Nonlinear Dynamics Behaviors of a Rotor Roller Bearing System with Radial Clearances and Waviness Considered

A rotor system supported by roller beatings displays very complicated nonlinear behaviors due to nonlinear Hertzian contact forces, radial clearances and bearing waviness. This paper presents nonlinear bearing forces of a roller bearing under four-dimensional loads and establishes 4-DOF dynamics equations of a rotor roller bearing system. The methods of Newmark-β and of Newton-Laphson are used to solve the nonlinear equations. The dynamics behaviors of a rigid rotor system are studied through the bifurcation, the Poincar è maps, the spectrum diagrams and the axis orbit of responses of the system. The results show that the system is liable to undergo instability caused by the quasi-periodic bifurcation, the periodic-doubling bifurcation and chaos routes as the rotational speed increases. Clearances, outer race waviness, inner race waviness, roller waviness, damping, radial forces and unbalanced forces-all these bring a significant influence to bear on the system stability. As the clearance increases, the dynamics behaviors become complicated with the number and the scale of instable regions becoming larger. The vibration frequencies induced by the roller bearing waviness and the orders of the waviness might cause severe vibrations. The system is able to eliminate non-periodic vibration by reasonable choice and optimization of the parameters.

Chaos Synchronization in Discrete-Time Dynamical Systems with Application in Population Dynamics

Study of chaotic synchronization as a fundamental phenomenon in the nonlinear dynamical systems theory has been recently raised many interests in science, engineering, and technology. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time dynamical system which has been coupled using convex link function, we introduce a synchronization threshold which passes that makes the drive-response system lose complete coupling and synchronized behaviors. We provide the application of this type of coupling in synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling makes this chaotic system and its corresponding coupled system starting from different initial conditions, quickly get synchronized.

Global Dynamic Characteristic of Nonlinear Torsional Vibration System under Harmonically Excitation

Torsional vibration generally causes serious instability and damage problems in many rotating machinery parts. The global dynamic characteristic of nonlinear torsional vibration system with nonlinear rigidity and nonlinear friction force is investigated. On the basis of the generalized dissipation Lagrange＇s equation, the dynamics equation of nonlinear torsional vibration system is deduced. The bifurcation and chaotic motion in the system subjected to an external harmonic excitation is studied by theoretical analysis and numerical simulation. The stability of unperturbed system is analyzed by using the stability theory of equilibrium positions of Hamiltonian systems. The criterion of existence of chaos phenomena under a periodic perturbation is given by means of Melnikov＇s method. It is shown that the existence of homoclinic and heteroclinic orbits in the unperturbed system implies chaos arising from breaking of homoclinic or heteroclinic orbits under perturbation. The validity of the result is checked numerically. Periodic doubling bifurcation route to chaos, quasi-periodic route to chaos, intermittency route to chaos are found to occur due to the amplitude varying in some range. The evolution of system dynamic responses is demonstrated in detail by Poincare maps and bifurcation diagrams when the system undergoes a sequence of periodic doubling or quasi-periodic bifurcations to chaos. The conclusion can provide reference for deeply researching the dynamic behavior of mechanical drive systems.

Synchronization Phenomena Investigation of a New Nonlinear Dynamical System 4D by Gardano’s and Lyapunov’s Methods

Synchronization is one of the most important characteristics of dynamic systems.For this paper,the authors obtained results for the nonlinear systems controller for the custom Synchronization of two 4D systems.The findings have allowed authors to develop two analytical approaches using the second Lyapunov(Lyp)method and the Gardanomethod.Since the Gardano method does not involve the development of special positive Lyp functions,it is very efficient and convenient to achieve excessive systemSYCR phenomena.Error is overcome by using Gardano and overcoming some problems in Lyp.Thus we get a great investigation into the convergence of error dynamics,the authors in this paper are interested in giving numerical simulations of the proposed model to clarify the results and check them,an important aspect that will be studied is Synchronization Complete hybrid SYCR and anti-Synchronization,by making use of the Lyapunov expansion analysis,a proposed control method is developed to determine the actual.The basic idea in the proposed way is to receive the evolution of between two methods.Finally,the present model has been applied and showing in a new attractor,and the obtained results are compared with other approximate results,and the nearly good coincidence was obtained.

