Modern Nonlinear Dynamics Research Articles

Research on Dynamic Characteristics of Rub-Impact of Progressive Cavity Pump with Rotation Speed

The dynamic characteristics of Progressive Cavity Pump (PCP) have an important influence on the stability of oil wells string system. A nonlinear dynamics model of rotor system was presented. The equations of unbalance rotor were established by the Lagrange approach and the vibration characteristics of the rotor system under rub-impact was analyzed by used the modern nonlinear dynamics and bifurcation theories, through numerical calculation. The bifurcation diagrams, time domain charts and Poincaré maps were obtained to analyze dynamic characteristics of the system and the results reveal the multiform complex nonlinear dynamic behavior of the rotor-bearing system under rub-impact. Moreover, the influence of the speed on the relatively stable speed of the rotor system was analyzed. These analysis results provide theoretical bases for the rotor system's status recognition, diagnosis as well as safe and economic operation.

Research on Dynamic Characteristics of Rub-Impact of Progressive Cavity Pump with Rotation Speed

The dynamic characteristics of Progressive Cavity Pump (PCP) have an important influence on the stability of oil wells string system. A nonlinear dynamics model of rotor system was presented. The equations of unbalance rotor were established by the Lagrange approach and the vibration characteristics of the rotor system under rub-impact was analyzed by used the modern nonlinear dynamics and bifurcation theories, through numerical calculation. The bifurcation diagrams, time domain charts and Poincaré maps were obtained to analyze dynamic characteristics of the system and the results reveal the multiform complex nonlinear dynamic behavior of the rotor-bearing system under rub-impact. Moreover, the influence of the speed on the relatively stable speed of the rotor system was analyzed. These analysis results provide theoretical bases for the rotor system's status recognition, diagnosis as well as safe and economic operation.

Synchronization structures in the chain of rotating pendulums

The collective behavior of the ensembles of coupled nonlinear oscillator is one of the most interesting and important problems in modern nonlinear dynamics. In this paper, we study rotational dynamics, in particular space-time structures, in locally coupled identical pendulum-type elements chains that describe the behavior of phase-locked-loop systems, distributed Josephson junctions, coupled electrical machines, etc. The control parameters in the considered chains are: dissipation, coupling strength, and number of elements. In the system under consideration, the realized modes are synchronous in frequency and synchronous (in-phase) or asynchronous (out-of-phase) in phase. In the low dissipation case, the in-phase synchronous rotational regime instability region boundaries are theoretically found and the bifurcations leading to the loss of its stability are determined. The analysis was carried out for chains of arbitrary length. The existence of various out-of-phase synchronous rotational modes types is revealed: completely asynchronous in phases and the cluster in-phase synchronization regime. Regularities of transitions from one type of out-of-phase synchronous mode to another are established. It was found that at certain coupling parameter values, the coexistence of stable in-phase and out-of-phase synchronous modes is possible. It was found that for arbitrary chain length, the number of possible stable out-of-phase modes is always one less than the chain elements number. Analytical results are confirmed by numerical simulations.

Vibration response characteristics of a dual-rotor with unbalance-misalignment coupling faults: Theoretical analysis and experimental study

In order to improve energy efficiency and compact structure, the dual-rotor structure, with low-pressure rotor and high-pressure rotor, has been widely used in aero-engines. In severe operation conditions, the high rotating speed dual-rotor structure is vulnerable to some faults, such as misalignment, which may cause strong vibration and even catastrophic accidents. Perfect balance of the dual-rotor system cannot be obtained in practice, and some amount of unbalance is almost always present. The rotors’ centrelines are not co-linear in the couplings and the rotors run in improper axial positions in a dual-rotor system. The differential equations of rotor system is derived by using modern nonlinear dynamics and dual-rotor dynamics principles. The governing equations of the dual-rotor system with unbalance-misalignment coupling faults are solved numerically by the Runge–Kutta method. The complicated vibration responses influenced by different rotational angular speeds, mass eccentricity, misalignment angle and parallel misalignment are analyzed by the cascade plot, time waveform and frequency spectrum. Second harmonic frequency and rotational frequency components of dual rotors are observed. To verify the validity of the dual-rotor system dynamic model, the unbalance-misalignment coupling faults are carried out on a dual-rotor test rig by adjusting the height and quantity of gaskets and installing a screw on the disk of the inner rotor. The simulation results are found to agree with the experimental results. These results provide important theoretical and engineering references for the safe operation of dual-rotor system and the exact identification of coupling faults.

