Karhunen–Loeve analysis and order reduction of the transient dynamics of linear coupled oscillators with strongly nonlinear end attachments

The Karhunen–Loeve (K–L) decomposition method has become a popular technique to create low-dimensional, reduced-order models of dynamical systems. In this paper this technique is applied to a multi-degree-of-freedom chain of linear coupled oscillators with a strongly nonlinear (nonlinearizable), lightweight end attachment. By performing K–L decomposition we show that the lightweight nonlinear attachment (possessing 0.5% of the total mass of the chain) can affect the global dynamics of the linear chain, exhibiting nonlinear energy-pumping phenomena; that is, irreversible passive targeted energy transfers from the linear chain to the nonlinear end attachment, where this energy is locally confined and dissipated without ‘spreading back’ to the primary system. It is shown that the occurrence of energy pumping can be identified by studying the dominant K–L modes of the dynamics, as well as, the energy distribution among them. Moreover, by comparing the action of the strongly nonlinear attachment to the classical linear vibration absorber, we show robustness of passive nonlinear energy absorption over wide parameter ranges. On the other hand, the case-sensitive nature of K–L-based reduced-order models has always been a constraint for K–L decomposition, since one cannot quantify a priori the error bound of such low-dimensional reduced-order models when different initial conditions are applied to the system. To alleviate this constraint, the paper proposes a multiple correlation coefficient (MCC) as a quantitative measure to effectively assess the applicability of a K–L-based reduced-order model derived for a specific set of initial conditions to a small neighborhood of initial conditions containing that initial state. The derived reduced-order models are validated through reconstruction of the system responses and comparisons to direct numerical integrations.     

Dynamics of linear discrete systems connected to local, essentially non-linear attachments

The dynamics of a linear periodic substructure, weakly coupled to an essentially non-linear attachment are studied. The essential (non-linearizable) non-linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena, e.g., passive, one-way, irreversible transfer of energy from the substructure to the attachment. As a specific application the dynamics of a finite linear chain of coupled oscillators with a non-linear end attachment is examined. In the absence of damping, it is found that the dynamical effect of the non-linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain. When damping exists energy pumping phenomena are realized in the system. It is shown that energy pumping strongly depends on the topological structure of the non-linear normal modes (NNMs) of the underlying undamped system. This is due to the fact that energy pumping is caused by the excitation of certain damped invariant NNM manifolds that are analytic continuations for weak damping of NNMs of the underlying undamped system. The bifurcations of the NNMs of the undamped system help explain resonance capture cascades in the damped system. This is a series of energy pumping phenomena occurring at different frequencies, with sudden lower frequency transitions between sequential events. The observed multi-frequency energy pumping cascades are particularly interesting from a practical point of view, since they indicate that non-linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure, in compatibility with the design objectives.     

Energy pumping for a larger span of energy

This paper is devoted to energy pumping: exhibiting a multi-degree-of-freedom (dof) nonlinear attachment coupled to a single-dof linear one, energy's area leading to energy pumping phenomenon is increased by involving different nonlinear modes in the process.     

Essentially nonlinear piezoelectric shunt circuits applied to mistuned bladed disks

An essentially nonlinear piezoelectric shunt circuit is proposed for the practical realization of nonlinear energy sink, and then applied to a mistuned bladed disk for blade vibration reduction. First, the global dynamics of a single degree-of-freedom linear mechanical oscillator, coupled to an essentially nonlinear shunted piezoelectric attachment, is studied. Under certain conditions, the nonlinear targeted energy transfer, i.e. a fast, passive energy transfer from the mechanical oscillator to the nonlinear attachment is observed. A numerical method, referred to as the variable-coefficient harmonic balance method, is developed to calculate quasi-periodic responses arising in the electromechanical system under harmonic forcing. Characterized by the nonexistence of a resonance frequency, the essentially nonlinear shunt circuit is able to work robustly over a broad frequency band with a smaller inductance requirement compared with the linear resonant shunt circuit.     

Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system

The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses. The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.     

Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber

This paper deals with the application of the concept of targeted energy transfer to the field of acoustics, providing a new approach to passive sound control in the low frequency domain, where no efficient dissipative mechanism exists. The targeted energy transfer, also called energy pumping, is a phenomenon that we observe by combining a pure nonlinear oscillator with a linear primary system. It corresponds to an almost irreversible transfer of vibration energy from the linear system to the auxiliary nonlinear one, where the energy is finally dissipated. In this study, an experimental set-up has been developed using the air inside a tube as the acoustic linear system, a thin circular visco-elastic membrane as an essentially cubic oscillator and the air inside a box as a weak coupling between those two elements. In this paper, which mainly deals with experimental results, it is shown that several regimes exist under sinusoidal forcing, corresponding to the different nonlinear normal modes of the system. One of these regimes is the quasi-periodic energy pumping regime. The targeted energy transfer phenomenon is also visible on the free oscillations of the system. Indeed, above an initial excitation threshold, the sound extinction in the tube follows a quasi-linear decrease that is much faster than the usual exponential one. During this linear decrease, the energy of the acoustic medium is irreversibly transferred to the membrane and then damped into this element called nonlinear energy sink. We present also the frequency responses of the system which shows a clipping of the original resonance peak of the acoustic medium and we finally demonstrate the ability of the nonlinear absorber to operate in a large frequency band, tuning itself to any linear system.     

Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments

We study complex damped and undamped dynamics and targeted energy transfers (TETs) in systems of coupled oscillators, consisting of single-degree-of-freedom primary linear oscillators (LOs) with vibro-impact attachments, acting, in essence, as vibro-impact nonlinear energy sinks (VI NESs). First, the complicated dynamics of such VI systems is demonstrated by computing the VI periodic orbits of underlying Hamiltonian systems and depicting them in appropriate frequency–energy plots (FEPs). Then, VI damped transitions and distinct ways of passive TETs from the linear oscillators to the VI attachments for various parameter ranges and initial conditions are investigated. As in the case of smooth stiffness nonlinearity [Y. Lee, G. Kerschen, A. Vakakis, P. Panagopoulos, L. Bergman, D.M. McFarland, Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Physica D 204 (1–2) (2005) 41–69], both fundamental and subharmonic TET can be realized in the VI systems under consideration. It is found that the most efficient mechanism for VI TET is through the excitation of highly energetic VI impulsive orbits (IOs), i.e., of periodic or quasiperiodic orbits corresponding to zero initial conditions except for the initial velocities of the linear oscillators. In contrast to NESs with smooth essential nonlinearities considered in previous works, VI NESs are capable of passively absorbing and locally dissipating significant portions of the energies of the primary systems to which they are attached, at fast time scale. This renders such devices suitable for applications, like seismic mitigation, where dissipation of vibration energy in the early, highly energetic regime of the motion is a critical requirement.     

An extension to the Wheeler phase-field model to allow decoupling of the capillary and kinetic anisotropies

The formulation of the phase-field problem due to Wheeler et al. [Physica D 66, 243 (1993)] has been adopted and extended as a tool for solidification research by many groups around the World. However, an intrinsic problem of this model is that it couples two physically distinct anisotropies, those associated with the surface energy of the solid-liquid interface and attachment kinetics, into a single anisotropy parameter. In this paper we present a simple extension to the Wheeler model in which we show that introducing a complex form of the anisotropy function allows these two physical parameters to be decoupled.Received: 16 June 2004, Published online: 21 October 2004PACS: 81.10.-h Methods of crystal growth; physics of crystal growth - 81.30.Fb Solidification - 64.70.Dv Solid-liquid transitions     

The influences of gas and electron temperatures on electron attachment in gas electrical discharges

Various electron attachment processes are reviewed, emphasising the way in which the rates and products of some selected reactions vary with the attaching gas temperatureT _g, the temperature,T _e, and the energy of the attaching electrons. The examples illustrating the variety of reactions are the efficient dissociative attachment reaction to CCl₄, attachment to SF₆ which involves both dissociative and non-dissociative attachment, attachment to CHCl₃ which requires activation energy, and attachment to CCl₃Br which results in both Cl- and Br- product ions. A model has been presented which is able to quantitatively explain the difference influences ofT _g andT _e on the rates of some of these reactions. Also described are the unusually efficient attachment properties of the fullerene molecules C₆₀ and C₇₀ as revealed by our FALP experiments, noting that these molecules have potential importance as efficient suppressers of electrical breakdown through gases such as those used to insulate high voltage devices. We emphasise throughout this paper the importance of an understanding of the separate influences of gas and electron temperature on attachment reactions for the modelling of practical gas discharge media such as etchant plasmas. We dedicate this paper to Professor Jan Janča on the occasion of his sixtieth birthday in recognition of his major contributions to gas discharge physics.     

