Some Research On Behaviors Of Bifurcation And Chaos In Several Mechanical Systems

Many problems can be described as nonlinear mechanical systems in nature and engineering.However,the complex structure of nonlinear mechanical systems can make them show more complicated dynamical behaviors.Therefore,stability and bifurcation analysis of nonlinear mechanical systems is of great practical significance for the engineering application.In this paper,the stability,bifurcation and chaotic dynamics of nonlinear mechanical systems are investigated employing the normal form theory,Melnikov method,global perturbation method,energy-phase method and the extended Melnikov method,respectively.It is shown that rich dynamical behaviors can occur.Numerical simulations are presented to verify the theoretical results.The paper is divided into the following sections.Chapter 1 provides an overview of research situation and research methods for nonlinear dynamical systems.It contains an introduction of stability,bifurcation and global perturbation methods.In chapter 2,the global bifurcations and chaotic dynamics of an aero-thermo-elastic functionally graded material(FGM)truncated conical shell under complex loads are investigated with the case of 1:2 internal resonance and primary parametric resonance.The method of multiple scales is utilized to obtain the averaged equations.The global perturbation method is employed to study the existence of single-pulse Shilnikov-type homoclinic orbits,and the critical conditions for the occurrence of chaotic motions are obtained.In addition,the energy-phase method developed by Haller and Wiggins is used to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the FGM truncated conical shell.The analytical results obtained here mean the existence of chaos in the sense of the Smale horseshoes for the FGM truncated conical shell.Numerical simulations are presented to show that the theoretical results are correct.It is demonstrated simultaneously that the structural-damping,the aerodynamic-damping,and the in-plane and transverse excitations have important influences on the nonlinear dynamics of the FGM truncated conical shell.In chapter 3,the chaotic dynamics of a reinforced composite plate with the carbon nanotubes(CNTs)under combined in-plane and transverse excitations are studied in the case of 1:1 internal resonance,principal parametric resonance,and 1/2 subharmonic resonance.The method of multiple scales is used to derive the averaged equations.The global perturbation method is utilized to study the existence of single-pulse Shilnikov-type homoclinic orbits,and the critical conditions for the occurrence of chaotic motions are obtained.The energy-phase method is utilized to examine the global bifurcations and chaotic dynamics of the CNT reinforced composite plate.And we prove the existence of the multi-pulse jumping orbits in the perturbed phase space for the dissipative perturbations.In order to verify the theoretical results,numerical simulations are given to show the multi-pulse Shilnikov-type chaotic motions in the CNT reinforced composite plate.In chapter 4,the multipulse global bifurcations and chaotic dynamics of a simply supported functionally graded piezoelectric(FGP)rectangular plate with bonded piezoelectric layer are investigated with the case of 1:2 internal resonance and primary parametric resonance.According to the explicit expressions of normal form,the extended Melnikov method developed by Camassa et al.is employed to discuss the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics in the sense of the Smale horseshoes.Two simple zeroes of the k-pulse Melnikov function are obtained.The analytical results indicate that there exists the Shilnikov-type multipulse chaotic dynamics for the FGP plate.The influence of the in-plane excitation and the piezoelectric voltage excitation on the system dynamical behaviors is also discussed by numerical simulations.In chapter 5,the bifurcations and chaotic motions of the nonlinear viscoelastic plates subjected to subsonic flow and external loads are studied by means of Melnikov method.The critical conditions for the occurrence of chaotic motions are obtained.The chaotic features on the system parameters are discussed in detail.The conditions for subharmonic bifurcations are also obtained by the subharmonic Melnikov function.For the system with no structural damping,we find that chaotic motions can occur through infinite subharmonic bifurcations of odd orders.Furthermore,we confirm our theoretical predictions by numerical simulations.In chapter 6,stability and bifurcation behaviors of an axially moving beam subjected to two frequency excitations are investigated both analytically and numerically.Based on the normal form theory,three types of degenerated equilibrium points are analyzed.The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are obtained.Possible bifurcations leading to 2-D tori are also studied.Moreover,numerical simulations are given,which agree with the analytical predictions.Chapter 7 ends the paper with conclusions and some open problems.