Complex nonlinear behaviors of a rotor dynamical system with non-analytical journal bearing supports

Nonlinear dynamic behaviors of a rotor dynamical system with finite hydrodynamic bearing supports were investigated. In order to increase the numerical accuracy and decrease computing costs, the isoparametric finite element method based on variational constraint approach is introduced because analytical bearing forces are not available. This method calculates the oil film forces and their Jacobians simultaneously while it can ensure that they have compatible accuracy. Nonlinear motion of the bearing-rotor system is caused by strong nonlinearity of oil film forces with respect to the displacements and velocities of the center of the rotor. A method consisting of a predictor-corrector mechanism and Newton-Raphson method is presented to calculate equilibrium position and critical speed corresponding to Hopf bifurcation point of the bearing-rotor system. Meanwhile the dynamic coefficients of bearing are obtained. The nonlinear unbalance periodic responses of the system are obtained by using Poincaré-Newton-Floquet method and a combination of predic- tor-corrector mechanism and Poincaré-Newton-Floquet method. The local stability and bifuration behaviors of periodic motions are analyzed by the Floquet theory. Chaotic motion of long term dynamic behaviors of the system is analyzed with power spectrum. The numerical results reveal such complex nonlinear behaviors as periodic, quasi-periodic, chaotic, jumped and coexistent solutions.

Controlof Orbitand Controlof Chaosina Class of Dynamic System

The problem of control of orbit for the dynamic system x ¨+x(1-x)(x-a)=0 is discussed. Any unbounded orbit of the dynamic system can be controlled to become a bounded periodic orbit by adding a periodic step excitation to the system. By using a nonlinear feedback control law presented in this paper the chaos of the dynamic system with excitation and damping is stabilized. This method is more effectual than the linear feedback control.

Analysis on Nonlinear Dynamic Properties of Dual-Rotor System

In order to clarify the effects of support structure on a dual-rotor machine,a dynamic model is established which takes into consideration the contact force of ball bearing and the cubic stiffness of elastic support. Bearing clearance,Hertz contact between the ball and race and the varying compliance effect are included in the model of ball bearing. The system response is obtained through numerical integration method,and the vibration due to the periodic change of bearing stiffness is investigated. The motions of periodic,quasiperiodic and even chaotic are found when bearing clearance is used as control parameter to simulate the response of rotor system. The results reveal two typical routes to chaos: quasi-periodic bifurcation and intermittent bifurcation. Large cubic stiffness of elastic support may cause jump and hysteresis phenomena in resonance curve when rotors run at the critical-speed region. The modeling results acquired by numerical simulation will contribute to understanding and controlling of the nonlinear behaviors of the dual-rotor system.

Criteria for Instability and Chaos in Nonlinear Systems

In the article, the methods of investigating the instability that were formulated earlier by the authors are systematized in the form of a set of criteria for the instability and chaos. The latest ones are used to study chaotic dynamics in the problems of Sprott and the nonlinear electronic generator of the CRC.

On Universality of Transition to Chaos Scenario in Nonlinear Systems of Ordinary Differential Equations of Shilnikov’s Type

Several nonlinear three-dimensional systems of ordinary differential equations are studied analytically and numerically in this paper in accordance with universal bifurcation theory of Feigenbaum-Sharkovskii-Magnitsky [1] [2]. All systems are autonomous and dissipative and display chaotic behaviour. The analysis confirms that transition to chaos in such systems is performed through cascades of bifurcations of regular attractors.

C-L METHOD AND ITS APPLICATION TO ENGINEERING NONLINEAR DYNAMICAL PROBLEMS

The C-L method was generalized from Liapunov-Schmidt reduction method, combined with theory of singularities, for study of non-autonomous dynamical systems to obtain the typical bifurcating response curves in the system parameter spaces. This method has been used, ar an example, to analyze the engineering nonlinear dynamical problems by obtaining the bifurcation programs and response curves which are useful in developing techniques of control to subharmonic instability of large rotating machinery.