On the Kinetics of the Active Zone of the Stationary SF6 Beam Discharge Plasma

The paper deals with the impact of intensive electron attachment on the kinetics of the electrons in the active zone of the stationary band-like beam discharge plasma in SF₆ which is an alternative useful plasma medium for “dry etching”. The energy distribution of the electrons in this plasma was obtained by numerically solving the Boltzmann equation which includes apart from elastic collisions, different exciting collision processes, attachment in electron collisions, direct ionization, the ambipolar loss of electrons, Coulomb interaction between electrons and of electrons with ions and the power input to the electrons by the turbulent electric field. In particular, due to the needed fulfilment of the consistent electron particle balance, for an extended region of the turbulence energy density in this plasma a large impact on the electron kinetics of the intensive electron attachment, which is the prevailing electron loss process, was found enforcing independent of the turbulence energy density always a large power input to the electrons, smooth and only slowly decreasing energy distributions even in the energy region of direct ionization.     

On the effects of system parameters on the response of a harmonically excited system consisting of weakly coupled nonlinear and linear oscillators

This paper discusses the dynamic behaviour of a nonlinear two degree-of-freedom system consisting of a harmonically excited linear oscillator weakly connected to a nonlinear attachment having linear and cubic restoring forces. The effects of the system parameters on the shape of the frequency-response curve are investigated, in particular those yielding the appearance and disappearance of outer and inner detached resonance curves. In contrast to the case when the linear stiffness of the attachment is zero, it is found that multivaluedness occurs at low frequencies as the resonant peak bends to the right. It is also found that as the coefficient of the linear term increases, the range of parameters yielding detached curves reduces. Compared to the case when the attached system has no linear stiffness term, this range of parameters corresponds to smaller values of the damping and nonlinear coefficients. Approximate analytical expressions for the jump-up and jump-down frequencies of the system under investigation are also derived.     

A self-consistent approach to measure preferential attachment in networks and its application to an inherent structure network

Preferential attachment is one possible way to obtain a scale-free network. We develop a self-consistent method to determine whether preferential attachment occurs during the growth of a network, and to extract the preferential attachment rule using time-dependent data. Model networks are grown with known preferential attachment rules to test the method, which is seen to be robust. The method is then applied to a scale-free inherent structure (IS) network, which represents the connections between minima via transition states on a potential energy landscape. Even though this network is static, we can examine the growth of the network as a function of a threshold energy (rather than time), where only those transition states with energies lower than the threshold energy contribute to the network. For these networks we are able to detect the presence of preferential attachment, and this helps to explain the ubiquity of funnels on potential energy landscapes. However, the scale-free degree distribution shows some differences from that of a model network grown using the obtained preferential attachment rules, implying that other factors are also important in the growth process.     

Ensemble energy average and energy flow relationships for nonstationary vibrating systems

This paper attempts to introduce a new point of view on energy analysis in structural dynamics with particular emphasis to its link with uncertainty and complexity. A linear, elastic system undergoing free vibrations, is considered. The system is subdivided into two subsystems and their respective energies together with the shared energy flow are analysed.First, the ensemble energy average of the two subsystems, assuming uncertain the natural frequencies, is investigated. It is shown how the energy averages follow a simple law when observing the long-term response of the system, obtained by a suitable asymptotic expansion. The second part of the analysis shows how the ensemble energy average of a set of random samples is representative even of the single case if the system is complex enough.The two previous points, combined, produce a result that applies to the energy sharing between two subsystems even independently of uncertainty: for complex systems, a simple energy sharing law is indeed stated. Moreover, in the case of absence of damping, a nonlinear relation between the energy flow and the energy (weighted) difference between the two subsystems is derived; on the other hand, when damping is present, this relationship becomes linear, including two terms: one is proportional to the energy (weighted) difference between the two subsystems, the other being proportional to its time derivative. Therefore, the approach suggests a way for deriving a general approach to energy sharing in vibration with results that, in some cases, are reminiscent of those met in Statistical Energy Analysis.Finally, computational experiments, performed on systems of increasing complexity, validate the theoretical results.     

Phase prediction of the response of choked nozzles to entropy and acoustic disturbances

The development and transmission of sound through the exit of an aero-engine combustor is often investigated by modelling the complex geometry as a convergent-divergent nozzle. However, these analytical acoustic predictions are usually limited to the compact case, where the length of the nozzle is insignificant compared to the wavelength of the flow perturbations, or to cases where the variation of the mean velocity through the nozzle may be treated as linear or piece-wise linear. Considering terms up to first order in frequency for the conservation of mass, momentum and energy, this paper investigates an alternative approach by deriving effective lengths for the passage of the flow perturbations through a supercritical convergent-divergent nozzle. The effects due to the presence of a normal shock wave are also studied using a linearised form of the Rankine-Hugoniot relations. The analyses lead to predictions for the phase and magnitude of the transmitted acoustic waves from finite-length nozzles, and are valid for low non-dimensional frequencies. It has been found that these predictions agree well with the numerical results from inviscid simulations.     

