lost if the system was reduced. Therefore the singularity theory is developed to the bifurcation analysis of the dynamical systems with two state variables. For the systems with multiple parameters, such as chemical systems and power systems, there are many physical parameters and some parameters are of the same importance, i.e. the change of each important parameter maybe arouse the change of the dynamical behavior—bifurcation. Therefore both two parameters may be considered as bifurcation parameters. In this dissertation the singularity theory is developed to the bifurcation analysis of the dynamical systems with two parameters and the transition sets are given.3. Singularity theory with two state variables is applied in the galloping of the transmission line. The model of the transmission line with two DOFs is constructed by using Hamilton principle after considering the initial location, the geometric nonlinearity caused by the deformation and the aerodynamic nonlinearity caused by the flow. The bifurcation equations are obtained by multiscale method. After singularity analysis the transition sets of the system are obtained. It is found that in different persistent regions there exist different bifurcation and hysteresis modals. As known that bifurcation and hysteresis modals maybe arouse the abrupt changes of the tension of the transimission line which are disadvantage. The model of the transmission line with one DOF is studied. The critical wind speed and the analytical solution of the amplitude of the transmission line are obtained. Except that, the effects of the torsional motion are considered. For anti-galloping, the optimal control of the masses, dynamic vibration absorber and detuning pendulum to the transmission line are studied, which can provide a theoretical basis for the design and control of the transmission line.4. The singularity theory with one bifurcation parameter and two parameters are both applied in Duffing-van der Pol system under multi-frequency excitations. After comparison, it can be found that the bifurcation properties of the system with two bifurcation parameters are much more than the system with one parameter. So for the system with multiple structural parameters, especially some parameters are of the same importance, only one parameter is considered as bifurcation parameter is not enough to bifurcation analysis. Additionally, the singularity theory with two bifurcation parameters is applied in a class of biochemical reaction model and the bifurcation properties are analyzed.