Nonlinear Deterministic Chaos in Benue River Flow Daily Time Sequence

The Various physical mechanisms governing river flow dynamics act on a wide range of temporal and spatial scales. This spatio-temporal variability has been believed to be influenced by a large number of variables. In the light of this, an attempt was made in this paper to examine whether the daily flow sequence of the Benue River exhibits low-dimensional chaos;that is, if or not its dynamics could be explained by a small number of effective degrees of freedom. To this end, nonlinear analysis of the flow sequence was done by evaluating the correlation dimension based on phase space reconstruction and maximal Lyapunov estimation as well as nonlinear prediction. Results obtained in all instances considered indicate that there is no discernible evidence to suggest that the daily flow sequence of the Benue River exhibit nonlinear deterministic chaotic signatures. Thus, it may be conjectured that the daily flow time series span a wide dynamical range between deterministic chaos and periodic signal contaminated with additive noise;that is, by either measurement or dynamical noise. However, contradictory results abound on the existence of low-dimensional chaos in daily streamflows. Hence, it is paramount to note that if the existence of low-dimension deterministic component is reliably verified, it is necessary to investigate its origin, dependence on the space-time behavior of precipitation and therefore on climate and role of the inflow-runoff mechanism.

Neutrality Criteria for Stability Analysis of Dynamical Systems through Lorentz and Rossler Model Problems

Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.

Three Routes to Chaos in Electrical Power Systems

Routes to chaos in power systems are studied. Using a three-bus simple system, three routes that can lead power system to chaos are presented, illustrated and discussed. They are cascading period doubling bifurcation, torus bifurcation and route directly initiated by a large disturbance. Period doubling bifurcation is caused by a real Floquet multiplier going out of the unit circle from point (-1,0), while torus bifurcation is caused by a couple of conjugated Floquet multipliers going out of the unit circle with a non-zero imaginary part in the complex plane. Cascading period doubling bifurcation and torus bifurcation are two typical routes to chaos in dynamic systems, which have been investigated in the previous studies. The last route, i.e. directly initiated by a large disturbance, is reported and studied. This phenomenon reveals that chaos is caused by external disturbances in power systems.

Classical Chaos on Double Nonlinear Resonances in Diatomic Molecules

Classical chaotic behavior in diatomic molecules is studied when chaos is driven by a circularly polarized resonant electric field and expanding up to fourth order of approximation the Morse’s potential and angular momentum of the system. On this double resonant system, we find a weak and a strong stationary (or critical) points where the chaotic characteristics are different with respect to the initial conditions of the system. Chaotic behavior around the weak critical point appears at much weaker intensity on the electric field than the electric field needed for the chaotic behavior around the strong critical point. This classical chaotic behavior is determined through Lyapunov exponent, separation of two nearby trajectories, and Fourier transformation of the time evolution of the system. The threshold of the amplitude of the electric field for appearing the chaotic behavior near each critical point is different and is found for several molecules.

Suppression of Chaotic Behaviors in a Complex Biological System by Disturbance Observer-based Derivative-Integral Terminal Sliding Mode

Coronary artery systems are a kind of complex biological systems. Their chaotic phenomena can lead to serious health problems and illness development. From the perspective of engineering, this paper investigates the chaos suppression problem. At first, nonlinear dynamics of coronary artery systems are presented. To suppress the chaotic phenomena, the method of derivative-integral terminal sliding mode control is adopted. Since coronary artery systems suffer from uncertainties, the technique of disturbance observer is taken into consideration. The stability of such a control system that integrates the derivative-integral terminal sliding mode controller and the disturbance observer is proven in the sense of Lyapunov. To verify the feasibility and effectiveness of the proposed strategy, simulation results are illustrated in comparison with a benchmark.

