Narrow Results

Equations governing the evolution of the hydrodynamic variables in a lattice-gas automaton, arbitrarily far from equilibrium, are derived from the micro-dynamical description of the automaton, under the condition that the local collision rules satisfy semi-detailed balance. This condition guarantees that a factorized local equilibrium distribution (for each node) of the Fermi–Dirac form is invariant under the collision step but not under propagation. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice-Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium. Linearization of the hydrodynamic equations derived here leads to Green–Kubo formulae for the transport coefficients.

The purpose of the present paper is to review briefly the quantum analysis and to apply it in deriving quantum correlation identities. These give the nonlinear response of isotropic magnetic systems in the presence of a finite magnetic field. Hybrid exponential product formulas are also discussed from the point of computationics, as an application of the quantum analysis.

Nonequilibrium systems driven by additive or multiplicative dichotomous Markov noise appear in a wide variety of physical and mathematical models. We review here some prototypical examples, with an emphasis on analytically-solvable situations. In particular, it has escaped attention till recently that the standard results for the long-time properties of such systems cannot be applied when unstable fixed points are crossed in the asymptotic regime. We show how calculations have to be modified to deal with these cases and present a few relevant applications — the hypersensitive transport, the rocking ratchet, and the stochastic Stokes' drift. These results reinforce the impression that dichotomous noise can be put on par with Gaussian white noise as far as obtaining analytical results is concerned. They convincingly illustrate the interplay between noise and nonlinearity in generating nontrivial behaviors of nonequilibrium systems and point to various practical applications.

We review some of our recent results on the dynamics of a dissipative quantum particle in periodic cosine potentials. The dissipation is modeled by the Caldeira-Leggett prescription. Exact real-time Wigner distributions for the particle alone can be obtained in powers of V₀ (the strength of the cosine potentials), which is suitable for calculations of various time-dependent averages. Special interest is devoted to summing over the whole series. For Ohmic dissipation, we have the following results: (a) The Kubo-Einstein relation for the linear response is shown to hold rigorously to all orders. (b) In the small viscosity limit, nearly exact analytic expressions for the nonlinear mobility and the nonlinear time-dependent response are found. (c) The resummation can also be performed in certain regimes of the dissipative Hafstadter problem in two dimensions. These results we believe can have useful applications in experiments in Josephson junctions and other mesoscopic systems.

The new features of the bound state’s formation due to the defect nonlinearity and stepwise medium nonlinearity are found. The Schrödinger equation containing two terms of different nonlinearity types is solved analytically. The exact solution found is described as the bound state near the nonlinear defect in the nonlinear medium. Properties of the bound state are analyzed in dependence on defect and medium parameters. The bound state existing only near nonlinear defect only is found. It is shown that the influence of a change in the effective defect power is qualitatively equivalent to a change in the defect power up to renormalization, and the effect of the defect nonlinearity is opposite.

In this paper, the nonlinear response and chaos of a cracked rotor with two disks are studied. Considering the breadth of crack in one rotor revolution, the motion equations of the system are derived and then solved. The results show that the rotor response is sensitive to the crack depth, rotating speed, damping ratio and imbalance. When a crack occurs, the frequency of swing vibration is a multiple of rotating speed (NΩ,N=2,3,…). There are three main routes for response to chaos, that is from quasi-periodic to chaos, from quasi-periodic to quasi-periodic bifurcation and then to chaos and the intermittence to chaos. The intermittence chaos occurs even for a small crack. With the intermittence chaos range there exists the periodic-doubling bifurcation with time. Larger imbalance parameter and damping ratio can suppress chaos. The diagram of time-phase is a useful way to analyze the nonlinear response.

Both the primary and superharmonic resonance responses of a rigid rotor supported by active magnetic bearings are investigated by means of the total harmonic balance method that does not linearize the nonlinear terms so that all solution branches can be studied. Two sets of second order ordinary differential equations governing the modulation of the amplitudes of vibration in the two orthogonal directions normal to the shaft axis are derived. Primary resonance is considered by six equations and superharmonic by eight equations. These equations are solved using the polynomial homotopy continuation technique to obtain all the steady state solutions whose stability is determined by the eigenvalues of the Jacobian matrix. It is found that different shapes of frequency-response and forcing amplitude-response curves can exist. Multiple-valued solutions, jump phenomenon, saddle-node, pitchfork and Hopf bifurcations are observed analytically and verified numerically. The new contributions include the foolproof multiple solutions of the strongly nonlinear system by means of the total harmonic balance. Some predicted frequency varying amplitudes could not be obtained by the multiple scales method.

We present an overview of the physics of self-written waveguides created in photosensitive optical materials, including the experimental observations and the corresponding theoretical models for describing the growth of both bright and dark self-written beams. We discuss in more details the properties of self-written waveguides created in photosensitive polymers, which have been discovered in the recent experimental and theoretical studies. The self-writing process is essentially a nonlinear phenomenon, since the temporal dynamics depends on the optical exposure. Under appropriate conditions, permanent large changes in the refractive index are induced along the propagation direction of an optical beam, so that optical channels, or "filaments", appear as waveguides becoming "frozen" in a photosensitive material. We describe the growth of individual filaments as well as the interaction of several filaments, also making a comparison between the physics of self-written waveguides and the concept of spatial optical solitons in self-focusing nonlinear media.

