Nonlinear dynamics of a rub-impact micro-rotor system with scale-dependent friction model

Scale effects in dry friction at microscale and the coefficients of friction due to adhesion force and two- and three-body deformations are considered. A rub-impact micro-rotor model with scaling nonlinear rub-impact force is presented and the nonlinear dynamic characteristics of the system in micro-electro-mechanical systems (MEMS) are investigated when the rotating speed, imbalance, damping coefficient, scale length and fractal dimension are regarded as the control parameters. Effects of scale length, fractal dimension, velocity-dependent impact factor and contact form on the coefficients of dry friction are investigated and discussed, and used to study the nonlinear behavior of rub-related vibrations with a large number of numerical simulations. The effects of rotating speed, imbalance, damping coefficient, and friction coefficient on the micro-rotor system responses are studied. It is indicated that the rub-impact micro-rotor system with the scale effects in friction alternates among the periodic, quasi-periodic and chaotic motions as the system parameters change. The results can be effectively used to diagnose the rub-impact fault, reduce the failure and improve the characteristics of a micro-rotor system, and optimize the design of micro-rotating machinery in MEMS.     

Parametrically excited helicopter ground resonance dynamics with high blade asymmetries

The present work is aimed at verifying the influence of high asymmetries in the variation of in-plane lead-lag stiffness of one blade on the ground resonance phenomenon in helicopters. The periodical equations of motions are analyzed by using Floquet's Theory (FM) and the boundaries of instabilities predicted. The stability chart obtained as a function of asymmetry parameters and rotor speed reveals a complex evolution of critical zones and the existence of bifurcation points at low rotor speed values. Additionally, it is known that when treated as parametric excitations; periodic terms may cause parametric resonances in dynamic systems, some of which can become unstable. Therefore, the helicopter is later considered as a parametrically excited system and the equations are treated analytically by applying the Method of Multiple Scales (MMS). A stability analysis is used to verify the existence of unstable parametric resonances with first and second-order sets of equations. The results are compared and validated with those obtained by Floquet's Theory. Moreover, an explanation is given for the presence of unstable motion at low rotor speeds due to parametric instabilities of the second order.     

Nonlinear dynamics of two harmonically excited elastic structures with impact interaction

In this paper, non-smooth dynamics of two elastic beams excited by harmonic force with impact interaction is studied through analyses, simulations, and experiments. A two degree-of-freedom vibro-impact model is improved by applying the Galerkin approach and Newton's impact law for the two cantilever beams with impact interaction. Numerical analysis is taken to investigate the vibro-impact motions of cantilever beams excited by harmonic force. The l-adding periodic motions and k=1/1, k=2/2, k=3/4, and k=4/4 type of stable periodic motions of the impacted cantilever beam are presented. Poincaré map is established and the Floquet multipliers of the periodic motions are obtained through semi-analytical method to determine the stability of the motions near the bifurcation point. Through associated experiments, typical bifurcations and chaos of the non-smooth system are examined, which are in good agreement with numerical results.     

ADDING ONE BIFURCATION IN A ROTOR SYSTEM WITH CLEARANCES

A mathematical model of a rotor system with clearances is analysed by the application of modern nonlinear dynamic theory. From the bifurcation diagrams, it is discovered that the rotor system alternates between periodic and chaotic motions at a supercritical rotational speed, and after undergoing a chaotic region the periodic number of the motion will increase by one. At the same time, the periodic number is equal correspondingly to the integral multiple of the critical rotational speed. At the subcritical rotational speed, it is discovered that the chaotic bands among successive orders of superharmonic responses return to the period one through a reversed period-doubling bifurcation, as a result of a period-doubling bifurcation. It is shown that the increase of damping may reduce the width of the chaotic bands and the amplitude of the periodic response; the increase of nonlinear degree also leads to the reduction of chaotic bandwidth, but makes the amplitude of the subharmonic response increase. So it is suggested that proper damping and correct material selection by considering the dynamic characteristics of the rotor system may reduce the proportion of faults and enhance the dynamic characteristics when designing the rotor system. The working speed should not be selected at N times its natural frequency and should not be set in the chaotic bands among the successive orders of periodic motion for the same purpose.     

