Recurrence in three dimensional magnetohydrodynamic plasma

Previous studies (e.g., Malara et al., Astrophys. J. 533, 523 (2000)) considered small-amplitude Alfven wave (AW) packets in Arnold-Beltrami-Childress (ABC) magnetic field using WKB approximation. They draw a distinction between 2D AW dissipation via phase mixing and 3D AW dissipation via exponentially divergent magnetic field lines. In the former case, AW dissipation time scales as S1∕3 and in the latter as log(S), where S is the Lundquist number. In this work, linearly polarised Alfven wave dynamics in ABC magnetic field via direct 3D magnetohydrodynamic (MHD) numerical simulation is studied for the first time. A Gaussian AW pulse with length-scale much shorter than ABC domain length and a harmonic AW with wavelength equal to ABC domain length are studied for four different resistivities. While it is found that AWs dissipate quickly in the ABC field, contrary to an expectation, it is found the AW perturbation energy increases in time. In the case of the harmonic AW, the perturbation energy growth is transient in time, attaining peaks in both velocity and magnetic perturbation energies within timescales much smaller than the resistive time. In the case of the Gaussian AW pulse, the velocity perturbation energy growth is still transient in time, attaining a peak within few resistive times, while magnetic perturbation energy continues to grow. It is also shown that the total magnetic energy decreases in time and this is governed by the resistive evolution of the background ABC magnetic field rather than AW damping. On contrary, when the background magnetic field is uniform, the total magnetic energy decrease is prescribed by AW damping, because there is no resistive evolution of the background. By considering runs with different amplitudes and by analysing the perturbation spectra, possible dynamo action by AW perturbation-induced peristaltic flow and inverse cascade of magnetic energy have been excluded. Therefore, the perturbation energy growth is attributed to a new instability. The growth rate appears to be dependent on the value of the resistivity and the spatial scale of the AW disturbance. Thus, when going beyond WKB approximation, AW damping, described by full MHD equations, does not guarantee decrease of perturbation energy. This has implications for the MHD wave plasma heating in exponentially divergent magnetic fields.

During the operation of rack railway on super-slope bridges (especially under temperature load), structural deformation of bridge-end is much larger than that of rack due to the difference of structural continuity between bridge and rack, which greatly increases local stress in rack, damages rack and fastener, and seriously threatens the operation reliability. To investigate the reliability of rack caused by temperature load and nonlinear dynamic interaction between super-slope bridge and rack, a detailed vehicle-rack(rail)-bridge coupled dynamic model is established with consideration of nonlinear dynamic meshing behaviour of gear-rack system and nonlinear dynamic contact behaviour of wheel-rail system. Then a field test is performed to obtain the basis dynamic behaviour of rack railway and validate the established numerical model. On this basis, the effects of temperature load, gear-rack meshing behaviour and wheel-rail contact behaviour on deformation and internal stress of rack-bridge system are investigated. Finally, the method to guarantee the structural reliability of rack railway on super-slope bridge is proposed. Results show that rack deformation nearby bridge gap is highly inconsistent with bridge deformation subject to different loads, and the internal stresses in rack and bolt exceed limit values written in relevant code, which seriously threatens the reliability of the rack structure. The maximum rack stress is 898 MPa, which exceeds the tensile strength of rack. The maximum shear stresses of bolts between fastener and rack and between fastener and sleeper are 1510 MPa and 656 MPa respectively, which exceed the shear strength of bolts. According to the calculation results in this paper, elastic fastener is suggested to be used to connect rack and sleeper, which can effectively reduce stress in rack and bolt, and improve the reliability of the rack structure.

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

This chapter introduces the basic concepts of dynamical systems theory and several basic mathematical methods for controlling chaos. The main goal of this chapter is to provide an introduction to and a summary of the theory of dynamical systems, with particular emphasis on fractal theory, chaos theory, and chaos control. We first define what is meant by a dynamical system, then we define an attractor, and then the concept of the fractal dimension of a geometrical object. We also define the Lyapunov exponents as a measure of the chaotic behavior of a dynamical system. On the other hand, the fractal dimension can be used to classify geometrical objects because it measures the complexity of an object. The chapter also describes mathematical methods for controlling chaos in dynamic systems. These methods can be used to control a real dynamic system; however, due to efficiency and accuracy requirements we were forced to use fuzzy logic to model the uncertainty, which is present when numerical simulations are performed. We also describe in this chapter a new theory of chaos using fuzzy logic techniques. Chaotic behavior in nonlinear dynamical systems is very difficult to detect and control. Part of the problem is that mathematical results for chaos are difficult to use in many cases, and even if one could use them there is an underlying uncertainty in the accuracy of the numerical simulations of the dynamical systems. For this reason, we can model the uncertainty of detecting the range of values where chaos occurs, using fuzzy set theory. Using fuzzy sets, we can build a theory of fuzzy chaos, where we can use fuzzy sets to describe the behaviors of a system. We illustrate our approach with two cases: Chua’s circuit and Duffing’s oscillator.

