Dynamics of vibrational chaos and entanglement in triatomic molecules: Lie algebraic model

In this paper, the dynamics of chaos and the entanglement in triatomic molecular vibrations are investigated. On the classical aspect, we study the chaotic trajectories in the phase space. We employ the linear entropy to examine the dynamical entanglement of the two bonds on the quantum aspect. The correspondence between the classical chaos and the quantum dynamical entanglement is also investigated. As an example, we apply our algebraic model to molecule H2O.

MOTION TYPES AND CHAOS OF MULTI-BODY SYSTEMS VIBRATING WITH IMPACTS

In this paper,the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts.We discover that if every motion of the system is bounded,it has only four different types:periodic motion 7 t,non-periodic recurrent motion γ2,and non-Poisson stable mo- tions γ3 and γ4 approaching γ1 and γ2, respectively.γ2 is the source of chaos.It is very interesting that cha- otic motions seem stochastic but possess the character of recurrence.By way of example,we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table.The result obtained by us is different from that obtained by Holmes.

GLOBAL BIFURCATION AND CHAOS IN A VAN DER POL-DUFFING-MATHIUE SYSTEM WITH A SINGLE-WELL POTENTIAL OSCILLATOR

The global bifurcation and chaos are investigated in this paper for a van der Pol-Duffing-Mathieu system with a single-well potential oscillator by means of nonlinear dynamics. The autonomous system corresponding to the system under discussion is analytically studied to draw all global bifurcation diagrams in every parameter space. These diagrams are called basic bifurcation ones. Then fixing parameter in every space and taking the parametrically excited amplitude as a bifurcation parameter, we can observe how to evolve from a basic bifurcation diagram to a chaos pattern in terms of numerical methods. The results are sufficient to show that the system has distinct dynamic behavior. Finally, the properties of the basins of attraction are observed and the appearance of fractal basin boundaries heralding the onset of a loss of structural integrity is noted in order to consider how to control the extent and the rate of the erosion in the next paper.

APPLICATION OF WAVELET TRANSFORM TO BIFURCATION AND CHAOS STUDY

The response of a nonlinear vibration system may be of three types, namely, periodic, quasiperiodic or chaotic,,when the parameters of the system are changed. The periodic motions can be identified by Poincare map, and harmonic wavelet transform (HWT) can distinguish quasiperiod from chaos, so the existing domains of different types of motions of the system can be revealed in the parametric space with the method of HWT joining with Poincare map.

NONLINEAR OSCILLATIONS AND CHAOS IN A RAILWAY VEHICLE SYSTEM

The study of nonlinear oscillations and chaos of a railway freight car is undertaken. The vehicle is considered as a multiple rigid body system with 9 degrees of freedom. The vehicle forward speed is taken as the control parameter of the system. Hopf bifurcation point, limit cycles, quasiperiodic and chaotic motions of the system are computed by use of numerical methods. The identification of periodic, quasiperiodic and chaotic motions of the system is carried out by using the methods of phase plane portrait and Poincare map. Numerical results show that chaotic motion appears via the route of quasiperiodicity when the vehicle runs over a certain speed.

Predicting vibration signals of automobile engine using chaos theory

Condition monitoring and life prediction of the vehicle engine is an important and urgent problem during the vehicle development process. The vibration signals that are closely associated with the engine running condition and its development trend are complex and nonlinear. The chaos theory is used to treat the nonlinear dynamical system recently. A novel chaos method in conjunction with SVD (singular value decomposition)denoising skill are used to predict the vibration time series. Two types of time series and their prediction errors are provided to illustrate the practical utility of the method.

Bifurcations of Resonant Solutions and Chaos in Physical Pendulum Equation with Suspension Axis Vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.

Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell:an experimental and numerical study

The snap-through behaviors and nonlinear vibrations are investigated for a bistable composite laminated cantilever shell subjected to transversal foundation excitation based on experimental and theoretical approaches.An improved experimental specimen is designed in order to satisfy the cantilever support boundary condition,which is composed of an asymmetric region and a symmetric region.The symmetric region of the experimental specimen is entirely clamped,which is rigidly connected to an electromagnetic shaker,while the asymmetric region remains free of constraint.Different motion paths are realized for the bistable cantilever shell by changing the input signal levels of the electromagnetic shaker,and the displacement responses of the shell are collected by the laser displacement sensors.The numerical simulation is conducted based on the established theoretical model of the bistable composite laminated cantilever shell,and an off-axis three-dimensional dynamic snap-through domain is obtained.The numerical solutions are in good agreement with the experimental results.The nonlinear stiffness characteristics,dynamic snap-through domain,and chaos and bifurcation behaviors of the shell are quantitatively analyzed.Due to the asymmetry of the boundary condition and the shell,the upper stable-state of the shell exhibits an obvious soft spring stiffness characteristic,and the lower stable-state shows a linear stiffness characteristic of the shell.

Modal analysis on transverse vibration of axially moving roller chain coupled with lumped mass

The modal characteristics of the transverse vibration of an axially moving roller chain coupled with lumped mass were analyzed.The chain system was modeled by using the multi-body dynamics theory and the governing equations were derived by means of Lagrange's equations.The effects of the parameters,such as the axially moving velocity of the chain,the tension force,the weight of lumped mass and its time-variable assign position in chain span,on the modal characteristics of transverse vibration for roller chain were investigated.The numerical examples were given.It is found that the natural frequencies and the corresponding mode shapes of the transverse vibration for roller chain coupled with lumped mass change significantly when the variations of above parameters are considered.With the movement of the chain strand,the natural frequencies present a fluctuating phenomenon,which is different from the uniform chain.The higher the order of mode is,the greater the fluctuating magnitude and frequency are.

Torsional Vibration of Roller Oscillating Tooth Gear Drive

Torsional vibration of roller oscillating tooth gear drive (ROTGD) is studied in this paper. On the basis of conservation law for kinetic energy and potential energy, the mathematical expressions are developed which describe transformation of moment of inertia of inertial components into input shaft. Also, the formula is derived which expresses transformation of contact stiffness of elastic components into input shaft torsional stiffness. Besides, torsional vibration model of ROTGD is presented by using the transfer matrix method, and natural frequencies and vibration mode shapes are determined. Eventually, an example is given.

Vibration sensitive admixture for roller compacted concrete

In order to further speed up the construction progress,shorten the rolling time and reduce the rolling times of roller compacted concrete after analyzing work of construction progress improvement measures of roller compacted concrete,research on vibration sensitive admixture is carried out and satisfactory results are acquired.The new type vibration sensitive admixture was used in the project of Jinghong hydropower station in Yunnan Province,China.The engineering experimental results show that various indexes have been verified.Meanwhile,the compactness of roller compacted concrete is improved under the same rolling times,the rolling times and the duration time can be reduced under the same compactness requirement in construction practice.

Analysis of Vibrations of Roller of the Polymer Composite Coating Equipment on Stitches of Tarpaulin Materials

The article presents a structural diagram and the principle of operation of the installation of a sewing machine for applying a polymer composition to the stitch lines of tarpaulin materials. The calculation schemes and the mathematical model of oscillations of the axis of the composite roller during the application of the polymer composition along the lines of tarpaulin materials are presented. Based on the numerical solution of the problem, the regularities of roller oscillations are presented. The main parameters of the system are substantiated.

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.

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.

