Showing papers on "Equations of motion published in 2000"

Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces

Abstract: Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the "peridynamic" formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.

Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation

Abstract: A thermodynamics-based variational method is developed to establish the equations of motion for three-dimensional (3D) interacting dislocation loops. The approach is appropriate for investigations of plastic deformation at the mesoscopic scale by direct numerical simulations. A fast sum technique for determination of elastic field variables of dislocation ensembles is utilized to calculate forces acting on generalized coordinates of arbitrarily curved loop segments. Each dislocation segment is represented by a parametric space curve of specified shape functions and associated degrees of freedom. Kinetic equations for the time evolution of generalized coordinates are derived for general 3D climb/glide motion of curved dislocation loops. It is shown that the evolution equations for the position $(\mathbf{P}),$ tangent $(\mathbf{T}),$ and normal $(\mathbf{N})$ vectors at segment nodes are sufficient to describe general 3D dislocation motion. When crystal structure constraints are invoked, only two degrees of freedom per node are adequate for constrained glide motion. A selected number of applications are given for: (1) adaptive node generation on interacting segments, (2) variable time-step determination for integration of the equations of motion, (3) dislocation generation by the Frank-Read mechanism in fcc, bcc, and dc crystals, (4) loop-loop deformation and interaction, and (5) formation of dislocation junctions.

An Introduction to the Navier-Stokes Initial-Boundary Value Problem

Abstract: The equations of motion of an incompressible, Newtonian fluid — usually called Navier-Stokes equations — have been written almost one hundred eighty years ago. In fact, they were proposed in 1822 by the French engineer C.M.L.H. Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than twenty years later that the same equations were rederived by the twenty-six year old G. H. Stokes 1845 in a quite general way, by means of the theory of continua.

Gravity in a lattice Boltzmann model

Abstract: In this paper we consider the introduction of a body force, in the incompressible limit, into the lattice Boltzmann model. A number of methods are considered and their suitability to our objectives determined. When there is no density variation across the fluid, gravity can be introduced in the form of an altered pressure gradient. This method correctly satisfies the Navier-Stokes equation; however, if there is a non-negligible density variation present (produced by the body force or otherwise) this method becomes less accurate as the density variation increases and the constant density approximation becomes less valid. Three other methods are also considered for application when there is a non-negligible density variation. The equations of motion satisfied by these models are found up to second order in the Knudsen number and it is seen that only one of these methods satisfies the true Navier-Stokes equation. Numerical simulations are performed to compare the different models and to assess the range of application of each.

Motion of dark solitons in trapped bose-einstein condensates

Abstract: We use a multiple time scale boundary layer theory to derive the equation of motion for a dark (or grey) soliton propagating through an effectively one-dimensional cloud of Bose-Einstein condensate, assuming only that the background density and velocity vary slowly on the soliton scale. We show that solitons can exhibit viscous or radiative acceleration (antidamping), which we estimate as slow but observable on experimental time scales.

Numerical Solution of Acoustic Propagation Problems Using Linearized Euler Equations

Abstract: The goal of this work is to study some numerical solutions of acoustic propagation problems using linearized Eider's equations. The two-dimensional Euler's equations are linearized around a stationary mean flow. The solution is obtained by using a dispersion-relation-preserving scheme in space, combined with a fourth-order Runge-Kutta algorithm in time. This numerical integration leads to very good results in terms of accuracy, stability and low storage. The radiation of a source hi a subsonic and supersonic uniform mean flow is investigated. The numerical estimates are shown to be in excellent agreement with the analytical solutions. Next, a typical problem in jet noise is considered, the propagation of acoustic waves in a sheared mean flow, and the numerical solution compares favorably with ray tracing. The final goal of this work is to improve and to validate the Stochastic Noise Generation and Radiation (SNGR) model. In this model, the turbulent velocity field is modeled by a sum of random Fourier modes through a source term in the linearized Euler's equations. The implementation of acoustic sources in the linearized Euler's equations is thus an important point. This is discussed with emphasis on the ability of the method to describe correctly the multipolar structure of aeroacoustic sources. Finally, a nonlinear formulation of Euler's equations is solved hi order to limit the growth of instability waves excited by the acoustic source terms.