The transient proper orthogonal decomposition (TPOD) method is applied to order reduction in the rotor-bearing system with the coupling faults in this chapter. A 24 degrees of freedom (DOFs) rotor model supported by a pair of sliding bearings with both crack and rub-impact faults is established with the discrete modeling method. The complexity of dynamic behaviors of the rotor system with the coupling faults is analyzed via the comparison of the rotor system with the single fault (crack or rub-impact). The proper orthogonal mode (POM) energy method is proposed to confirm DOF number of the reduced model, and TPOD method is used in the coupling faults system to obtain the optimal order reduction model based on POM energy. The presented order reduction method is verified by comparing the bifurcation behaviors between the original and the reduced system. As one conclusion, it can be drawn that TPOD method could provide the optimal order reduction model to study the nonlinear dynamic behaviors of the complex rotor system with the coupling faults.

Cadmium has been selected as a metal with which to investigate the detailed flow behaviour of a hexagonal close packed metal under constant shear stress σ, some of the results having been confirmed with zinc. A new type of flow, here called subnormal flow, has been found to prevail at low stresses, the shear strain γ being accurately represented by γ = M(1 - e-mt) + KMt, the transient flow strain thus reaching a definite limit, which for long times of flow is small compared to the total strain. The mechanism of the M flow is grain boundary sliding, of a type discussed, which leads to the observed variation of M with grain size. When the stress exceeds a transition stress, σc, the flow is w hat is here called normal, being represented by γ = A + Bt1/3 + Kt. The exponential flow exists for these larger stresses, but is in general negligibly small compared to the normal flow. There is a small t1/3 contribution to the subnormal flow at stresses just below σc. In the region of subnormal flow the flow constants show a sudden increase with increase of temperature at a transition temperature Tc, which depends upon grain size and is the temperature which, if exceeded, leads to grain growth in the stress-free metal. The dependence on grain size, which is marked, of all the constants involved in both types of flow has been investigated in some detail. It has been found that the transition stress is proportional to d-1/2 while the excess of the transition temperature over a certain fixed temperature T0, which occurs in another relation, is proportional to d1/2, where d is the average grain diameter. In the transition stress, and in the constants which express normal flow, the expression σd1/2 which it is proposed to call the shove, plays a prominent part. It has been found that the flow linear with time, both in the normal region, where logK varies linearly with a σ, and in the subnormal region, where KM is proportional to σ2, is connected with grain boundary migration, to the extent of which it is proportional. In the normal region the B flow is governed by intragranular glide, Bt1/2 being proportional to the number of glide bands visible in photomicrographs. The B flow is conditioned by twinning, which takes place immediately on application of stress and is responsible for the immediate strain. The question of flow under reversals of stress has been investigated, among the results being that, under certain conditions, the strain in reversed flow varies with time in the same manner as it does in forward flow, both with normal and subnormal flow. The various regularities which have been found are discussed in terms of the mechanical properties of the grains and of the boundary layer. The contrast between the flow properties of hexagonal and cubic metals is largely due to the smallness of the intragranular hardening with strain of the former as compared with the latter, which finds expression in the behaviour of individual slip bands.