This work focuses on understanding the nonlinear-optical response of a 1-D quantum wire embedded in 2-D space when quantum-size effects in the transverse direction are minimized using an extremely weighted delta function potential. Our aim is to establish the fundamental basis for understanding the effect of geometry on the nonlinear-optical response of quantum loops that are formed into a network of quantum wires. It is shown that in the limit of full confinement, the sum rules are obeyed when the transverse infinite-energy continuum states are included. While the continuum states associated with the transverse wavefunction do not contribute to the nonlinear optical response, they are essential to preserving the validity of the sum rules. This work is a building block for future studies of nonlinear-optical enhancement of quantum graphs (which include loops and bent wires) based on their geometry. These properties are important in quantum mechanical modeling of any response function of quantum-confined systems, including the nonlinear-optical response of any system in which there is confinement in at least one dimension, such as nanowires, which provide confinement in two dimensions.

Quasi-1D quantum structures with spectra scaling faster than the square of the eigenmode number (superscaling) can generate intrinsic, off-resonant optical nonlinearities near the fundamental physical limits, independent of the details of the potential energy along the structure. The scaling of spectra is determined by the topology of the structure, while the magnitudes of the transition moments are set by the geometry of the structure. This paper presents a comprehensive study of the geometrical optimization of superscaling quasi-1D structures and provides heuristics for designing molecules to maximize intrinsic response. A main result is that designers of conjugated structures should attach short side groups at least a third of the way along the bridge, not near its end as is conventionally done. A second result is that once a side group is properly placed, additional side groups do not further enhance the response.

This paper presents a simplified model for predicting the nonlinear random response of flat and buckled plates. Based on a single mode representation of vibration response, r.m.s. values of the strain response to broadband excitation are evaluated for different static buckled configurations using the equivalent linearization technique. The dynamic effects on the overall strain response due to instability motion of snap-through are included. Parametric studies are performed in which the influences of the clamped and simply-supported boundaries, aspect ratio of the plate, thickness and length of the plate are considered. Using a simple single-model formula, the results of dynamic buckling motion were compared with finite-element models as well as experimental results. The comparisons between analytical results and experimental results help to assess the accuracy of the theory and the conditions under which deviations from the theory due to effects of imperfection and higher modes are significant. It is found that the theoretical model is useful for design and checking of computer results for curved plates in a slightly deflected form (initial deflection less than twice the thickness of plate) and the prediction accuracy on nonlinear analysis is higher than existing design formula based on linear response.

Owing to their superior mechanical and thermal properties, carbon nanotube (CNT) reinforced composite materials have wide range of applications in various technical areas such as aerospace, automobile, chemical, structural and energy. In this paper, the nonlinear axisymmetric dynamic behavior of sandwich spherical and conical shells made up of CNT reinforced facesheets is studied. The shell is subjected to thermal loads and discretized with three-noded axisymmetric curved shell element based on field consistency approach. The in-plane and the rotary inertia effects are included within the transverse shear deformation theory in the element formulation. The present model is validated with the available analytical solutions from the literature. A detailed parametric study is carried out to showcase the effects of the shell geometry, the volume fraction of the CNT, the core-to-face sheet thickness and the environment temperature on the dynamic buckling thermal load of spherical caps.

A marine riser operated in a deep-water field could be substantially affected by large amounts of movement of the floating platform, which is more complicated and very challenging to analyze. This paper presents a mathematical model involving nonlinear dynamic response analysis of a marine riser caused by sways and heave motions at the top end, which are treated as the constraint conditions. The nonlinear equation of motion, arising from the nonlinearity of the ocean current and wave loadings, is derived and written in general matrix form using the finite element method. The excitation caused by platform movement is imposed on the riser system through the time-dependent constrained condition using the penalty method. The advantages of this method are that it is easily implemented on the nonlinear equation of motion and it requires no additional unknown variable, and thus consumes less computational time. By this method, the stiffness matrix and the force vector of the system are then modified, enforcing top-end vessel motion. The dynamic responses are evaluated by using numerical time integration based on Newmark’s method with direct iteration. The effects of the oscillation frequency of top-end vessel sway and heave motions on the nonlinear dynamic characteristics of the riser are investigated. The numerical results reveal that the riser responses to the top-end vessel excitation behave like a periodic motion, which is conformable to the characteristics of vessel movements. The increase in the oscillation frequency of the top-end vessel increases the maximum displacement amplitude for both the horizontal and vertical directions. The directional motion of the vessel also significantly influences the response amplitude of the riser.