Dynamic Model of Mineralization Enrichment and Its Applications

This paper studies the chaos dynamic mechanism of the migration, enrichment and mineralization of elements in the crust. The research shows that the interaction of the nonlinear process in the geological environment is an essential factor for the uneven distribution of elements and the mineralization in the crust, determining the element contents and the fractal structure of the distribution of the large and small sized mineral deposits. The logistic map is a better mathematical model describing the behavior of the chaos dynamic. The parameter μ , i.e., the mineralizing potential, is employed to divide the region into non mineralization region or mineralization region. The value of the parameter μ in model (3) with true data (in Xinjiang Au tomatio region, China) is obtained with the statistical method. The forecasting results are generally in accordance with those obtained with other methods, for example, with the characteristic analysis.

Detecting Unstable Periodic Orbits in Hyperchaotic Systems Using Subspace Fixed-Point Iteration

We present a numerical method for efficiently detecting unstable periodic orbits(UPO’s)embedded in chaotic attractors of high-dimensional systems.This method,which we refer to as subspace fixed-point iteration, locates fixed points of Poincare maps using a form of fixed-point iteration that splits the phase space into appropriate subspaces.In this paper,among a number of possible implementations,we primarily focus on a subspace method based on the Schmelcher-Diakonos(SD)method that selectively locates UPO’s by varying a stabilizing matrix,and present some applications of the resulting subspace SD method to hyperchaotic attractors where the UPO’s have more than one unstable direction.

Signature of chaos in the semi quantum behavior of a classically regular triple well heterostructure

We analyze the phenomenon of semiquantum chaos in the classically regular triple well model from classical to quantum. His dynamics is very rich because it provides areas of regular be-havior, chaotic ones and multiple quantum tun-neling depending on the energy of the system as the Planck’s constant varies from 0 to 1. The Time Dependent Variational Principle TDVP using generalized Gaussian trial wave function, which, in many-body theory leads to the Hartree Fock Approximation TDHF, is added to the tech-niques of Gaussian effective potentials and both are used to study the system. The extended classical system with fluctuation variables non- linearly coupled to the average variables exhibit energy dependent transitions between regular behavior and semi quantum chaos monitored by bifurcation diagram together with some numerical indicators.

SLIDING MODE CONTROL FOR ACTIVE AUTOMOBILE SUSPENSIONS

Nonlinear control methods are presented based on theory of sliding mode control (SMC) or variable structure control (VSC) for application to active automobile suspensions. Requirements of reducing manufacturing cost and energy consumption of the active suspension system may be satisfied by reasonable design of the sliding surface and hydraulic servo system. Emphasis is placed on the study of the discrete sliding mode control method (DSMC) applicable for a new sort of speed on off solenoid valves of anti dust capability and low price. Robustness and effectiveness of the feedback linearized controller in typical road conditions are demonstrated by numerical results for a quarter car suspension model.

基于混沌云量子蝙蝠CNN-GRU大坝变形智能预报方法研究

针对大坝变形影响因素复杂、精准预报难度较大问题,为了提高在大坝安全管理过程中大坝变形的预报精度,本文从大坝变形非线性动力系统时间序列的强非线性出发,引入深度卷积神经网络,对大坝变形及其空间影响特性进行挖掘,引入门控循环单元,对大坝变形的时域特性进行挖掘,构建应用于大坝变形预报的深度卷积神经网络-门控循环单元大坝变形组合深度学习网络;同时,为了获取深度卷积神经网络-门控循环单元组合网络的最佳超参,引入了混沌云量子蝙蝠算法,建立了基于混沌云量子蝙蝠算法算法的深度卷积神经网络-门控循环单元组合网络超参优选方法;最后,提出了深度卷积神经网络-门控循环单元-混沌云量子蝙蝠算法大坝变形组合深度学习智能预报方法。基于实测数据开展预报研究,对比结果表明:与对比模型相比,提出的深度卷积神经网络-门控循环单元-混沌云量子蝙蝠算法预报方法取得了更精确的预报结果,混沌云量子蝙蝠算法算法用于超参优选获得了更佳的超参组合。