Velocity measurement of a settling sphere

Abstract: We study experimentally the motion of a solid sphere settling under gravity in a fluid at rest. The particle velocity is measured with a new acoustic method. Variations of the sphere size and density allow measurements at Reynolds numbers, based on limit velocity, between 40 and 7 000. At all Reynolds numbers, our observations are consistent with the presence of a memory-dependent force acting on the particle. At short times it has a t -1/2 behaviour as predicted by the unsteady Stokes equations and as observed in numerical simulations. At long times, the decay of the memory (Basset) force is better fitted by an exponential behaviour. Comparison of the dynamics of spheres of different densities for the same Reynolds number show that the density is an important control parameter. Light spheres show transitory oscillations at Re∼ 400, but reach a constant limit speed.

Dynamic density functional theory of fluids

Abstract: We present a new time-dependent density functional approach for studying the relaxational dynamics of an assembly of interacting particles, subject to thermal noise. Starting from the Langevin stochastic equations of motion for the velocities of the particles, we are able by means of an approximate closure to derive a self-consistent deterministic equation for the temporal evolution of the average particle density. The closure is equivalent to assuming that the equal-time two-point correlation function out of equilibrium has the same properties as its equilibrium version. The changes over time of the density depend on the functional derivatives of the grand canonical free-energy functional F [ ] of the system. In order to assess the validity of our approach, we performed a comparison between the Langevin dynamics and the dynamic density functional method for a one-dimensional hard-rod system in three relevant cases and found remarkable agreement. In addition, we consider the case where one is forced to use an approximate form of F [ ].

Singular Direction Avoidance Steering for Control-Moment Gyros

Abstract: A new form of the equations of motion for a spacecraft with Single Gimbal Control Moment Gyros is developed using a momentum approach. This set of four vector equations describing the rotational motion of the system is of order 2N + 7 where N is the number of CMGs. The control input is an N x I column vector of torques applied to the gimbal axes. A modification to the singularity robust Lyapunov control law presented by Oh and Vadali is examined and compared to their control law. Specifically, the attempt to avoid singular gimbal configurations is abandoned in favor of simply avoiding movement in the singular direction. The singular value decomposition is used to compute a pseudoinverse which prevents large gimbal rate commands near or at actual singularities.

Liquid drops and surface tension with smoothed particle applied mechanics

Abstract: Smoothed particle applied mechanics (SPAM), also referred to as smoothed particle hydrodynamics, is a Lagrangian particle method for the simulation of continuous flows. Here we apply it to the formation of a liquid drop, surrounded by its vapor, for a van der Waals (vdW) fluid in two dimensions. The cohesive pressure of the vdW equation of state gives rise to an attractive, central force between the particles with an interaction range which is assumed to exceed the interaction range of all the other smoothed forces in the SPAM equations of motion. With this assumption, stable drops are formed, and the vdW phase diagram is well reproduced by the simulations. Below the critical temperature, the surface tension for equilibrated drops may be computed from the pressure excess in their centers. It agrees very well with the surface tension independently determined from the vibrational frequency of weakly excited drops. We also study strongly deformed drops performing large-amplitude oscillations, which are reminiscent of the oscillations of a large ball of water under microgravity conditions. In an appendix we comment on the limitations of SPAM by studying the violation of angular momentum conservation, which is a consequence of noncentral forces contributed by the full Newtonian viscous stress tensor.

Aerodynamic coupling effects on flutter and buffeting of bridges

Abstract: The effects of aerodynamic coupling among modes of vibration on the flutter and buffeting re- sponse of long-span bridges are investigated. By introducing the unsteady, self-excited aerodynamic forces in terms of rational function approximations, the equations of motion in generalized modal coordinates are transformed into a frequency-independent state-space format. The frequencies, damping ratios, and complex mode shapes at a prescribed wind velocity, and the critical flutter conditions, are identified by solving a complex eigenvalue problem. A significant feature of this approach is that an iterative solution for determining the flutter conditions is not necessary, because the equations of motion are independent of frequency. The energy increase in each flutter motion cycle is examined using the work done by the generalized aerodynamic forces or by the self-excited forces along the bridge axis. Accordingly, their contribution to the aerodynamic damping can be clearly identified. The multimode flutter generation mechanism and the roles of flutter derivatives are investigated. Finally, the coupling effects on the buffeting response due to self-excited forces are also discussed.

Nonlinear dynamics and bifurcations of an axially moving beam

Abstract: The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem; a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied.

