The Research Of Nonlinear Dynamic Behavior Of Some Mechanical Systems

The main research content of nonlinear science is the common law of nonlinear phenomena invarious systems. Nonlinear dynamics, which is relatively mature, provides reliable theoretical basisfor nonlinear science. The research focus of nonlinear dynamics is bifurcation and chaos. Thegoverning equations of motion for many mechanical models can be described by high-dimensionalnonlinear systems. The investigations of mode bifurcations and chaos play an important part in thisarea.By using normal form method, global perturbation method, energy-phase method and numericalsimulation, the present dissertation is devoted to the stability, bifurcations and chaos of somehigh-dimensional nonlinear mechanical systems. The paper is divided into the following six chapters.In the first chapter, some important concepts, theories and methods of bifurcation and chaos innonlinear dynamic systems are introduced. They are the theoretical foundations of the later chapters.In the second chapter, stability and local bifurcation of a functionally graded material plate undertransversal and in-plane excitations in the uniform thermal environment are investigated bycombination of analytical and numerical method. Three types of degenerated equilibrium points arestudied in detail, which are characterized by a double zero eigenvalues, a simple zero and a pair ofpure imaginary eigenvalues and two pairs of pure imaginary eigenvalues in nonresonant case,respectively. Each case of the degenerated equilibrium is divided into two parts to considered, whichare parameters （μ₁,μ₂） and detuning parameters. The stability regions for the initial equilibriumsolution and the explicit expressions of the transition curves leading to bifurcations are obtained foreach case. The abundant dynamical behaviors are presented. Numerical simulations are carried out toverify the analytic results.In the third chapter, the local bifurcation and global bifurcation of a shallow cable subjected toharmonic excitations are investigated with the case of1:1internal resonance. The stability of theinitial equilibrium solution is investigated for the case of two pairs of pure imaginary eigenvalues.Numerical simulations are carried out to verify the analytic results. The existence of Silnikov-typesingle-pulse homoclinic orbits and chaotic behaviors are studied using the global perturbation methodwhich is proposed by Kovacic and Wiggins. The existence of Silnikov-type multi-pulse homoclinicorbits is investigated by applying the energy-phase method which is developed by Haller and Wiggins.The results indicate that Smale-type chaos may occur in the system.In the fourth chapter, the local bifurcation and global bifurcation of a taut string which is a limitingcase of shallow cable subjected to harmonic excitations are investigated with the case of1:2internal resonance. The stability of the initial equilibrium solution is investigated for the case of two zeros anda pair of pure imaginary eigenvalues. Numerical simulations are carried out to verify the analyticresults. By employing the global perturbation method which is proposed by Kovacic and Wiggins, theexistence of Silnikov-type single-pulse homoclinic orbits and chaotic behaviors are studied. With theenergy-phase method which is developed by Haller and Wiggins, the existence of Silnikov-typemulti-pulse homoclinic orbits is investigated. The results indicate that Smale-type chaos may occur inthe system.In the fifth chapter, the stability and bifurcation behaviors of a two-dimensional nonlinearviscoelastic panel in supersonic flow are investigated by combination of analytical and numericalmethod. The study includes single degree of freedom model analysis and two degree of freedommodel analysis.The last chapter is the summary and the outlook of this paper.