基于碰撞能量模型的非线性碰摩评估研究

提出了一种用于评估非线性碰摩的碰撞能量模型(IEM),根据模型特性可得出IEM指标η用于评估碰摩故障的严重程度或发生概率. 通过对在不同阻尼比ζ状态下IEM指标η随转速比Ω的响应变化趋势研究,给出了η随ζ和Ω的响应联合分布. 结合IEM模型特性提出了"敏感区域"的概念. 此区域概念的提出,不仅对于转子设备的操作运行具有较大的参考价值,对于转子系统的设计更有重要的指导意义. 关键词： 碰撞能量模型 碰摩 转子系统 故障诊断     

The application of Duffing oscillator in characteristic signal detection of early fault

This paper presents a study of application of Duffing oscillator for extracting the features of early mechanical failure signal. By analysis of global solutions and bifurcation set of Duffing equation, we conclude that the bifurcation threshold, which corresponds to the maximum orbit outside the homoclinic orbit of Duffing equation, can be used to detect weak signal, such as the characteristic signal of early machinery fault. Therefore, the Duffing oscillator is suitable to be a model for weak signal detection, although some relative aspects for practical uses should be considered carefully. In this paper, the strategies for testing whether the weak signal is exists and how to estimate its amplitude are discussed in detail. Moreover, an example is presented here to demonstrate the utility of this method by analyzing the early rub-impact signal appearing in a rotor. The results show that the method is effective for early detection of fault described by specific periodic-motion components such as rub-impact fault of rotor in aeroengine.     

Polarization fields and phase space densities in storage rings: Stroboscopic averaging and the ergodic theorem

A class of orbital motions with volume preserving flows and with vector fields periodic in the “time” parameter θ is defined. Spin motion coupled to the orbital dynamics is then defined, resulting in a class of spin-orbit motions which are important for storage rings. Phase space densities and polarization fields are introduced. It is important, in the context of storage rings, to understand the behavior of periodic polarization fields and phase space densities. Due to the 2π time periodicity of the spin-orbit equations of motion the polarization field, taken at a sequence of increasing time values θ,θ+2π,θ+4π,…, gives a sequence of polarization fields, called the stroboscopic sequence. We show, by using the Birkhoff ergodic theorem, that under very general conditions the Cesàro averages of that sequence converge almost everywhere on phase space to a polarization field which is 2π-periodic in time. This fulfills the main aim of this paper in that it demonstrates that the tracking algorithm for stroboscopic averaging, encoded in the program SPRINT and used in the study of spin motion in storage rings, is mathematically well-founded. The machinery developed is also shown to work for the stroboscopic average of phase space densities associated with the orbital dynamics. This yields a large family of periodic phase space densities and, as an example, a quite detailed analysis of the so-called betatron motion in a storage ring is presented.     

Transition to instability in a kicked bose-einstein condensate

A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor. For weak interactions between atoms, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.     

Experimental evidence for chaotic behaviour of a parametrically forced pendulum

Experimental evidence is presented for chaotic type nonperiodic motions of a parametrically forced pendulum. A bifurcation diagram is measured directly, showing successive subharmonic bifurcations to ?/4, onset of a periodic motion and the appearance of periodic motions via intermittency. The experimentally determined threshold values of the amplitude of the driving force for the first period doublings and the onset of a periodic motion are found to be in good agreement with the theoretical predictions.     

Paul阱中共线三离子体系的经典动力学

研究了在Paul阱囚禁场赝势作用下共线构形的三离子体系经典动力学特性.尽管这是一个非线性体系，但不存在混沌，即体系在任何能量下运动都是规则的，而相空间则由两个轨迹为对称和反对称周期(或准周期)轨道的KAM不变环面构成.体系的两条最简单的周期轨道S和A的周期随能量E的下降而增大，并在E趋于体系的最小值E_min=3.0时分别为反对称和对称谐振动. 关键词：     

3-D spatial chaos in the elastica and the spinning top: Kirchhoff analogy

The existence of spatially chaotic deformations in an elastica and the analogous motions of a free spinning rigid body, an extension of the problem originally examined by Kirchhoff are investigated. It is shown that a spatially periodic variation in cross sectional area of the elastica results in spatially complex deformation patterns. The governing equations for the elastica were numerically integrated and Poincare maps were created for a number of different initial conditions. In addition, three dimensional computer images of the twisted elastica were generated to illustrate periodic, quasiperiodic, and stochastic deformation patterns in space. These pictures clearly show the existence of spatially chaotic deformations with stunning complexity. This finding is relevant to a wide variety of fields in which coiled structures are important, from the modeling of DNA chains to video and audio tape dynamics to the design of deployable space structures.     