Modeling and set point control of closed-chain mechanisms: theory and experiment

Abstract: We derive a reduced model, that is, a model in terms of independent generalized coordinates, for the equations of motion of closed-chain mechanisms. We highlight the fact that the model has two special characteristics which make it different from models of open-chain mechanisms. First, it is defined locally in the generalized coordinates. We therefore characterize the domain of validity of the model in which the mechanism satisfies the constraints and is not in a singular configuration. Second, it is an implicit model, that is, parts of the equations of motion are not expressed explicitly. Despite the implicit nature of the equations of motion, we show that closed-chain mechanisms still satisfy a skew symmetry property, and that proportional derivative (PD)-based control with so-called simple gravity compensation guarantees (local) asymptotic stability. We discuss the computational issues involved in the implementation of the proposed controller. The proposed modeling and PD control approach is illustrated experimentally using the Rice planar delta robot which was built to experiment with closed-chain mechanisms.

Real-time kadanoff-baym approach to plasma oscillations in a correlated electron Gas

Abstract: A nonequilibrium Green's functions approach to the collective response of correlated Coulomb systems at finite temperatures is presented. It is shown that solving Kadanoff-Baym-type equations of motion for the two-time correlation functions including the external perturbing field allows one to compute the plasmon spectrum with collision effects in a systematic and consistent way. The scheme has a ``built-in'' sum-rule preservation and is simpler to implement numerically than the equivalent equilibrium approach based on the Bethe-Salpeter equation.

Three-Dimensional Mathematical Model of Suspended-Sediment Transport

Abstract: This paper presents the basic equations for a mathematical model of sediment-laden flow in a nonorthogonal curvilinear coordinate system. The equations were derived using a tensor analysis of two-phase flow and incorporate a natural variable-density turbulence model with nonequilibrium sediment transport. Correspondingly, a free-surface and the bottom sediment concentration are employed to provide the boundary conditions at the river surface and the riverbed. The finite analytic method is used to solve the equations of mass and momentum conservation and also the transport equation for suspended sediment. To demonstrate the method, the sediment deposition for the Three Gorges Project is considered. The mathematical model specifies the boundary conditions for the inlet and outlet using data from physical model experiments. The results for the mathematical model were tested against laboratory measurements from the physical model experiment. Good agreement and accuracy were obtained.

Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations: Foundations

Abstract: We present a self-contained framework called Direct Integration of the Relaxed Einstein Equations (DIRE) for calculating equations of motion and gravitational radiation emission for isolated gravitating systems based on the post-Newtonian approximation. We cast the Einstein equations into their ``relaxed'' form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve the equations formally as a retarded integral over the past null cone of the field point (chosen to be within the near zone when calculating equations of motion, and in the far zone when calculating gravitational radiation). The ``inner'' part of this integral(within a sphere of radius $\cal R \sim$ one gravitational wavelength) is approximated in a slow-motion expansion using standard techniques; the ``outer'' part, extending over the radiation zone, is evaluated using a null integration variable. We show generally and explicitly that all contributions to the inner integrals that depend on $\cal R$ cancel corresponding terms from the outer integrals, and that the outer integrals converge at infinity, subject only to reasonable assumptions about the past behavior of the source. The method cures defects that plagued previous ``brute-force'' slow-motion approaches to motion and gravitational radiation for isolated systems. We detail the procedure for iterating the solutions in a weak-field, slow-motion approximation, and derive expressions for the near-zone field through 3.5 post-Newtonian order in terms of Poisson-like potentials.

Essentials of classical brane dynamics

Abstract: This paper provides a self-contained overview of the geometry and dynamics of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation in an ordinary 4-dimensional spacetime background. This category also includes “brane world” models that treat the observed universe as a 3-brane in 5 or higher dimensional background. The first sections are concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor. The later sections show how—to lowest order in the thin limit—the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor T¯µv with the (geometrically defined) second fundamental tensor Kμνρ, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane.

A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion

Abstract: Computational fluid mechanics techniques for examining free surface problems in two-dimensional form are now well established. Extending these methods to three dimensions requires a reconsideration of some of the difficult issues from two-dimensional problems as well as developing new formulations to handle added geometric complexity. This paper presents a new finite element formulation for handling three-dimensional free surface problems with a boundary-fitted mesh and full Newton iteration, which solves for velocity, pressure, and mesh variables simultaneously. A boundary-fitted, pseudo-solid approach is used for moving the mesh, which treats the interior of the mesh as a fictitious elastic solid that deforms in response to boundary motion. To minimize mesh distortion near free boundary under large deformations, the mesh motion equations are rotated into normal and tangential components prior to applying boundary conditions. The Navier–Stokes equations are discretized using a Galerkin–least square/pressure stabilization formulation, which provides good convergence properties with iterative solvers. The result is a method that can track large deformations and rotations of free surface boundaries in three dimensions. The method is applied to two sample problems: solid body rotation of a fluid and extrusion from a nozzle with a rectangular cross-section. The extrusion example exhibits a variety of free surface shapes that arise from changing processing conditions. Copyright © 2000 John Wiley & Sons, Ltd.

