Normal modes of oscillations of one-dimensional monoatomic and diatomic lattices

A polynomial equation is obtained for the solutions of the vibrational frequencies of one-dimensional monoatomic and diatomic lattices with particles connected by identical springs, but with arbitrary springs connecting the end particles to rigid walls. The exact expressions of the different normal modes of oscillations of the linear chain of particles for monoatomic, diatomic and defective lattices are derived in a straightforward way. As special cases of our problem we have considered the effects of different end springs on the vibrational frequencies. One interesting result is that very high frequencies are allowed when the ends of the diatomic lattice are rigidly fixed with the boundary walls.     

Static criteria for the existence of Coulomb strings in storage rings

We derive four rigorous conditions for the stability of Coulomb strings in circular storage rings. These criteria are well met by the existing data from experiments in SIS, ESR, and CRYring but not by the NAP-M experiment. We calculate the potential of the joint transverse zigzag excitation and the longitudinal motion against each other of a string of charged particles as a function of their amplitudes and with the linear density as parameter. This potential exhibits a saddle point in amplitude space which, if overcome, destroys the order of the string. The conditions of stability are derived from the position and height of the saddle point which are fairly independent of the linear density. Our findings confirm the supposition that only the Coulomb interaction in the immediate vicinity of very close encounters of particles is important for the existence of strings.     

Computational analysis for dynamic response of a rotating shaft on flexible support structure with clearances

A finite element-based formulation for modelling the dynamic behavior of a rotating flexible shaft supported by a flexible support structure is presented. The coupling effect between the rigid-body rotation and the flexible deformation of the shaft is considered and represented by non-linear coupling terms in the mass matrix and forcing vectors in the global system of equations. The rigid-body rotation is treated as one of the degrees of freedom (d.o.f.) of the entire system. The interaction between the rotating shaft and the flexible support is modelled by either linear or non-linear springs distributed around the circumference of the shaft. The coupling between the flexibility of the shaft and the flexibility of the support structure are considered. The flexible d.o.f. of both the shaft and the support structure are represented as a set of retained and internal d.o.f. of a Craig-Bampton formulation. An additional transformation is performed when the rigid-body d.o.f. is coupled with the internal and the retained d.o.f. in a Craig-Bampton basis. The equations of motion are solved in the time domain using a modified Newmark method for time integration, in which the Newton-Raphson method is used for handling the non-linear behavior within each time step. Analyses are performed to validate the new development for different combinations of load condition, spring type, and rigid-body rotation.     

Thermodynamic limit in the elastic triangle: perturbation theory

The equilibrium state of a triangular pile of particles, interconnected by linear springs and subjected to the force of gravity, is explored algebraically. A subset of the springs is treated as weak in relation to the others and perturbation theory is used to obtain the zero order and first order expressions for the equilibrium positions of the particles. The zero order results prove to satisfy a thermodynamic limit. In contrast, the first order results fail to exhibit a thermodynamic limit because the perturbation expansion parameter proves to be a function of the number of rows of particles in the triangle.     

TIME INTEGRATION OF NON-LINEAR DYNAMIC EQUATIONS BY MEANS OF A DIRECT VARIATIONAL METHOD

Non-linear dynamic problems governed by ordinary (ODE) or partial differential equations (PDE) are herein approached by means of an alternative methodology. A generalized solution named WEM by the authors and previously developed for boundary value problems, is applied to linear and non-linear equations. A simple transformation after selecting an arbitrary interval of interest T allows using WEM in initial conditions problems and others with both initial and boundary conditions. When dealing with the time variable, the methodology may be seen as a time integration scheme. The application of WEM leads to arbitrary precision results. It is shown that it lacks neither numerical damping nor chaos which were found to be present with the application of some of the time integration schemes most commonly used in standard finite element codes (e.g., methods of central difference, Newmark, Wilson-θ, and so on). Illustrations include the solution of two non-linear ODEs which govern the dynamics of a single-degree-of-freedom (s.d.o.f.) system of a mass and a spring with two different non-linear laws (cubic and hyperbolic tangent respectively). The third example is the application of WEM to the dynamic problem of a beam with non-linear springs at its ends and subjected to a dynamic load varying both in space and time, even with discontinuities, governed by a PDE. To handle systems of non-linear equations iterative algorithms are employed. The convergence of the iteration is achieved by takingn partitions of T. However, the values of T/n are, in general, several times larger than the usual Δt in other time integration techniques. The maximum error (measured as a percentage of the energy) is calculated for the first example and it is shown that WEM yields an acceptable level of errors even when rather large time steps are used.     