Finite-Time Fluctuations in the Degree Statistics of Growing Networks

This paper presents a comprehensive analysis of the degree statistics in models for growing networks where new nodes enter one at a time and attach to one earlier node according to a stochastic rule. The models with uniform attachment, linear attachment (the Barabási-Albert model), and generalized preferential attachment with initial attractiveness are successively considered. The main emphasis is on finite-size (i.e., finite-time) effects, which are shown to exhibit different behaviors in three regimes of the size-degree plane: stationary, finite-size scaling, large deviations.     

A Weighted Evolving Network with Community Size Preferential Attachment

Community structure is an important characteristic in real complex network. It is a network consists of groups of nodes within which links are dense but among which links are sparse. In this paper, the evolving network include node, link and community growth and we apply the community size preferential attachment and strength preferential attachment to a growing weighted network model and utilize weight assigning mechanism from BBV model. The resulting network reflects the intrinsic community structure with generalized power-law distributions of nodes' degrees and strengths.     

Growth kinetics of polymer crystals in bulk

Temperature-dependent measurements of spherulite growth rates carried out for i-polystyrene, poly(ε -caprolactone) and linear polyethylene show that the controlling activation barrier diverges at a temperature which is 14K, 22K and 12K, respectively, below the equilibrium melting points. We discuss the existence of such a “zero growth temperature” T _zg in the framework of a recently introduced thermodynamic multiphase scheme and identify T _zg with the temperature of a (hidden) transition between the melt and a mesomorphic phase which mediates the crystal growth. The rate-determining step in our model of crystal growth is the attachment of chain sequences from the melt onto the lateral face of a mesomorphic layer at the growth front. The necessary straightening of the sequence prior to an attachment is the cause of the activation barrier. A theory based on this view describes correctly the observations. With a knowledge of T _zg it is possible to fully establish the nanophase diagram describing the stability ranges of crystalline and mesomorphic layers in a melt. An evaluation of data from small-angle X-ray scattering, calorimetry and optical growth rate measurements yields heats of transition and surface free energies of crystals and mesophase layers, as well as the activation barrier per monomer associated with the chain stretching. According to the theory, the temperature dependence of the crystallization rate is determined by both the activation energy per monomer and the surface free energy of the preceding mesomorphic layer. Data indicate that the easiness of crystallization in polyethylene is first of all due to a particularly low surface free energy of the mesomorphic layer.     

Hydration,entropy, and energy relationships for periodate oxidations

Periodate oxidations of ethanediol and pinacol each occur in two phases; these are (1) formation and (2) decomposition of the intermediate complex. In phase (1), an increase in acidity gives . The rate of oxidation of ethanediol decreases with increasing acidity, whereas the rate of oxidation of pinacol maximizes with H₅IO₆. For both glycols, the activation energy increases and ΔS_act decreases with increasing acidity. In phase (2), the energy of activation is essentially constant with pH, whereas the rate decreases, and the entropy of activation decreases modestly as pH decreases. The latter correlates with the nonhomogeniuty of product formation. Rates for 3‐chloro‐1,2‐propanediol are also listed. Pentaerythritol forms an inactive complex with or H₅IO₆ indicating the importance of chelation in the formation of the intermediate complex. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamics of plasma blobs in a shear flow

The global dynamic of plasma blobs in a shear flow is investigated in a simple magnetized torus using the spatial Fourier harmonics (k-space) framework. Direct experimental evidence of a linear drift in k space of the density fluctuation energy synchronized with blob events is presented. During this drift, an increase of the fluctuation energy and a production of the kinetic energy associated with blobs are observed. The energy source of the blob is analyzed using an advection-dissipation-type equation that includes blob-flow exchange energy, linear drift in k space, nonlinear processes, and viscous dissipations. We show that blobs tap their energy from the dominant ExB vertical background flow during the linear drift stage. The exchange of energy is unidirectional as there is no evidence that blobs return energy to the flow.     

Hydrostatic pressure and temperature effects of an exciton-donor complex in quantum dots

Using the matrix diagonalization method and the compact density-matrix approach, we studied the combined effects of hydrostatic pressure and temperature on the electronic and optical properties of an exciton-donor complex in a disc-shaped quantum dot. We have calculated the binding energy and the oscillator strength of the intersubband transition from the ground state into the first excited state as a function of the dot radius. Based on the computed energies and wave functions, the linear, third-order nonlinear and total optical absorption coefficients as well as the refractive index have been examined. We find that the ground state binding energy and the oscillator strength are strongly affected by the quantum dot radius, hydrostatic pressure and temperature. The results also show that the linear, third-order nonlinear and total absorption coefficients and refractive index changes strongly depend on temperature and hydrostatic pressure.