Study On The Global Dynamical Behavior For Several Kinds Of Nonlinear Elastic Rods Based Inertial Manifolds With Delay

With the development of science and technology and the increasing demand of people’s life, the application of nonlinear elastic structure in infinite dimensional dynamical system are very common in the aerospace, shipbuilding, construction and machinery manufacturing, but also the load capacity of these structure often need to be considered in the design of application. Therefore dynamic stability of nonlinear elastic structure under impact loads is a research topic with important theoretical value and many engineering application.From the dynamic point of view, the nonlinear elastic structure is an infinite dimensional dynamical system, however, Galerkin truncation is always used to dynamic analysis of these structure evolutions. The infinite dimensional dynamical system can be transformed to the finite dimensional system by directly selecting one or several modes in this method. The rationality of truncation problem, whether international or domestic, has be verified by experimental method, but not be proved in theory. We believe that the reduction theory of infinite dimensional dynamical systems must be used to solve the rationality of truncation problem. It may make some dynamical behavior of the system is lost, and some strange phenomenon is difficult to explain by the truncation of higher-order modes. Therefore, the dynamical systems will be studied by the method based upon inertial manifolds with delay in this paper.In this paper, there kinds of nonlinear elastic rod have be studied by the nonlinear Galerkin method based upon inertial manifolds with delay(IMD), also some imperfections of nonlinear problem at present have be paid more attention. By this method, the higher-order modes were expressed by the lower-order modes and a time delay was introduced. Thus the same precision is kept and long computation time is saved. First of all, theoretical basis of the rationality about truncation problem was established by proving that these systems have a unique solution and a global attractor; secondly, numerical simulations of various modes were presented by IMD.The thesis consists of five chapters, and the detailed content included following aspects.In Chapter1, we not only exposit the research background of this paper and the current situation of the development of infinite dimensional dynamical system, but also introduce the inertial manifold with delay. In addition, we simply introduced the main research problems discussed in this thesis.In Chapter2, some basic concept and theory that we will use in the thesis are presented.In chapter3, a kind of Kirchhoff type wave equation with strong damping utt-α△u1-M(｜▽u|2)△u+∫0’λ(t-s)△u(s)ds+g(u)ut+6(ut)=f(x), was considered, and we study the existence of this equation’s global attractor in the space X1=(H2(Ω)∩H01(Ω))×H01(Ω). Finally, numerical simulations of various modes for this system were presented by IMD.In Chapter4, we consider a class of strongly damped wave equations with critical growth exponent utt-α△u-μ△u1-β△utt+h(ut)+g(u)=f(x), and prove that the weak solution for this equation has a global attractor. Finally, numerical simulations of various modes for this system were presented by IMD.In Chapter5,we consider a class of nonlinear elastic rod equation with memory term utt-△u-r△ut-β△utt-φ(0)△u-∫0∞φ’(s)△u(t-s)ds=f(u), and prove that the weak and strong solutions for this equation have global attractors.Finally,numerical simulations of various modes for this system were presented by IMD.