Philosophy of Mathematics and Natural Science,by H. Weyl

The paper begins with a general discussion of the properties of elastic liquids and with the use of models, consisting of springs and dashpots, to describe their behaviour. Next the linear and non-linear behaviour of these liquids are described together with some of the more striking observed properties. Experimental measurements are next discussed. The paper ends with a brief account of theoretical work that has been done on elastic liquids.     

Status of the AMANDA experiment

The AMANDA high energy neutrino telescope has successfully been increased in size from four detector strings to ten detector springs during the 1996/1997 season. The first upward going muon-neutrino candidates have been reconstructed from the 1996 year's four-string data. Three new detector strings will be deployed during 1997/1998 to 2350 metres depth.     

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl’s projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.     

Simplified models of the vibration of mannequins in car seats

A simplified two-dimensional modelling approach to predict the vibration response of mannequin occupied car seats about a static settling point is demonstrated to be feasible. The goal of the research is to develop tools for car seat designers. The two-dimensional model, consisting of interconnected masses, springs and dampers is non-linear due to geometric effects but, under the excitations considered, the model behaviour is linear. In this approach to modelling, the full system is initially broken down into subsystems, and experiments are conducted with subsystems to determine approximate values for the stiffness and damping parameters. This approach is necessary because of the highly non-linear behaviour of foam where stiffness changes with compression level, and because the simplified model contains more structure than is necessary to model the relatively simple measured frequency response behaviour, thus requiring a good initial starting point from which to vary parameters. A detailed study of the effects of changing model parameters on the natural frequencies, the mode shapes and resonance locations in frequency response functions is given, highlighting the influence of particular model parameters on features in the seat-mannequin system's vibration response. Reasonable qualitative as well as good quantitative agreement between experimental and simulation frequency response estimates is obtained. In particular, the two-dimensional motions at the peaks in the frequency response, a combination of up and down and rotational behaviour is predicted well by the model. Currently research is underway to develop a similar model with non-linear springs, surface friction effects and viscoelastic elements, that predicts the static settling point, a necessary step to aid in the subsystem modelling stage in this dynamic modelling approach.     

Dynamics of Strings and Branes

We study Nambu–Goto strings and branes. It is shown that they can be considered as continuous limits of ordered discrete sets of relativistic particles for which the tangential velocities are excluded from the action. The linear in unphysical momenta constraints are found. It allows to derive the evolution operators for the objects under consideration from the “first principles.”     

A fluctuation problem in particle transport theory II

A balance equation is formulated for the probability that a particle injected into an infinite, amorphous medium will have suffered N collisions and have given rise to n new particles in a given energy range at time t. The method of regeneration points has been employed and this leads, in the case of two particle production, to a non-linear, integro-differential equation for the probability generating function. This equation is solved for the case of foreign particles slowing down, in which case it becomes linear and results are obtained which include the effects of electronic stopping and absorption, thus generalizing the work in part I. In the cascade problem, a single particle gives rise to two new particles in every collision and it is shown, for a simple hard-sphere model with 1/v scattering and absorption, how the non-linear equation may be solved. The probability for the number of particles and the number of collisions suffered to absorption is obtained in the case of zero absorption, the probability law is shown to obey a Furry distribution. The limitations of the method described in part I for dealing with cascades are highlighted.     

A numerical model for the 3-D non-linear vibrations of an N-string

We present a non-linear numerical model describing the 3-D vibrations of a planar network of N sections of string which are connected together at one common mobile extremity. We call such a network N-string. For small-amplitude vibrations perpendicular to the N-string equilibrium plane, the numerical results coincide with the already known analytical solutions of the linear model. This non-linear model makes it possible to describe small- or large-amplitude 3-D vibrations of any kind of N-string subjected to an initial plucking. The equations of motion are also presented in a dimensionless form and a vast dimensionless physical parameter space is identified. The numerical model can be extended to more complex networks of strings.     

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

A new frequency-time domain procedure, the dynamic Lagrangian mixed frequency-time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.     

