Linearly Coupled,Slowly Varying Oscillators

A dynamical system is considered whose normal frequencies and normal modes vary slowly with time in such a way that two frequencies come into close coincidence. When this occurs the corresponding normal modes undergo a drastic change in their physical properties. Away from coincidence, each normal mode conserves its action. A multiple-time-scale asymptotic procedure is employed to derive equations which describe the mode coupling at coincidence. These equations are solved exactly using parabolic cylinder functions. It is found that in general, action is exchanged between modes at coincidence, but that except for very strong coupling the amount of action exchanged is quite small.     

Linearly Coupled,Slowly Varying Oscillators: The Interaction of a Positive-Energy Mode With a Negative-Energy Mode

A linear dynamical system is considered whose normal frequencies and normal modes come into close coincidence. The case when both modes have positive energy has been discussed by Grimshaw and Allen (1979). Here the case when one mode has positive energy and the other mode has negative energy is discussed. Coupled equations are derived and solved exactly using parabolic cylinder functions. It is found that the action in both modes grows during the coupling.     

Nonstationary energy localization vs conventional stationary localization in weakly coupled nonlinear oscillators

In this paper we study the effect of nonstationary energy localization in a nonlinear conservative resonant system of two weakly coupled oscillators. This effect is alternative to the well-known stationary energy localization associated with the existence of localized normal modes and resulting from a local topological transformation of the phase portraits of the system. In this work we show that nonstationary energy localization results from a global transformation of the phase portrait. A key to solving the problem is the introduction of the concept of limiting phase trajectories (LPTs) corresponding to maximum possible energy exchange between the oscillators. We present two scenarios of nonstationary energy localization under the condition of 1:1 resonance. It is demonstrated that the conditions of nonstationary localization determine the conditions of efficient targeted energy transfer in a generating dynamical system. A possible extension to multi-particle systems is briefly discussed.     

The coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features

The purpose of this paper is to investigate the coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features. There are some new phenomena due to the coupling effects of shaft-bending, shaft-torsion, disk-transverse and blade-bending. In this investigation, this paper mainly focuses on the influence of mistuned features of the blade's length and the stagger angle. It is found that there are four types of coupling modes: the coupling mode of shaft bending, disk transverse and blade bending (SDB), the coupling mode of shaft torsion disk transverse-blade bending (TDB), the coupling mode of disk transverse and blade bending (DB), the repeated mode of blade bending-blade bending (BB). With the effect of mistuned features, the natural frequencies and the coupling mode type will change correspondingly. With the mistuning value of blade length employed in this study, the TDB mode in the tuned system will disappear and shift into TSDB mode instead, and one of the repeated SDB modes will be replaced by STDB modes. Due to this mistuned features, the blades and disk experience a certain degree of vibration localization phenomenon. Different from the length feature, the influence of mistuning values of blade's stagger angle mainly take effect on the coupling modes. At last, by inspection on the Campbell diagrams, the influence of rotational speed on the transformation of natural frequencies is illustrated on the tuned/mistuned flexible shaft-disk-blades coupling structure.     

Decentralized optimal control of a group of dynamical objects

The problem of optimal control of a group of coupled dynamical objects is considered. The cases are examined in which the centralized control of a group of objects is impossible. Fast real-time optimal control algorithms of each of the dynamical systems are described that use information exchanged between group members in the course of control. The proposed methods supplement the earlier developed real-time optimal control methods for an individual dynamical system. The results are illustrated using optimal control of two coupled mathematical pendulums as an example.     

Stochastic model order reduction in randomly parametered linear dynamical systems

This study focuses on the development of reduced order models for stochastic analysis of complex large ordered linear dynamical systems with parametric uncertainties, with an aim to reduce the computational costs without compromising on the accuracy of the solution. Here, a twin approach to model order reduction is adopted. A reduction in the state space dimension is first achieved through system equivalent reduction expansion process which involves linear transformations that couple the effects of state space truncation in conjunction with normal mode approximations. These developments are subsequently extended to the stochastic case by projecting the uncertain parameters into the Hilbert subspace and obtaining a solution of the random eigenvalue problem using polynomial chaos expansion. Reduction in the stochastic dimension is achieved by retaining only the dominant stochastic modes in the basis space. The proposed developments enable building surrogate models for complex large ordered stochastically parametered dynamical systems which lead to accurate predictions at significantly reduced computational costs.     

