The Application Of Complex Normal Form Method In Strongly Nonlinear Vibration With Multi-Degree Of Freedom

Normal form theory is one of the important methods for the research of the nonlinear constant differential equations, and it plays an important role in the study of bifurcation and stability behaviors of the nonlinear dynamical systems. In recent years, the methodology of normal form and its application in the dynamical systems have made great progress with the domestic and foreign scholars'unceasing research in this field. This dissertation mainly pays attention to the implication of normal form theory in the solution of strongly nonlinear oscillation system. The method of the conventional normal form in solving the weakly nonlinear oscillation system has been successfully developed, but it may not be available under the strongly nonlinear circumstances. The advanced conventional normal form method has been only used in the one and two dimensional systems and never in the research of multi-degree freedom system by far. So there are great needs for us to develop the methodology to a higher level in view of the fact that more engineering problems require to be predigested as the multi-degree of freedom systems. In order to deal with those problems, the results were shown as follows:(1) Conventional normal form method has the limitation of dealing with the strongly nonlinear vibration systems. By introducing the undecided instantaneous frequency, the necessary improvements have been made in this paper, and extend the researching field into the multi-degree freedom systems. The stable asymptotic solutions and limit cycles of the strongly nonlinear oscillation system with three-degree freedom and uncoupled in the linear parts have been researched. Those results testify the validity of the refined theory in such condition, which great coincide with the solutions of numerical integration more than the initial normal form approach without the new undecided instantaneous fundamental frequencies.(2) The advanced method has also been used to analyze the strongly nonlinear oscillation system with three-degree freedom and coupled in its linear parts. The complex normal form method in view of this melt-degree freedom is introduced and the expression of normal form is also offered here. The validity of the refined theory is testified in a more complicated actual system by comparing the asymptotic solutions with the numerical imitation.