Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.     

On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

Bifurcation and stability analysis for a non-smooth friction oscillator

Summary Friction-induced self-sustained oscillations, also known as stick-slip vibrations, occur in mechanical systems as well as in everyday life. On the basis of a one-dimensional map, the bifurcation behaviour including unstable branches is investigated for a friction oscillator with simultaneous self-and external excitation. The chosen way of mapping also allows a simple determination of Lyapunov exponents.Dedicated to Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. mult. Erwin Stein on the occasion of his 65th birthday.     

Bifurcation sequences of a Coulomb friction oscillator

In some parameter ranges, the dynamics of a forced oscillator with Coulomb friction dependent on both displacement and velocity is reducible to the dynamics of a one-dimensional map. In numerical simulations, period-doubling bifurcations are observed for the oscillator. In this bifurcation procedure, the map arising from the Coulomb model may not have standard form. The bifurcation sequence of the Coulomb model is compared to that of the standard one-dimensional maps to see if it exhibits universal behavior. All observed components of the bifurcation sequence fit the universal sequence, although some universal events are not witnessed.     

Dynamics of pseudo-rigid ball impact on rigid foundation

This paper concerns the dynamics induced by the ideally elastic normal impact of a linearly elastic pseudo-rigid sphere on a rigid, stationary foundation. An impact map is derived and studied by numerical and analytical means. Periodic, quasi-periodic, and chaotic response is observed consistently with the symplectic nature of the map.     

Chaos in the Convective Flow of a Fluid Mixture in a Porous Medium

The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.     

Lyapunov spectrum determination from the FEM simulation of a chaotic advecting flow

The problem of the determination of the Lyapunov spectrum in chaotic advection using approximated velocity fields resulting from a standard FEM method is investigated. A fourth order Runge–Kutta scheme for trajectory integration is combined with a third order Jacobian matrix method with QR ‐factorization. After checking the algorithm on the standard Lorenz and coupled quartic oscillator systems, the method is applied to a model 3‐D steady flow for which an analytical expression is known. Both linear and quadratic approximated velocity fields succeed in predicting the Lyapunov exponents as well as describing the chaotic or regular regions inside the flow with satisfactory accuracy. A more realistic flow is then studied in order to delineate the possible limitations of the approach. Copyright © 2005 John Wiley & Sons, Ltd.     

圆板振子超谐分岔和混沌运动的实验研究

设计了非线性圆板混沌实验振动台,就轴对称圆板在简谐载荷作用下的非线性动力学行为进行了较为系统的实验研究,理论分析和数值计算,对基础作简谐运动,周边固支圆板,进行了实验。通过测量时间－中心挠度的加速度曲线,进行快速傅立叶变换（FFT）分析,实验发现了对称破缺,超谐分岔,调幅调相等复杂现象;对基础作简谐运动,周边固支,中心加质量块的圆板,进行了实验,通过测量时间－周边应变曲线,进行FFT分析,实验发现了混沌,对称破缺和恢复及超谐分岔等复杂动力学行为。     

Two-plane automatic balancing: A symmetry breaking analysis

We present an analysis of a two-plane automatic balancing device for rotating machinery. The mechanism consists of a pair of races that contain balancing balls which move to eliminate imbalance due to rotor eccentricity or principal axis misalignment. A model is developed that includes the effect of support anisotropy and rotor acceleration. The symmetry of the imbalance is considered, and techniques from equivariant bifurcation theory are used to derive a necessary condition for the stability of balanced operation. The unfolding of the solution structure is explored and we investigate mechanical systems in which either the supports or the automatic ball balancer is asymmetric. Here it is shown that, provided the imbalance is small, the balanced state is robust to the considered asymmetries.     

Dynamics of a Simple Damped Oscillator Undergoing Stick-Slip Vibrations

The dynamics of a simple dynamical system subjected to an elastic restoring force, viscous damping and dry friction forces is investigated. Self-sustained oscillations occur with non-standard attracting properties. Discontinuity of the governing equations leads to non-standard bifurcations, which are studied here, with analytical and numerical tools. This revised version was published online in July 2006 with corrections to the Cover Date.     