Hybrid dynamical systems with controlled discrete transitions

This invited survey focuses on a new class of systems–hybrid dynamical systems with controlled discrete transitions. A type of system behavior referred to as the controlled infinitesimal dynamics is shown to arise in systems with widely divergent dynamic structures and application domains. This type of behavior is demonstrated to give rise to a new dynamic mode in hybrid system evolution–a controlled discrete transition. Conceptual and analytical frameworks for modeling of and controller synthesis for such transitions are detailed for two systems classes: one requiring bumpless switching among controllers with different properties, and the other–exhibiting single controlled impacts and controlled impact sequences under collision with constraints. The machinery developed for the latter systems is also shown to be capable of analysing the behavior of difficult to model systems characterized by accumulation points, or Zeno-type behavior, and unique system motion extensions beyond them in the form of sliding modes along the constraint boundary. The examples considered demonstrate that dynamical systems with controlled discrete transitions constitute a general class of hybrid systems.     

On Meeting Energy Balance Errors in Co-Simulations

The term co-simulation denotes the coupling of some simulation tools for dynamical systems into one big system by having them exchange data at points of a fixed time grid and extrapolating the received data into the interval, while none of the steps is repeated for iteration. From the global perspective, the simulation thus has a strong explicit component. Frequently, among the data passed across subsystem boundaries there are flows of conserved quantities, and as there is no iteration of steps, system-wide balances may not be fulfilled: the system is not solved as one monolithic equation system. If these balance errors accumulate, simulation results become inaccurate. Balance correction methods which compensate these errors by adding corrections for the balances to the signal in the next coupling time step have been considered in past research. But establishing the balance of one quantity a posteriori due to the time delay in general cannot establish the balances of quantities that depend on the exchanged quantities, usually energy. In most applications from physics, the balance of energy is equivalent to stability. In this paper, a method is presented which allows users to choose the quantity that should be balanced to be that energy, and to accurately balance it. This establishes also numerical stability for many classes of stable problems.     

Newtonian Dynamical Systems Admitting Normal Blow-Up of Points

A construction of normal shift along trajectories of a Newtonian dynamical system in the case of an extremely singular initial hypersurface collapsing to a single point is considered. A new class of Newtonian dynamical systems is introduced: systems admitting normal blow-up of points. The class is described in terms of differential equations of normality. Bibliography: 20 titles.     

Modelling of nonlinear modal interactions in the transient dynamics of an elastic rod with an essentially nonlinear attachment

We perform system identification and modelling of the strongly nonlinear modal interactions in a system composed of a linear elastic rod with an essentially nonlinear attachment at its end. Our method is based on slow/fast decomposition of the transient dynamics of the system, combined with empirical mode decomposition (EMD) and Hilbert transforms. The derived reduced order models (ROMs) are in the form of sets of uncoupled linear oscillators (termed intrinsic modal oscillators – IMOs), each corresponding to a basic frequency of the dynamical interaction and forced by transient excitations that represent the nonlinear modal interactions between the rod and the attachment at each of these basic frequencies. A main advantage of our proposed technique is that it is nonparametric and multi-scale, so it is applicable to a broad range of linear as well as nonlinear dynamical systems. Moreover, it is computationally tractable and conceptually meaningful, and it leads to reduced order models of rather simple form that fully capture the basic strongly nonlinear resonant interactions between the subsystems of the problem.     

The equations of the perturbed motion of a rocket as a thin-walled structure with a liquid

The perturbed motion of a rocket as an elastic thin-walled structure with compartments partially filled with liquid propellant is considered. It is assumed that the normal modes of the hydroelastic oscillations of the rocket are determined under the condition that the velocity potential on the free surface of the liquid is equal to zero and with standard remaining conditions. Certain features of these modes with zero fundamental frequencies are pointed out and the “loss” of mass effect associated with this is explained. Equations are derived for the perturbed motion of a rocket taking account of the hydroelastic oscillations of its structure and the oscillations of the liquid with deviations of the free surface from the equilibrium position under the action of mass forces. The coefficients of these equations, characterizing the relation between the different type of oscillations, are expressed in terms of known hydrodynamic parameters and the values of the oscillation modes at certain points.     

Small perturbations of quasistochastic dynamical systems

Stability questions of invariant measures of quasistochastic dynamical systems are investigated under the action of small perturbations. Both purely random and deterministic perturbations, small in C, are considered. The connection between the stability properties of the invariant measures with respect to a given family of perturbations and the concept, introduced in the paper, of the intrinsic stability of these measures, determined by the dynamical system itself, is elucidated. The connection between the considered questions and the problems of the modeling of dynamical systems on computers is discussed.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 166–189, 1986.The author is greatful to Ya. G. Sinai for the formulation of the problem and for numerous useful discussions.     