APPLICATION OF UKRAINIAN GRID INFRASTRUCTURE FOR INVESTIGATION OF NONLINEAR DYNAMICS IN LARGE NEURONAL NETWORKS

<p>In present work the Ukrainian National Grid (UNG) infrastructure was applied for investigation of synchronization in large networks of interacting neurons. This application is important for solving of modern neuroscience problems related to mechanisms of nervous system activities (memory, cognition etc.) and nervous pathologies (epilepsy, Parkinsonism, etc.). Modern non-linear dynamics theories and applications provides powerful basis for computer simulations<br />of biological neuronal networks and investigation of phenomena which mechanisms hardly could be clarified by other approaches. Cubic millimeter of brain tissue contains about 105 neurons, so realistic (Hodgkin-Huxley model) and phenomenological (Kuramoto-Sakaguchi, FitzHugh-Nagumo, etc. models) simulations require consideration of large<br />neurons numbers.</p>

Dynamic characteristics of multi-degrees of freedom system rotor-bearing system with coupling faults of rub-impact and crack

Extensive studies on rotor systems with single or coupled multiple faults have been carried out. However these studies are limited to single-span rotor systems. A finite element model for a complex rotor-bearing system with coupled faults is presented. The dynamic responses of the rotor-bearing system are obtained by using the rotor dynamics theory and the modern nonlinear dynamics theory in connection with the continuation-shooting algorithm(commonly used for obtaining a periodic solution for a nonlinear system) for a range of rub-impact clearances and crack depths. The stability and Hopf instability of the periodic motion of the rotor-bearing system with coupled faults are analyzed by using the procedure described. The results indicate that the finite element method is an effective way for determining the dynamic responses of such complex rotor-bearing systems. Further for a rotor system with rub-impact and crack faults, the influences of the clearances are significantly different for different rub-impact stiffness. On the contrary, the influence of crack depths is rather small. The instability speeds of the rotor-bearing system increase due to the presence of the crack fault. The results obtained using the new finite element model, presented for computation and analysis of dynamic responses of the rotor-bearing systems with coupled faults, are in accordance with measurements in experiment. The formulations given can be used for diagnosis of faults, vibration control, and safe and stable operations of real rotor-bearing systems.

Dynamic characteristics analysis of a rub-impact rubber bearing-shaft system

Shaft/bearing rubbing systems may undertake a number of quite different responses. Recent experiments on shaft/bearing rubbing have revealed that two or three different responses may exist under the same frequency ratio with different initial conditions and system parameters. In the present paper, the rubber bearing/shaft system is simplified into a general bearing/shaft system with nonlinear elastic support of bearing, which is established based on the classic impact theory and is analyzed by the modern nonlinear dynamics and bifurcation theories. The boundaries for the existence of the no-rub responses are determined analytically, and the synchronous full annular rub solution are derived and the stability is analyzed to determine the region of the synchronous full annular rub response. The results provide an opportunity for a better understanding of the dynamical characteristics, such as the synchronous full annular rub motion, the jump phenomena as well as the transition between the partial rub motion, the dry whip and the full annular periodic rub motion. A systematic study on the influence of the system parameters on the dynamics characteristics is carried out. An overall picture of the response characteristics of this model is then obtained by drawing the existence boundaries in the parameter space. The present results provide a good understanding of the relationship between the different rubbing responses and the system parameters.