This paper adopts both explicit and implicit finite element methods in a specialist package LS-DYNA to investigate the nonlinear, dynamic response of a long span shell roof structure when subjected to blast loading. Parametric studies have been carried out on blast loaded laminated glass plates with reference to experimental results obtained by European researchers. A case study that has been chosen is a light rail station in The Netherlands called The Erasmusline. Following the detonation of 15kg TNT charge, explicit analysis showed breakage surrounding the rigid supports along the edge beam where modal vibrations are restrained. An implicit analysis has confirmed the resonances in global eigen-frequencies where most blast damage is localized around the roof canopy hence producing cracking and potential glass detachment from the panels without full structural demolition. This insight from this study will inform structural engineers about the potential modes of failure and preventative strategies to minimize further secondary losses of life or assets from a terrorist attack.

This paper investigates nonlinear damage response and ultimate collapse of laminates under in-plane and lateral pressure loadings. The in-plane loading was in the form of end-shortening strain, while the lateral pressure was sinusoidal. The plates had initial geometric imperfection to which simply-supported boundary conditions were applied. Ritz techniques with nonlinear strain terms in kinematic relations as well as the first-order shear deformation theory were applied. Hashin and Rotem failure criteria were used for failure analysis. Two models were also employed for degradation of material properties in the plates. The complete ply degradation model was implemented along with the ply region degradation model, in which stiffness reduction was applied only to one region of the ply in which failure had occurred. Note that the stiffness degradation after the failure was investigated as both instantaneous and linear models. In both complete ply and region ply degradation models with instantaneous degradation of material properties, at any location in a ply or region, which has exceeded the given stress criterion, the corresponding stiffness properties are instantaneously degraded throughout that ply or region but with linear material degradation model, the stiffness diminishes gradually and linearly. Finally, the results were then validated against the findings of different references as well as finite element analysis. According to the results, it was seen that in the ply region degradation model, last ply failure loads are generally larger than those of the complete ply degradation model.

This paper presents the results of a study aimed at assessing the effects of higher modes of vibration on the nonlinear dynamic response of tall framed-buildings subjected to narrow-banded motions. For this purpose, analytical models of a 20-story building were developed under the consideration of two types of hysteretic behavior and subjected to 20 seismic excitations having a dominant period of motion of 1s. While the fundamental period of vibration of the building equals 2.7s, its second period is close to 1s; as a result, the selected seismic excitations over-stimulate the participation of the second mode in the overall dynamic response of the building. The circumstances under which the contribution of higher modes gives place to an excessive response of the upper stories are identified, and quantitative measures are presented. From an engineering point of view, parameters that predict an excessive response of higher-level floors are proposed.

The validity of linear response theory to describe the response of a nonlinear stochastic system driven by an external periodical time dependent force is put to a critical test. A variety of numerical and analytical approximations is used to compare its predictions with numerical solutions over an extended parameter regime of driving amplitudes and frequencies and noise strengths. The relevance of the driving frequency and the noise value for the applicability of linear response theory is explored for single and multi-frequency input signals.

A model for a Brownian particle is introduced which shows multiple current reversals in response to an external static force, i.e. the average particle velocity reverses its direction several times upon continuously increasing or decreasing the external force. The model is studied analytically, and the physical mechanism behind the current reversals is discussed.

The computation of constant ductility (or isoductile) response spectra for single-degree-of-freedom systems can require numerous individual response history analyses. Recognising that the same ductility response may be obtained for different strength oscillators of a given period, greater computational effort is required to reduce the possibility that a desired solution is not overlooked. Even a single solution may not exist if a local discontinuity in the strength-ductility relationship coincides with the desired value of ductility. This paper describes a two-phase algorithm to identify the highest strength solution for which the corresponding ductility equals (or does not exceed) the desired ductility. The first phase adopts a "check-reject" approach to reject intervals of strength where the possibility of unidentified higher-strength solutions is considered to be remote, thereby narrowing the strength interval in which the solution will be found. The second phase identifies a solution within this interval as rapidly as possible using a bisection approach. The algorithm is implemented in the USEE software program. The efficiency and accuracy of the algorithm are demonstrated by comparison to results obtained with other software programs.

This paper deals with nonlinear response of smart two-phase nanocomposite plates with surface-bonded piezoelectric layers under a combined mechanical, thermal and electrical loading. The governing equations of the carbon nanotube reinforced composite plate are derived based on first order shear deformation plate theory (FSDT) and von Kármán geometric nonlinearity. The material properties of the nanocomposite host are assumed to be graded in the thickness direction. The single-walled carbon nanotubes (SWCNTs) are assumed aligned, straight and a uniform layout. The Galerkin method is employed to derive the nonlinear governing equations of the problem. A perturbation scheme is employed to determine the nonlinear vibration response and the nonlinear natural frequencies of the plates with immovable simply supported boundary conditions. Post-buckling load–deflection and maximum transverse load–deflection relations have been obtained for the plate under consideration. The effects of the applied voltage, temperature change, plate geometry, and the volume fraction and distribution pattern of the SWCNTs on the linear and nonlinear natural frequencies of the smart two-phase composite plates are investigated through a detailed parametric study.