被检测系统的可靠性和最优检测策略

研究被检测系统的一个模型,假定系统有4种运行状态(正常工作、异常工作、正常故障和异常故障)．系统故障时不需检测,系统工作时必须经过检测才能知道它是正常还是异常．系统开始工作后,每隔一段随机时间对它检测一次,直到系统故障或检测出系统处于异常状态为止．利用概率分析和随机模型的密度演化方法,导出了系统的一些新的可靠性指标和最优检测策略．     

Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses

Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an ??optimized?? DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of?60. Time-varying modes computed from the DMD variants yield low projection errors.     

On the effect of damping on the stabilization of mechanical systems via parametric excitation

The effect of damping on the re-stabilization of statically unstable linear Hamiltonian systems, performed via parametric excitation, is studied. A general multi-degree-of-freedom mechanical system is considered, close to a divergence point, at which a mode is incipiently stable and n ? 1 modes are (marginally) stable. The asymptotic dynamics of system is studied via the Multiple Scale Method, which supplies amplitude modulation equations ruling the slow flow. Several resonances between the excitation and the natural frequencies, of direct 1:1, 1:2, 2:1, or sum and difference combination types, are studied. The algorithm calls for using integer or fractional asymptotic power expansions and performing nonstandard steps. It is found that a slight damping is able to increase the performances of the control system, but only far from resonance. Results relevant to a sample system are compared with numerical findings based on the Floquet theory.     

Static duality and a stationary-action application

Conservative dynamical systems propagate as stationary points of the action functional. Using this representation, it has previously been demonstrated that one may obtain fundamental solutions for two-point boundary value problems for some classes of conservative systems via solution of an associated dynamic program. Further, such a fundamental solution may be represented as a set of solutions of differential Riccati equations (DREs), where the solutions may need to be propagated past escape times. Notions of “static duality” and “stat-quad duality” are developed, where the relationship between the two is loosely analogous to that between convex and semiconvex duality. Static duality is useful for smooth functionals where one may not be guaranteed of convexity or concavity. Some simple properties of this duality are examined, particularly commutativity. Application to stationary action is considered, which leads to propagation of DREs past escape times via propagation of stat-quad dual DREs.     

Adaptive synchronization in tree-like dynamical networks

Synchronization in an array of coupled identical nonlinear dynamical systems have attracted increasing attention from various fields of science and engineering. In this paper, we investigate the synchronization phenomenon in tree-like dynamical networks. Based on the LaSalle invariant principle, a simple and systematic adaptive control scheme with variable coupling strength is proposed for the synchronization of tree-like dynamical networks without any knowledge of the concrete structure of isolate system. This result indicates that synchronization can be achieved for strong enough coupling if there exists a system (located at the root of the tree) which directly or indirectly influences all other systems. Furthermore, the main result is applied to several Lorenz chaotic systems coupled by a tree. And numerical simulations are also given to show the effectiveness of the proposed synchronization method.     

Dynamical modeling and attitude analysis for the spacecraft with lateral solar arrays

The dynamics and attitude motion of the three-axis stabilized spacecraft installed with lateral solar arrays is investigated in terms of the rigid-flexible coupled global modes of the system. The spacecraft consists of a rigid platform with small moment of inertia and two groups of flexible solar arrays with relatively large moment of inertia installed on the rigid rotation shafts. The rigid-flexible coupled dynamic model of the spacecraft is established by using the Hamiltonian Principle. The global mode method is employed to work out the natural frequency and global modal shapes of the rigid-flexible coupled dynamic model combined with corresponding boundary conditions. To validate the effectiveness of the analytical results obtained by global mode method, the natural frequencies and mode shapes obtained from finite element model using MSC.Patran software are used as a reference. A numerical example is given to show that the results obtained from both methods are matched very well (the relative errors of the corresponding frequencies are small enough) and the rigid motion of the platform is coupled with the vibration mode of the flexible solar arrays. This implies that the global analytical modes can be used to accurately describe the rigid-flexible coupled motion of the spacecraft. By comparing with the finite element model, the reduced dynamical model derived in terms of the global modes of the system has a lower dimension. Numerical simulations for the system with variations of parameters and dynamic responses analysis for different applied forces are performed to illustrate that, the characteristics of the model are affected by inner and external factors.     

Noise and coupling induced synchronization in a network of chaotic neurons

The synchronization in four forced FitzHugh–Nagumo (FHN) systems is studied, both experimentally and by numerical simulations of a model. We show that synchronization may be achieved either by coupling of systems through bidirectional diffusive interactions, by introducing a common noise to all systems or by combining both ingredients, noise and coupling together. Here we consider white and colored noises, showing that the colored noise is more efficient in synchronizing the systems respect to white noise. Moreover, a small addition of common noise allows the synchronization to occur at smaller values of the coupling strength. When the diffusive coupling in the absence of noise is considered, the system undergoes the transition to subthreshold oscillations, giving a spike suppression regime. We show that noise destroys the appearance of this dynamical regime induced by coupling.