Showing papers on "Equations of motion published in 2001"

Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture

Abstract: We study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques. First, we transform the equations of motion using the action-angle variables of the underlying Hamiltonian system and bring them into the form where two-frequency averaging can be applied. We then show that energy pumping is due to resonance capture in the 1:1 resonance manifold of the system, and perform a perturbation analysis in an O (√e) neighborhood of this manifold in order to study the attracting region responsible for the resonance capture. The second method is based on the assumption of 1:1 internal resonance in the fast dynamics of the system, and utilizes complexification and averaging to develop analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The results compare favorably to numerical simulations. The practical implications of the energy pumping phenomenon are discussed.

Energy pumping in nonlinear mechanical oscillators : Part I : Dynamics of the underlying Hamiltonian systems

Abstract: The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) channeling of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

Lattice Boltzmann Simulations of Liquid Crystal Hydrodynamics

Abstract: We describe a lattice Boltzmann algorithm to simulate liquid crystal hydrodynamics The equations of motion are written in terms of a tensor order parameter This allows both the isotropic and the nematic phases to be considered Backflow effects and the hydrodynamics of topological defects are naturally included in the simulations, as are non-Newtonian flow properties such as shear thinning and shear banding

Constrained molecular dynamics approach to fermionic systems

Abstract: We propose a constrained molecular dynamics model for a fermionic system In this approach the equations of motion of the centroids related to the single-particle phase-space distributions are solved by imposing that the one-body occupation probability ${f}_{i},$ evaluated for each particle, can assume only values less than or equal to 1 This condition reflects the fermionic nature of the studied systems, and it is implemented with a fast algorithm which allows also the study of the heaviest colliding system The parameters of the model have been chosen to reproduce the average binding energy and radii of nuclei in the mass region $A=30--208$ Some comparison to the data is given

The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions

Abstract: Strong turbulence at a water–air free surface can lead to splashing and a disconnected surface as in a breaking wave. Averaging to obtain boundary conditions for such flows first requires equations of motion for the two-phase region. These are derived using an integral method, then averaged conservation equations for mass and momentum are obtained along with an equation for the turbulent kinetic energy in which extra work terms appear. These extra terms include both the mean pressure and the mean rate of strain and have similarities to those for a compressible fluid. Boundary conditions appropriate for use with averaged equations in the body of the water are obtained by integrating across the two-phase surface layer.A number of ‘new’ terms arise for which closure expressions must be found for practical use. Our knowledge of the properties of strong turbulence at a free surface is insufficient to make such closures. However, preliminary discussions are given for two simplified cases in order to stimulate further experimental and theoretical studies.Much of the turbulence in a spilling breaker originates from its foot where turbulent water meets undisturbed water. A discussion of averaging at the foot of a breaker gives parameters that may serve to measure the ‘strength’ of a breaker.

General relativistic dynamics of compact binaries at the third post-Newtonian order

Abstract: The general relativistic corrections in the equations of motion and associated energy of a binary system of pointlike masses are derived at the third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian expansion of the metric in harmonic coordinates at the 3PN approximation. The metric is parametrized by appropriate nonlinear potentials, which are evaluated in the case of two point particles using a Lorentzian version of a Hadamard regularization which has been defined in previous works. Distributional forms and distributional derivatives constructed from this regularization are employed systematically. The equations of motion of the particles are geodesiclike with respect to the regularized metric. Crucial contributions to the acceleration are associated with the nondistributivity of the Hadamard regularization and the violation of the Leibniz rule by the distributional derivative. The final equations of motion at the 3PN order are invariant under global Lorentz transformations, and admit a conserved energy (neglecting the radiation reaction force at the 2.5PN order). However, they are not fully determined, as they depend on one arbitrary constant, which probably reflects a physical incompleteness of the point-mass regularization. The results of this paper should be useful when comparing theory to the observations of gravitational waves from binary systems in future detectors VIRGO and LIGO.