一种值得注意的动力学行为

在讨论一类以ｐａｔｅｒｎ动力学为背景的二维平面映射时，发现了一种具有两个正Ｌｙａｐｕｎｏｖ特征指数的动力学行为．分析表明这种行为可能来自于ｓｎａｐｂａｃｋｒｅｐｅｌｅｒ．进一步的理论工作有待于深入     

Mechanical modelling and stability investigation of a non-material system in rotor dynamics

For the first time the behaviour of a Timoshenko-rotor-model with a non-material constraint is investigated. The constraint is caused by an axially-moving disc guided by the flexible shaft. Both, the development of the equations of motion (including the additionally occuring jump conditions) and the analysis of stability are essentially influenced by the non-classical character of the system. As result some stability diagrams are shown. They are based on statistical methods of theory of stability. The results allow the conclusion that most of the non-material constraints lead to a system behaviour as well-known from parametric excitations.     

Practical Evaluation of Invariant Measures for the Chaotic Response of a Two-Frequency Excited Mechanical Oscillator

This paper presents results which characterize the chaotic response of alow-dimensional mechanical oscillator. An experimental system based on acart rolling on a two-well potential surface has been shown to closelyapproximate a modified form of Duffing's equation. Two-frequency forcingis applied, providing a useful means of varying the dimension of theresponse. Computation of correlation dimension and Lyapunov spectra areperformed on both experimental and numerical data in order to assess theutility of these measures in a practical setting. A specific focus isthe distinction between subharmonic and quasi-periodic forcing, sincethis has a subtle, and interesting, effect on the subsequent dynamics.The results tend to highlight the statistical nature of the measures andthe caution that should be used in their interpretation.     

Nonlinear Responses of a Magnetoelastic Beam in a Step-Pulsed Magnetic Field

A theoretical study is carried out on the dynamics of a magnetoelastic beam being in a step-pulsed magnetic field. For this aim, the magnetic potential and elastic energies are determined for the beam and partial differential equations are established according to Hamilton's principle. It is proven that the magnetoelastic beam can give a variety of complex behavior in the case of step-pulsed field excitations. An intermediate regime of two-well chaos is observed. Theoretical findings were found to be in a good agreement with the experimental results for the specific system parameters. On leave from Institute of Physics, University of Bayreuth, 65440 Bayreuth, Germany An erratum to this article is available at .     

Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations

In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.     

Local and Global Lyapunov exponents

Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias. These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established. Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved. Along the way a simple relation between topological entropy and the fractal dimension is obtained.     

Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures

The dynamic pull-in instability of double clamped microscale beams actuated by a suddenly applied distributed electrostatic force and subjected to non-linear squeeze film damping is investigated. A reduced order model is built using the Galerkin decomposition with undamped linear modes as base functions and verified through comparison with numerical finite differences solution. The stability analysis of a beam actuated by one and two electrodes symmetrically located at two sides of the beam and operated by a step-input voltage is performed by evaluating the largest Lyapunov exponent, the sign of which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of dynamic pull-in instability, especially when combined with compact reduced order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated.     

Identification of regions of fastest mixing in a system of point vortices

This paper describes a numerical method for efficiently identifying the regions of fastest mixing of a passive dye in a flow due to a system of point vortices. Results obtained from computations are presented for systems of three and four point vortices, both in the unbounded domain and inside a circular cylinder. The flow is two‐dimensional and the fluid is incompressible. The regions where mixing is possible are found by studying the largest Lagrangian Lyapunov exponent distribution with respect to various initial positions of tracer particles. The regions of fastest mixing are then identified from the Lyapunov exponent distribution at small times. The results of the method are verified by quantifying the mixing by using a traditional box counting technique. The technique is then applied to several different initial configurations of vortices and some interesting results are obtained. Some numerical findings about the nature of the exponents computed are also discussed. Copyright © 2002 John Wiley Sons, Ltd.     

Dynamic stability of a viscoelastic rotating shaft under parametric random excitation

This paper deals with dynamic stability of a viscoelastic rotating shaft subjected to a parametric random axial compressive thrust, by using moment Lyapunov exponents and the largest Lyapunov exponents as indicators. The equation of motion for the shaft is derived, which is a system of gyroscopic stochastic differential equations. The method of stochastic averaging is used to decouple the governing equations into Itô equations, from which the moment Lyapunov exponent is obtained by using mathematical transformations only. The largest Lyapunov exponent is obtained through its relation with moment Lyapunov exponents. The effects of various parameters on the stochastic dynamic stability are discussed. The approximate analytical results are confirmed by Monte Carlo simulation.