Effects of tension on vortex-induced vibration(ⅥⅤ) responses of a long tensioned cylinder in uniform flows

The effects of tension on vortex-induced vibration (VIV) responses for a tension-dominated long cylinder with an aspect ratio of 550 in uniform flows are experimentally investigated in this paper. The results show that elevated tension suppresses fluctuations of maximum displacement with respect to flow velocity and makes chaotic VIV more likely to appear. With respect to periodic VIV, if elevated tension is applied, the dominant vibration frequency in the in-line (IL) direction will switch from a fundamental vibration frequency to twice the value of the fundamental vibration frequency, which results in a ratio of the dominant vibration frequency in the IL direction to that in the cross-flow direction of 2.0. The suppression of the elevated tension in the fluctuation of the maximum displacement causes the axial tension to become an active control parameter for the VIV maximum displacement of a tension-dominated long riser or tether of an engineering structure in deep oceans. However, the axial tension must be optimized before being used since the high dominant vibration frequency due to the elevated tension may unfavorably affect the fatigue life of the riser or tether.

Complicated motion of particles on vibration screen surface

The particles’ motion on screen surface was studied by the way of nonlinear dynamics. It was found that the particles’ motion may be changed from one form to another with its vibration strength. When the vibration strength K is bigger than 1 and smaller than 1. 33, the particles’ motion is the bifurcating type;when K is bigger than 1. 33 and smaller than 1. 67, its motion becomes the double bifurcation type; when K is bigger than 1. 67, its motion changes into chaos motion type. On basis of studying all effects of large vibration strength on particles penetrating, it was pointed out that a larger vibration strength K of screen surface will increase the strength of particles’ motion, and then increase their probability of penetrating screen surface. A primary theory of particles’ motion under large vibration strength for moisture fine coal was established.

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study,the first-order shear deformation theory(FSDT)is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC).The vibration analyses of the FG-GPLRC truncated conical shell are presented.Considering the graphene platelets(GPLs)of the FG-GPLRC truncated conical shell with three different distribution patterns,the modified Halpin-Tsai model is used to calculate the effective Young’s modulus.Hamilton’s principle,the FSDT,and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell.The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell.Then,the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method.The effects of the weight fraction and distribution pattern of the GPLs,the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed.This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.

基于ADAMS/Vibration的立式辊磨机粉磨装置的振动分析

针对立式辊磨机在运行过程中存在的振动现象,联合ADAMS/View与ANSYS建立立式辊磨机的刚柔耦合模型,导入ADAMS/Vibration创建立式辊磨机的振动分析模型,对该模型进行模态分析和强迫振动分析,得到系统的各阶固有频率、模态主振型及频率响应曲线,为其结构优化改进提供了理论依据。

Nonlinear parametrically excited vibration and active control of gear pair system with time-varying characteristic

In the present work, we investigate the nonlinear parametrically excited vibration and active control of a gear pair system involving backlash, time-varying meshing stiffness and static transmission error. Firstly, a gear pair model is established in a strongly nonlinear form, and its nonlinear vibration characteristics are systematically investigated through different approaches. Several complicated phenomena such as period doubling bifurcation, anti period doubling bifurcation and chaos can be observed under the internal parametric excitation. Then, an active compensation controller is designed to suppress the vibration, including the chaos. Finally, the effectiveness of the proposed controller is verified numerically.

Theoretical analysis of nonlinear vibration characteristics of gear pair with shafts

The circumferential vibration of a gear pair is a parametric excitation caused by nonlinear tooth stiffness,which fluctuates with meshing.In addition,the vibration characteristics of the gear pair become complicated owing to the tooth profile error and backlash.It is considered that the circumferential vibration of the gear pair is affected by the torsional vibration of the shafts.It is important to understand quantitatively the vibration characteristics of the gear system considering the shafts.Therefore,the purpose of this research was to clarify the nonlinear vibration characteristics of a gear pair considering the influence of the shafts using theoretical methods.To achieve this objective,calculations were performed using equations of motion in which the circumferential vibration of the gear pair and the torsional vibration of the shafts were coupled.The nonlinear tooth stiffness was represented by a sine wave.The influence of tooth separation was considered by defining a nonlinear function using backlash and the tooth profile error.For the numerical calculations,both stable and unstable periodic solutions were obtained by using the shooting method.The effect of the shafts on the gear system vibration were clarified by comparing the results in the cases in which the shaft was not considered,one shaft was considered,and both shafts were considered.