Controlling General Polynomial Networks

We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions, we give sufficient conditions for Hörmander’s “bracket condition” to hold, which implies the uniqueness of the steady state (if it exists), as well as the controllability of the associated system in control theory. These conditions are constructive; they are formulated in terms of inequivalence of the forces (modulo translations) and/or conditions on the topology of the connections. We illustrate our results with examples, including “conducting chains” of variable cross-section. This then extends the results for a simple chain obtained in Eckmann et al. in (Commun Math Phys 201:657–697, 1999).     

Non-commutative geometry and string field theory

An attempt is made to interpret the interactions of bosonic open strings as defining a non-cummulative, associative algebra, and to formulate the classical non-linear field theory of such strings in the language of non-commulative geometry. The point of departure is the BRST approach to string field theory. A setting is given in which there is a unique gauge invariant action, whose linearized approximation reproduces the conventional Veneziano spectrum. A derivation of conventional Veneziano model amplitudes from this gauge invariant action is sketched. Some brief comments are made about attempts to extend these results to open superstrings and to closed strings.     

Higher order contribution to the propagation characteristics of low frequency transverse waves in a dusty plasma

Characteristic features of low frequency transverse wave propagating in a magnetised dusty plasma have been analysed considering the effect of dust-charge fluctuation. The distinctive behaviours of both the left circularly polarised and right circularly polarised waves have been exhibited through the analysis of linear and non-linear dispersion relations. The phase velocity, group velocity, and group travel time for the waves have been obtained and their propagation characteristics have been shown graphically with the variations of wave frequency, dust density and amplitude of the wave. The change in non-linear wave number shift and Faraday rotation angle have also been exhibited with respect to the plasma parameters. It is observed that the effects of dust particles are significant only when the higher order contributions are considered. This may be referred to as the ‘dust regime’ in plasma.     

Hawking Radiation of Charged Particles from a Rotating Black String

Using Damour-Ruffini method, Hawking radiation of rotating black strings is studied. Under the condition that the total energy, total angular momentum and total charge are conservative, the transition probability from initial state (energy M+ω, charge Q+e and angular momentum J+m) to final state (energy M, charge Q and angular momentum J) for black strings is derived considering the reaction of radiation particles to spacetime. That is, the probability that black strings radiate particles with energy ω, charge e and angular momentum m is obtained. The real spectrum is not a strictly pure thermal spectrum. Our result is consistent with Parikh and Wilczek’s result. It satisfies the unitary principle of quantum mechanics. However, in our result there are not only the term that denotes effect of energy and charge of radiation particles but also the term that denotes effect of radiation particles angular momentum on rotating black strings angular momentum. We provide a new way for investigating radiation of black strings.     

NORMAL MODE LOCALIZATION FOR A TWO DEGREES-OF-FREEDOM SYSTEM WITH QUADRATIC AND CUBIC NON-LINEARITIES

The non-similar normal modes of free oscillations of a coupled non-linear oscillator are examined. So far, the study of non-linear vibrations has been based on the assumption that the system is admissible. This requirement is satisfied when the stiffness of the springs are odd functions of their displacement. In this work, a two-degrees-of-freedom tuned system is considered with stiffness elements having linear, quadratic and cubic non-linearities. The potential energy function of this system is not symmetric with respect to the origin (equilibrium point) of the configuration space due to the presence of the quadratic non-linearity. Hence, the system considered is no longer admissible. A study of the balancing diagrams is performed to determine the “degenerate” and “global” similar modes of the system. Manevich-Mikhlin asymptotic methodology is used for solving the singular differential equation describing the non-similar modes and approximate analytical expressions are derived. For this system, with weak coupling, localized non-similar modes are detected in a small neighborhood of degenerate similar modes of the tuned system. Numerical integration is used to verify theoretically predicted non-similar normal modes. It is found that these modes pass periodically through a non-zero point in the configuration space.     

Fundamental cosmic strings

Cosmic strings are linear concentrations of energy that may be formed at phase transitions in the very early universe. At one time they were thought to provide a possible origin for the density inhomogeneities from which galaxies eventually develop, though this idea has been ruled out, primarily by observations of the cosmic microwave background (CMB). Fundamental strings are the supposed building blocks of all matter in superstring theory or its modern version, M-theory. These two concepts were originally very far apart, but recent developments have brought them closer. The ‘brane-world’ scenario in particular suggests the existence of macroscopic fundamental strings that could well play a role very similar to that of cosmic strings.

In this paper, we outline these new developments, and also analyse recent observational evidence, and prospects for the future.