Experimental and Numerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics

Abstract: A model based upon a finite element procedure is introduced for analyzing the influence of tooth friction on spur and helical gear dynamics. The equations of motion are solved by combining a time-step integration method with several iterative algorithms aimed at satisfying normal and tangential contact conditions. Comparisons between simulated and measured quasi-static bearing forces are satisfactory and largely validate the theoretical developments. Results also reveal the potentially significant contribution of tooth friction to gear vibration and noise. Simulations are then extended to high speeds and the interest of considering both transmission error and tooth friction excitations to achieve silent gears is discussed.

Axiomatic approach to radiation reaction of scalar point particles in curved spacetime

Abstract: Several different methods have recently been proposed for calculating the motion of a point particle coupled to a linearized gravitational field on a curved background. These proposals are motivated by the hope that the point particle system will accurately model certain astrophysical systems which are promising candidates for observation by the new generation of gravitational wave detectors. Because of its mathematical simplicity, the analogous system consisting of a point particle coupled to a scalar field provides a useful context in which to investigate these proposed methods. In this paper, we generalize the axiomatic approach of Quinn and Wald in order to produce a general expression for the self force on a point particle coupled to a scalar field following an arbitrary trajectory on a curved background. Our equation includes the leading order effects of the particle's own fields, commonly referred to as ``self force'' or ``radiation reaction'' effects. We then explore the equations of motion which follow from this expression in the absence of non-scalar forces.

Numerical investigation of gas‐driven flow in 2‐D bubble columns

Abstract: Gas-liquid bubbly flow in 2-D bubble columns was studied by numerical simulation. A Eulerian-Eulerian two-fluid model used describes the time-dependent motion of liquid driven by small, spherical gas bubbles injected at the bottom of the columns. Such equations, numerically implemented in this work, were derived by Zhang and Prosperetti. A distinctive feature of this method is the derivation of the disperse-phase momentum equation by averaging the particle (here, the bubble) equation of motion directly, not the macroscopic equation for the particle phase. Both the time-averaged quantities and dynamic characteristics of the macroscopic coherent structures agree with the experimental data of Lin et al. and Mudde et al. The comparison of simulated results with data demonstrates that this physical model and numerical approach can provide the key features of the time-dependent behavior of dispersed bubbly flows qualitatively with reasonable quantitative accuracy. Effects of the number of injectors, magnitude of bubble-induced viscosity, and various parameters in the interphase momentum exchange were also studied by simulating various cases and comparing with measurements. The applicability of different boundary conditions and the sensitivity to the mesh system used are also examined.

Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids

Abstract: The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified ...

Granular dynamics of gas-solid two-phase flows

Abstract: The study reported in this thesis is concerned with the granular dynamics of gas-solid two-phase flows. In granular dynamics simulations the Newtonian equations of motion are solved for each individual particle in the system while taking into account the mutual interaction between particles and between particles and walls. The gas-phase hydrodynamics is described by the volume averaged Navier-Stokes equations for twophase flow.

Linear Dynamics of an Elastic Beam Under Moving Loads

Abstract: The dynamic response of an Euler-Bernoulli beam under moving loads is studied by mode superposition. The inertial effects of the moving load are included in the analysis. The time-dependent equations of motion in modal space are solved by the method of multiple scales. Instability regions of parametric resonance are identified and the moving mass effect is shown to significantly affect the transient response of the beam. Importance of modal interaction arising out of the possible internal resonance is highlighted. While the external resonance is due to the gravity effects of the moving load, the parametric and internal resonance solely depends on the load mass parameter-ratio of the moving load mass to the beam mass. Numerical results show the influence of the load inertia terms on the beam response under either a single moving load or a series of moving loads.

Linear Theory of a Dual-Spin Projectile in Atmospheric Flight

Abstract: : The equations of motion for a dual-spin projectile in atmospheric flight are developed and subsequently utilized to solve for angle of attack and swerving dynamics. A combination hydrodynamic and roller bearing couples forward and aft body roll motions. Using a modified projectile linear theory developed for this configuration, it is shown that the dynamic stability factor, S(g), and the gyroscopic stability factor, S(g) are altered compared to a similar rigid projectile, due to new epicyclic fast and slow arm equations. Swerving dynamics including aerodynamic jump are studied using the linear theory.