Stability Analysis of a Prey-Predator Model by Using Homotopy Analysis Method

A prey-predator model was considered. Using the methods of the modern nonlinear dynamics and homotopy analysis method (HAM), its stability was discussed. Firstly, we found the system’s positive equilibrium point and shifted it to zero point through transformation. Secondly, we analyzed the stability of the system at the equilibrium point. Lastly, we analyzed the transformed system by HAM. We support our analytical findings with numerical simulation.

Bifurcation analysis of unstable ship motions resulting from strong nonlinear coupling

The present paper employs modern nonlinear dynamics tools in order to investigate the dynamic characteristics governing the complex coupling of the heave, roll and pitch modes in head seas at some regions of the numerical stability map of a fishing vessel. Bifurcation diagrams and Poincare mappings are computed and employed to investigate the appearance of multistability and chaos associated with increased values of the selected control parameter, the wave amplitude. The connection between these nonlinear characteristics and the coupled nature of the mathematical model are analyzed.

Stability, bifurcation and chaos of a high-speed rub-impact rotor system in MEMS

A general model of a Jeffcott micro-rotor system with rub-impact is established based on the classic impact theory and is analyzed by the modern nonlinear dynamics and bifurcation theories. An investigation is carried out on the stability of the rub solutions of the rotor system and the results provide a better understanding of the dynamical characteristics, such as the transition among periodic, quasi-periodic and chaotic responses, of the rotor system with rub-impact under different system parameter combinations. Nonlinear behaviors of rub-related vibrations are studied with a large number of numerical simulations. The effects of rotating speed, imbalance, damping coefficient, impact stiffness, and friction coefficient on the micro-rotor responses are investigated in detail. It is demonstrated that the system goes through an extraordinary route to chaos and alternates among the periodic, quasi-periodic and chaotic motions as the system parameters change. These forms of dynamical characteristics can be effectively used to diagnose the rub-impact fault. The results help engineers to optimize the design of micro-rotating machinery in MEMS, as well as to develop sensitive monitoring/diagnosis systems assessing machinery conditions.

Stability and bifurcation behaviors analysis in a nonlinear harmful algal dynamical model

A food chain made up of two typical algae and a zooplankton was considered. Based on ecological eutrophication, interaction of the algal and the prey of the zooplankton, a nutrient nonlinear dynamic system was constructed. Using the methods of the modern nonlinear dynamics, the bifurcation behaviors and stability of the model equations by changing the control parameter r were discussed. The value of r for bifurcation point was calculated, and the stability of the limit cycle was also discussed. The result shows that through quasi-periodicity bifurcation the system is lost in chaos.

Analysis on complicated characteristics of a high-speed rotor system with rub-impact

A general model of a rub-impact rotor is founded in the light of the classic impact theory, which is analyzed by applying modern nonlinear dynamics theory. The numerical results given in the form of the projection of Poincaré section, the rotor trajectory, the bifurcation diagram, and the power spectrum indicate that the motion of the rotor system alternates among the periodic, chaotic and quasi-periodic vibrations, as rotating speed increases. Meanwhile, the different chaotic regions show different configurations of attractors. When damping and nonlinear stiffness increase, respectively, complicated motion of the rotor system is cut down and boosted up, respectively. The phenomena have great benefits in diagnosing accurately rub-impact failure, reducing the failure ratio, and improving the characteristics of a rotor system.