Density functional theory study on antiferromagnetic interactions in a silver (I) complex of nitronyl nitroxide

A one-dimensional (1D) silver (I) complex of nitronyl nitroxide with fairly strong antiferromagnetic interaction, in which the metal ions are diamagnetic, is investigated by means of the ab initio crystal orbital method based on the density functional theory. The calculated values of the magnetic coupling constant (J) are close to the experimental measured J value in the periodic system. The magneto-structural correlation reveals that the existence of an antiferromagnetic coupling pathway along nitronyl nitroxide units via silver (I) ion in this system. The spin population distribution also shows the existence of spin delocalization along the ONCNO–Ag–ONCNO, which affords a rational interpretation for the antiferromagnetic interaction mechanism.     

Thermally induced transitions to chaos in plate vibrations

The geometrically nonlinear vibrations of linear elastic and isotropic plates under the combined effect of thermal fields and mechanical excitations are analysed. With this purpose, a model based on a p-version, hierarchical, first-order shear deformation finite element is employed. The equations of motion are solved in the time domain by Newmark's implicit time integration method. The temperature and the amplitude of the mechanical excitation are varied, and transitions from periodic to non-periodic motions are found.     

Study of a dynamical system with a strange attractor and invariant tori

A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.     

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Resonance formation in few (3) body Coulomb systems

The standard physical picture of resonance formation breaks down for three-body Coulomb systems close to the threshold for total fragmentation. Near this threshold the complex as a whole performs collective motions localised essentially along potential energy surface ridges. Above threshold these motions are nonperiodic and describe total fragmentation, below threshold unstable periodic motions lead to new resonances. A semiclassical theory which describes these phenomena for two-electron atoms in excellent agreement with experiments is generalised to the ppp system.     

Explicit determination of certain periodic motions of a generalized two-field gyrostat

The case of motion of a generalized two-field gyrostat found by V. V. Sokolov and A.V. Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. For this system, we find some special periodic motions at which the momentum mapping has rank 1. For such motions, all phase variables can be expressed in terms of algebraic functions of a single auxiliary variable and a set of constants. This auxiliary variable satisfies a differential equation which can be integrated in elliptic functions of time. As an application, the explicit formulas of characteristic exponents for determining the Williamson type of the special periodic motions are obtained.     

CHAOTIC AND NON-LINEAR DYNAMIC ANALYSIS OF A TWO-AXIS RATE GYRO WITH FEEDBACK CONTROL MOUNTED ON A SPACE VEHICLE

An analysis is presented in this paper for a two-axis rate gyro subjected to linear feedback control mounted on a space vehicle, which is spinning with uncertain angular velocity ω_z(t) about its spin of the gyro. For the autonomous case in which ω_z(t) is steady, the stability analysis of the system is studied by Routh-Hurwitz theory. For the non-autonomous case in which ω_z(t) is sinusoidal function, this system is a strongly non-linear damped system subjected to parametric excitation. By varying the amplitude of sinusoidal motion, periodic and chaotic responses of this parametrically excited non-linear system are investigated using the numerical simulation. Some observations on symmetry-breaking bifurcations, period-doubling bifurcations, and chaotic behavior of the system are investigated by various numerical techniques such as phase portraits, Poincaré maps, average power spectra, and Lyapunov exponents. In addition, some discussions about chaotic motions of this system can be suppressed and changed into regular motions by a suitable constant motor torque are included.     

Watt regulator dynamics

The nonlinear problem on Watt regulator motion mounted on a machine subjected to set vertical harmonic small-amplitude vibrations is considered. The problem on the existence and stability of the periodic motions of the regulator is investigated by the methods of the classical perturbation theory and the Kolmogorov?Arnold?Moser (KAM) theory. Under additional assumptions that the regulated-machine vibrations are high-frequency, an integrable approximate set of equations is obtained and it is shown that the majority of its trajectories are retained also in the complete set.     

Existence and analytical predictions of periodic motions in a periodically forced,nonlinear friction oscillator

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.