This work investigates how viscoelasticity affects the dynamic behavior of a lumped-parameter model of a bistable von Mises truss. The system is controlled by a linear first-order equation and a second-order nonlinear Duffing equation with a quadratic nonlinearity that governs mechanical behavior. The second-order equation controls mechanical oscillations, while the linear first-order equation controls viscoelastic force evolution. Combined, the two equations form a third-order jerk equation that controls system dynamics. Viscoelasticity adds time scales and degrees-of-freedom to material behavior, distinguishing it from viscosity-only systems. Due to harmonic excitation, the system exhibits varied dynamic responses, from periodic to quasi-periodic to chaotic. We explore the dynamics of a harmonically forced von Mises truss with viscous damping to address this purpose. We demonstrate this system's rich dynamic behavior due to driving amplitude changes. This helps explain viscoelastic system behavior. A viscoelastic unit replaces the viscous damper, and we show that, although viscous damping merely changes how fast the trajectory decays to an attractor, viscoelasticity modifies both the energy landscape and the rate of decay. In a conventional linear solid model, three viscoelastic parameters control the system's behavior instead of one, as in pure viscous damping. This adds degrees-of-freedom that affect system dynamics. We present the parameter space for chaotic behavior and the shift from regular to irregular motion. Finally, Melnikov's criteria identify the regular-chaotic threshold. The system's viscous and elastic components affect the chaotic threshold amplitude

In a previous article (1) the time-discretization of those relations mod- eling a class of dynamical systems with friction was discussed. The main goal of this article is to address similar problems using a more sophisticated friction model giving a better description of the system behavior particularly when the velocities are close to zero. These investigations are motivated by the need for more accu- rate friction models in the software simulating the motion of mechanical systems, such as the remote manipulators of the Space Shuttle or of the International Space Station. In this article, we discuss the methods in the case of higher number of degrees of freedom elasto-dynamical systems, and the special case of one degree of freedom. The content can be summarized as follows: We discuss rst models of the constrained motions under consideration, including a rigorous formulation involv- ing a kind of dynamical multiplier. An iterative method allowing the computation of this multiplier will be discussed. Next, in order to treat friction, we introduce an implicit/explicit numerical scheme which is unconditionally stable, and easy to implement and generalize to more complicated systems. Indeed the above scheme can be coupled, via operator-splitting, to schemes classically used to solve dieren- tial equations from frictionless elasto-dynamics. The above schemes are validated through numerical experiments.

Linear and nonlinear spectroscopy techniques are widely used to study numerous important chemical and physical processes. However, the interpretation of these experimental spectra often becomes very complicated because a particular spectrum constitutes a mere footprint of a host of various, possibly intertwined, effects. In this spirit, calculations in the time-dependent picture provide a useful tool for decoding such spectra. Nevertheless, the ultimate challenge is to devise a theoretical framework that could yield sufficient efficiency as well as accuracy to describe the molecular system of interest in a satisfactory way. The strategy is relatively straightforward for low dimensional systems that are directly tractable, e.g., with exact quantum dynamics performed on an equidistant grid. Despite the formidable overall exponential scaling, these calculations can be significantly accelerated by using higher-order split-operator propagation schemes. In general however, one is forced to seek an affordable balance between physical accuracy on one hand and computational efficiency on the other by employing, for instance, some of the techniques from the broad family of semiclassical methods based on classical trajectories. To this end, the thawed Gaussian approximation (TGA) is combined with an on-the-fly ab initio scheme (OTF-AI). The resulting OTF-AI-TGA algorithm is efficient enough to treat all vibrational degrees of freedom (DOFs) on an equal footing even in case of larger molecules such as pentathiophene (105 DOFs). Moreover, in sharp contrast to popular approaches based on global harmonic approximation, OTF-AI-TGA reproduces almost perfectly the absorption and photoelectron spectra of ammonia, i.e., spectra with strong dependence on large amplitude motions. In addition to the mere reproduction of experimental spectra, a novel systematic approach is introduced to assess the importance and the dynamical couplings of individual vibrational DOFs. This is in turn used to gain a deeper insight into the associated physical and chemical processes by attributing specific spectral features to the underlying molecular motion. Specifically, in the case of oligothiophenes, this approach was used to assign the dynamical interplay between quinoid and aromatic characters of individual rings to particular spectral patterns and, furthermore, to explain the changes in the vibrational line shape with an increasing number of rings. Furthermore, in systems that are too large to be treated with accurate quantum methods, efficient methods such as OTF-AI-TGA are expected to be useful as a preliminary tool for identification of the subspace of the important DOFs. On this subspace, one can then unleash some of the less efficient yet better-suited methods. In summary, OTF-AI-TGA combined with this novel analysis approach is intended to provide the first crucial step in a hierarchical computational protocol for studying large molecules such as dyes.

Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

The cosmic magnetic fields in regions of low plasma pressure and large currents, such as in interstellar space and gaseous nebulae, are force-free in the sense that the Lorentz force vanishes. The three-dimensional Arnold-Beltrami-Childress (ABC) field is an example of a force-free, helical magnetic field. In fluid dynamics, ABC flows are steady state solutions of the Euler equation. The ABC magnetic field lines exhibit a complex and varied structure that is a mix of regular and chaotic trajectories in phase space. The characteristic features of field line trajectories are illustrated through the phase space distribution of finite-distance and asymptotic-distance Lyapunov exponents. In regions of chaotic trajectories, an ensemble-averaged variance of the distance between field lines reveals anomalous diffusion—in fact, superdiffusion—of the field lines. The motion of charged particles in the force-free ABC magnetic fields is different from the flow of passive scalars in ABC flows. The particles do not necessarily follow the field lines and display a variety of dynamical behavior depending on their energy, and their initial pitch-angle. There is an overlap, in space, of the regions in which the field lines and the particle orbits are chaotic. The time evolution of an ensemble of particles, in such regions, can be divided into three categories. For short times, the motion of the particles is essentially ballistic; the ensemble-averaged, mean square displacement is approximately proportional to t2, where t is the time of evolution. The intermediate time region is defined by a decay of the velocity autocorrelation function—this being a measure of the time after which the collective dynamics is independent of the initial conditions. For longer times, the particles undergo superdiffusion—the mean square displacement is proportional to tα, where α > 1, and is weakly dependent on the energy of the particles. These super-diffusive characteristics, both of magnetic field lines and of particles moving in these fields, strongly suggest that theories of transport in three-dimensional chaotic magnetic fields need a shift from the usual paradigm of quasilinear diffusion.

Fuzzy decoupled-states multi-model identification of gas turbine operating variables through the use of their operating data

High-temperature superconducting (HTS) pinning maglev trains, with their self-stabilizing levitation and low magnetic resistance, are promising candidates for future transportation. This paper investigates the nonlinear vibration behaviors of the single levitation system of the world’s first HTS high-speed maglev train under different load-levitation ratios (LLRs). First, vertical free vibration experiments were conducted under various load conditions, obtaining response curves for stable levitation height, main vibration frequency, stiffness, and damping with various loads. Based on these results, a set of equivalent stiffness and damping parameters for different LLRs was compiled. Second, focusing on three specific LLR conditions, levitation force curves were fitted using experimental data. The system’s stability and nonlinear vibration characteristics were analyzed through theoretical, numerical, and experimental methods. The experimental results show that the stable levitation height decreases nonlinearly as the load increases and stabilizes at 10 mm under rated load conditions. The system’s natural frequency remains nearly constant, showing minimal sensitivity to load changes. Both equivalent stiffness and damping increase with load. The analysis reveals that the HTS levitation system exhibits inherent stability and can recover balance after transient disturbances. It also shows nonlinear dynamics behavior such as period-doubling bifurcation and harmonic resonance, influenced significantly by excitation frequency, amplitude, and LLRs. These phenomena are more pronounced with larger LLR. Under certain conditions, the system exhibits jump phenomena. Resonance primarily occurs in the primary and double-frequency bands. This study enhances the understanding of HTS system dynamics and provides a reference for the dynamic research of HTS pinning maglev trains.

Article Effects of Preload and Number of Balls on Nonlinear Dynamic Behavior of Ball Bearing System was published on September 1, 2003 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 4, issue 3).

In this article, a semi-analytical and analytical method for the nonlinear static and dynamic thermal buckling behavior of imperfect multilayer FG cylindrical shells with an FG porous core in the thermal environment is investigated. The structure is embedded within a generalized nonlinear elastic foundation which is composed of a two-parameter Winkler-Pasternak foundation augmented by a nonlinear cubic stiffness. Two types of multilayer FG cylindrical shells with an FG porosity core distribution consist of a uniform porosity distribution and a non-uniform porosity distribution core are considered. Using the Galerkin method with regard to the von Kármán equations, the discretized motion equation is obtained. The fourth-order Runge-Kutta method is utilized to obtain the nonlinear dynamic thermal buckling responses. The effects of various geometrical characteristics, material parameters, and elastic foundation coefficients are investigated on the nonlinear static thermal buckling and dynamic thermal buckling behavior of the multilayer FG system with an FG porous core. The results show that the various types of porosity core, imperfection and the elastic foundation parameters have strongly effect on the thermal buckling behaviors of the multilayer FG cylindrical shells with an FG porous core.