Nonlinearity in Structural Dynamics: Detection, Identification and Modeling

3R12. Nonlinearity in Structural Dynamics: Detection, Identification and Modeling. - K Worden and GR Tomlinson (Univ of Sheffield, UK). Inst Phys Publ, Bristol, UK. 2001. 659 pp. ISBN 0-7503-0356-5. $130.00.Reviewed by K Yagasaki (Dept of Mech and Syst Eng, Gifu Univ, 1-1 Yanagido, Gifu, 501-1193, Japan).This book provides backgrounds in some techniques for analyzing nonlinear structural dynamical systems and is intended for engineers and scientists in the fields of structural dynamics and nonlinear systems. It is also suitable for postgraduate and senior undergraduate students in these disciplines as a textbook and for expert structural dynamicists as a survey. Many engineering examples are given. As for the mathematics of the reader, calculation of vectors and matrices, and basic knowledge of linear differential equations and Fourier analysis are almost sufficient. The most enthusiastic topics in modern nonlinear dynamics are chaos and bifurcation. There have also been remarkable advances in the field of nonlinear control. However, these subjects are not treated in this book at all. Instead, it explains useful techniques, which include the frequency response functions (FRFs) and Hilbert transformations, for detection of nonlinearity and system identification (ie, estimation of the governing equations) when the systems do not exhibit very different motions, like chaos, from ones of linear systems. The techniques are especially important from a point of view of structural engineering because really nonlinear behavior like chaos is thought to be rather uncommon in the problems of that field. The book begins by describing the relevant backgrounds including the frequency response functions (FRFs) in linear dynamics for discrete- and continuous-time systems in Chapter 1. The discussion is first given for single-degree-of-freedom (SDOF) systems and finally generalized to multi-degree-of-freedom (MDOF) systems with an outline of modal analysis. Chapter 2 gives fundamental results and classical approaches to nonlinear systems in structural dynamics. An idea of FRF distortion is also stated there. Chapter 3 discusses FRFs for nonlinear systems in detail and describes how they are used to obtain information about nonlinearity.In Chapters 4 and 5, the Hilbert transformation, which can not only detect nonlinearity but also solve system identification problems, is explained. A mathematical exposition of the method by complex analysis is also presented. In Chapters 6 and 7, several techniques of system identification for discrete- and continuous-time equations are discussed. In Chapter 8, the concept of FRFs is generalized to higher-order FRFs with the assistance of the Volterra series. The generalization presents a method of system identification for MDOF systems. Finally, three experimental examples of a built-in beam rig, automotive shock absorber, and bilinear beam rig (the last of which is motivated from constructing a system with localized damage in the benchmark of fault detection algorithms) are given, and the techniques of the earlier chapters are applied to demonstrate their effectiveness in Chapter 9. This chapter is especially interesting for the reader from a practical point of view. A substantial set of appendices is also valuable for not only the beginners, but also ordinary researchers in the field. In summary, this reviewer recommends Nonlinearity in Structural Dynamics: Detection, Identification and Modeling for students and researchers in structural dynamics who want to study techniques for detection of nonlinearity and system identification in realistic problems.

Classical specific heat of an atomic lattice at low temperature, revisited

We present results of a standard (constant energy) molecular dynamics simulation of a Lennard-Jones lattice at low temperature. The kinetic energy fluctuations exhibit an anomalous behavior, due to a dynamics which is only weakly chaotic. Such a dynamics does not allow the use of the usual microcanonical equilibrium formula to compute the specific heat. We devise a different method for computing the specific heat, which exploits just the weak chaos at low temperature. The result is that at low temperature this ``revisited'' specific heat is lower than the classical value, and approaches zero when the temperature goes to zero. Only for exceedingly long trajectories does the specific heat approach the classical value, with the exception of the very low temperature range. These results prompt a reconsideration, in the frame of modern nonlinear dynamics, of early intuitions by Nernst and Jeans.

Arterial Thrombosis as a Consequence of 'Quasi-Chaotic' Coordination of Mechanical and Chemical Events

Contrary to a popular notion, thrombosis is neither associated with 'stasis' nor with low-shear flow or turbulence. Instead, a regular eddy formation or flow separation as universal condition favoring thrombotic events is the fluid-dynamic anomaly initiating thrombotic events. Flow is discussed in terms of modern nonlinear fluid dynamics and is appreciated as a highly deterministic event ('low dimensional turbulence'). Moreover, with respect to the biochemistry and cytology of white thrombus formation, transient or permanent flow separation provides ideal conditions for the manifestation of self-organized procoagulatory reactions: these have marked positive feedback. The holistic reaction can be taken as a combination of deterministic yet stochastic events. Rather than calling such an event chaotic, it is proposed to accept it as a consequence of pseudochaotic coordination. At any rate, a combination of nonlinear biophysical and biochemical concepts provides an appropriate new paradigm for the interpretation of the pathogenesis of thrombotic events conventionally paraphrased as Virchow's triad.

