Study On Dynamic Behavior Of A Class Of Cylindrical Geometrical Nonlinear Systems

Simple pendulum is a typically geometrical nonlinear system with cylindrical characteristics,which plays a very important role in nonlinear dynamics.In recent years,a class of spring-mass system with the geometrical nonlinearity named SD(Smooth and Discontinuous)oscillator is presented which has attracted extensive attentions due to its originality.Particularly,this geometrical nonlinear system can accurately restore the essential nonlinearity.The aim of this dissertation is to inherit and develop the original theory of SD oscillator and provide the primitive theory basis for engineering application of the cylindrical geometrical nonlinearity.This cylindrical geometrical nonlinear system is composed of two classical geometrical nonlinear dynamical models of traditional pendulum and SD oscillator,which exhibits both smooth and discontinuous dynamics of multistable characteristics depending on a smoothness parameter.It is worth pointing out that this system is a strongly nonlinear system with the trigonometric function and irrational terms.A series of cylindrical geometrical nonlinear systems are constructed based upon SD oscillator in this dissertation.It is quite clear that the motivations of developing the analytical methods for the proposed systems are to show completely the global dynamic behaviors and describe precisely the local dynamic characteristics.Furthermore,the bifurcation and chaos of this cylindrical geometrical nonlinear system are reviewed with the help of the nonlinear dynamical techniques.The global bifurcations of large amplitude motions and the resonance response of small amplitude motions are clarified and investigated by using numerical and analytical methods.The main content and achievement of this thesis are the following:Firstly,based on SD oscillator,we propose a novel mechanical rotating model named rotating SD oscillator which consists of a rotating disk linked by a pair of oblique springs with fixed ends.This proposed system is a tipically cylindrical geometrical nonlinear system.For the free vibration system,a new type heteroclinic orbit connecting a standard saddle equilibrium and a non-standard saddle equilibrium(saddle-like)is proposed and studied in the discontinuous system.By means of Hamilton function,the analytical solutions of the heteroclinic-like orbits can be obtained.The mathematic characteristics and their corresponding movement laws of the heteroclinic-like orbits are reported via the analytical solutions.For the forced vibration system,the conventional and extended Melnikov methods are employed to detect the chaotic thresholds for a pair of the homoclinic orbits and a pair of the heteroclinic-like orbits for both smooth and discontinuous systems,respectively.Numerical simulations show the efficiency of the proposed method and the results presented herein demonstrate the predicated bifurcation,periodic solution and chaotic attractors and so on.Secondly,we present a rotating pendulum model which exhibits the coupling dynamic behavior of inverted pendulum and SD oscillator.Because the traditional approximate method can't accurately reflects the global features of rotary motion of the cylindrical geometrical nonlinear system,based upon the analytical expression of equilibria and the double pitchfork bifurcation,a cylindrical approximate system is introduced to successfully reflect the nature of the original system of simple pendulum,inverted pendulum and bistate characteristics.Under the perturbations of linear damping and harmonic excitation,the analytic al chaotic thresholds can be obtained by means of Melnikov method.Through theoretical analysis and numerical comparison,it is found that the dynamic behavior of the approximate system is topologically equivalent to the original system,such as the bifurcation diagram,the periodic solution and the structure chaotic attractor,the numerical characteristic of chaos and so on.Furthermore,we consider the cylindrical geometrical nonlinear system under a nonlinear perturbation which allows us to investigate various kinds of bifurcations and limit cycles.This system undergoes various kinds of dynamic bifurcations including the pitchfork bifurcation,the homo-heteroclinic orbits transition,Hopf bifurcation,the homoclinic bifurcations,the saddle node bifurcation of periodic orbits,the homoclinic-like bifurcations,Hopf-homoclinic bifurcation,Hopf-saddle node bifurcation of periodic orbits and so on.The number,position and stability of all the oscillating and rotational limit cycles are given as the parameters vary.It is found that there exists five limit cycles in the smooth system due to two saddle-node bifurcation of periodic orbits existing.Unlike the smooth system,the discontinuous system undergoes Hopf bifurcation and the homoclinic-like bifurcation without any saddle-node bifurcation of periodic orbits.Especially,it is found that the formation mechanism of two saddle node bifurcations of periodic orbit are caused by the homoclinic orbit of the first type and the characteristics of equilibrium point,respe-ctively.It is worth pointing out that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles.Finally,a novel parametrically excited pendulum with irrational nonlinearity is proposed,which comprises of a simple pendulum linked by an oblique spring under base excitation.Due to the geometry configuration,this parametric excited system bears both the bistable state and discontinuous characteristics.For the small oscillations,this parametric excited system can be described by Mathieu equation coupled with SD oscillator of which dynamic response is detected analytically in smooth and discontinuous case by means of the averaging method.Numerical simulations is carried out to describe the complicated dynamics behavior of multiple periodic motions and different types of chaotic motions,such as the oscillating periodic and chaotic motions,the oscillating-rotational chaotic and periodic motions,the rotational chaotic and periodic motions,coexisting periodic solutions and coexisting chaotic solutions.Additionally,we present a more complicated cylindrical geometrical nonlinear system with tristable feature which undergones the rich static bifurcations and the atypical equilibrium including supercritical pitchfork bifurcation,subcritical pitchfork bifurcation,saddle-node bifurcation,sharp equilibrium,saddlelike equilibrium,tangent-saddle equilibrium and so on.