A Chaotic Motion Of Nonlinear Structure

Converted by Galerkin principle, problems of forced vibration of nonlinear structure such as beam, arch, slab and shell can be converted into the following nonlinear dynamic equationResearches on dynamic properties of equation (1) are the leading work of present research activities of Solid Mechanics. Through the analytical research result of equation (1), evolution rules of nonlinear elastic dynamic system such as beam, slab, shell and arch can be effectively analyzed, calculated and mastered. Therefore prediction and control of this sort of nonlinear system can be more effectively recognized, understood and achieved.Equation (1) is the dynamic equation that contains quadratic nonlinear term and cubic non-linear term. When =0, this equation becomesEquation (2) is the Duffing equation, whose characteristic is that the nonlinear terms on the left side of the equation are all cubic power. Many studies have been carried out on the system described by Duffing equation (2), but very few on equation (1) by using the analytical method. The quadratic nonlinear term and the cubic nonlinear term that coexist in equation (1), thus causes difficulty in studying this sort of system through the analytical method and correspondingly makes the solution of analytical expression of homoclinic orbit or heteroclinic orbit in Hamilton system extremely difficult. By using Melinkov Method, the chaotic movement of equation (1) is thoroughly studied in this paper. The main work and results are as follows:1. Nature of the singular point of plane Poincare mapping that is established by equation (1) is analyzed. Relationship between the three parameters and homoclinic orbit and heteroclinic orbit of the Hamilton system corresponding to this sort of equation is discussed. The adequate and essential condition of the existing homoclinic orbit or heteroclinic orbit for the Hamilton system is presented.2. An analytical expression of homoclinic orbit or heteroclinic orbit is worked out. By using Melnikov method, the Melnikov function of homoclinic orbit or heteroclinic orbit is calculated and established. The critical value when Poincare mapping taken the form of Smale horseshoes chaos is presented3. Within the homoclinic orbit or heteroclinic orbit, the analytical expression of one set of periodical track surrounding the center-type singular point is worked out. The Melnikov function of subharmonic orbits is calculated and established and the criterion of appearing periodical m point of Poincare mapping is presented.4. The specific route of the system going into Smale horse-hoof chaos through subharmonic bifurcation is discussed.Every result in the paper is presented through specific analytic expression, including the analytic expression of homoclinic orbit or heteroclinic orbit and its Melnikov function, analytical expression of periodical track surrounding the center-type singular point within the homoclinic orbit or heteroclinic orbit and its Melinkov function, the critical value when periodical m point appears, the critical value when Smale horseshoes chaos appears, etc.. These results are of great significance for studying and analyzing the Smale horseshoes chaos of equation (1).Influence of the three parameters in the equation is persistently considered in the research work conducted in this paper. Because the determination of the dynamic behavior of the system is decided by the parameters , every result in this paper has general and universal meaning. At this time, the criterion concerning the Smale horse-hoes chaos of equation (1) and its related problems are basically solved in this paper.