Topological fluid dynamics of interfacial flows

The topological description of flows in the vicinity of a solid boundary, that is familiar from the aerodynamics literature, has recently been extended to the case of flow at a liquid–gas interface or a free surface by Lugt [Phys. Fluids 30, 3647 (1987)]. Lugt’s work is revisited in a more general setting, including nonconstant curvature of the interface and gradients of surface tension, using tools of modern nonlinear dynamics. Bifurcations of the flow pattern occur at degenerate configurations. Using the theory of unfolding, this paper gives a complete description of the bifurcations that depend on terms up to the second order. The general theory of this paper is applied to the topology of streamlines during the breaking of a wave and to the flow below a stagnant surface film.

A nonlinear probabilistic method for predicting vessel capsizing in random beam seas

The capsizing of vessels in random beam seas is investigated using a single degree-of-freedom model which is limited to the roll motion. Several factors, including sea wave spectrum, nonlinear righting moment characteristics, and nonlinear damping, are taken into account in the analysis. A nonlinear probabilistic method is developed by combining ideas from modern nonlinear dynamics (specifically, the Melnikov function and the phase space area flux) and random vibrations. Conditions for the onset of vessel capsizing are obtained in terms of the sea state characteristics (significant wave height and characteristic wave period) and the vessel parameters (damping and stiffness coefficients). Extensive numerical simulations are carried out to demonstrate the validity of the analytical results. It is found that there exists an excellent correlation between the rate of phase space flux and the probability of capsizing.

Fractional differentiation of devil's staircases

Modern nonlinear dynamics abounds in mathematical objects with bizarre shapes and properties. It is argued that fractional calculus provides powerful tools for the description of such “monstrosities”. The application of generalized differintegration to devil's staircases is discussed in detail. On the basis of these results, an extension of the conventional classification scheme for phase transitions, introducing fractional order, is proposed.

30] fractal applications in biology: Scaling time in biochemical networks

Publisher Summary This chapter describes some of the terms and concepts of modern dynamical systems theory, including chaotic dynamics and fractals, with suggested applications. Although chaos and fractals are different concepts that should not be confounded, they intersect in the field of modern nonlinear dynamics. The general motivation for biologists to examine chaotic dynamics and fractal geometries arises from recognition that most of physics addresses only simple systems, no matter how elaborate the mathematical apparatus seems to the uninitiated. The chapter explores the possible relevance of fractals to the understanding of both structure and function in biology. It is hypothesized that biological systems are not inherently noisy but follow state-determined dynamics (motion and change) of the deterministic chaotic class. Their chaotic dynamics produce their marginal stability, many of their structures, and their fractal organization of time. Fractals can pertain to the organization of both space and time. In the case of generating fractal trees as computer images of this recursion, if the parameters remain constant through various generations of branching, a somewhat regular tree that looks like a bracken fern is obtained, but if the parameters for the branchings are random, then very irregular shapes are obtained. In the mathematical generation of fractals by recursion rules, the orders of magnitude from zero to infinity are permitted. Classic Fourier-based spectral analysis consists of fitting time history data with a linear series of sines and cosines.

Stability of particle orbits

The correspondence between the stability of particle orbits in synchrotrons and storage rings and the standard maps of modern nonlinear dynamics is discussed. The problem is exposed in the language of particle accelerator physics. Examples are given how the equations for synchrotron and betatron oscillations can be brought into the form of standard maps.