Third post-Newtonian dynamics of compact binaries: Noetherian conserved quantities and equivalence between the harmonic-coordinate and ADM-Hamiltonian formalisms

Abstract: A Lagrangian from which one can derive the third post-Newtonian (3PN) equations of motion of compact binaries (neglecting the radiation reaction damping) is obtained. The 3PN equations of motion were computed previously by Blanchet and Faye in harmonic coordinates. The Lagrangian depends on the harmonic-coordinate positions, velocities and accelerations of the two bodies. At the 3PN order, the appearance of one undetermined physical parameter ? reflects the incompleteness of the point-mass regularization used when deriving the equations of motion. In addition the Lagrangian involves two unphysical (gauge-dependent) constants r'1 and r'2 parametrizing some logarithmic terms. The expressions of the ten Noetherian conserved quantities, associated with the invariance of the Lagrangian under the Poincar? group, are computed. By performing an infinitesimal `contact' transformation of the motion, we prove that the 3PN harmonic-coordinate Lagrangian is physically equivalent to the 3PN Arnowitt-Deser-Misner Hamiltonian obtained recently by Damour, Jaranowski and Sch?fer.

Real-time simulation of deformation and fracture of stiff materials

Abstract: Existing techniques for real-time simulation of object deformation are well suited for animating soft materials like human tissue or two-dimensional systems such as cloth. However, simulation of deformation in malleable materials and fracture in brittle materials has only been done offline because the underlying equations of motion are numericaly stiff, requiring many small steps in explicit integration schemes. In contrast, the better-behaved implicit integration techniques are more expensive per time step, particularly for volumetric meshes. We present a stable hybrid method for simulating deformation and fracture of materials in real-time. In our system, the effects of impact forces are computed only at discrete collision events. At these impacts, we treat objects as if they are anchored and compute their static equilibrium response using the Finite Element technique. Static analysis is not time-step bound and its stability is independent of the stiffness of the equations. The resulting deformations, or possible fractures, are computed based on internal stress tensors. Between collisions, disconnected objects are treated as rigid bodies. The simulator is demonstrated as part of a system that provides the user with physically-based tools to interactively manipulate 3D models.

Gravitational field and equations of motion of spinning compact binaries to 2.5 post-Newtonian order

Abstract: We derive spin-orbit coupling effects on the gravitational field and equations of motion of compact binaries in the 2.5 post-Newtonian approximation to general relativity, one PN order beyond where spin effects first appear. Our method is based on that of Blanchet, Faye, and Ponsot, who use a post-Newtonian metric valid for general (continuous) fluids and represent pointlike compact objects with a $\ensuremath{\delta}$-function stress-energy tensor, regularizing divergent terms by taking the Hadamard finite part. To obtain post-Newtonian spin effects, we use a different $\ensuremath{\delta}$-function stress-energy tensor introduced by Bailey and Israel. In a future paper we will use the 2.5PN equations of motion for spinning bodies to derive the gravitational-wave luminosity and phase evolution of binary inspirals, which will be useful in constructing matched filters for signal analysis. The gravitational field derived here may help in posing initial data for numerical evolutions of binary black hole mergers.

Equations of motion for a two-axes gimbal system

Abstract: Equations of motion for the two-axes yaw-pitch gimbal configuration are discussed on the assumption that the gimbals are rigid bodies and have no mass unbalance. The equations are derived and different terms are brought together into separate groups for the sake of clarity and to facilitate certain interpretations. Different kinds of disturbances and their elimination or reduction are discussed, as is the yaw gain dependence on the pitch angle. Inertia cross couplings are also given. The purpose is twofold: to simplify the picture of the two-axes gimbals and to further illustrate the properties of this configuration.

Renormalization in self-consistent approximation schemes at finite temperature: Theory

Abstract: Within finite temperature field theory, we show that truncated non-perturbative selfconsistent Dyson resummation schemes can be renormalized with local counter terms defined at the vacuum level. The requirements are that the underlying theory is renormalizable and that the self-consistent scheme follows Baym’s �-derivable concept. The scheme generates both, the renormalized self-consistent equations of motion and the closed equations for the infinite set of counter terms. At the same time the corresponding 2PI-generating functional and the thermodynamical potential can be renormalized, in consistency with the equations of motion. This guarantees the standard �-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes to hold. The proof uses the techniques of BPHZ-renormalization to cope with the explicit and the hidden overlapping vacuum divergences.

Electronic transitions with quantum trajectories. II

Abstract: The quantum trajectory method (QTM) is applied to nonadiabatic electronic transitions. Equations of motion in a Lagrangian framework are derived for the probability density, velocity, position, and action functions for a discretized wave packet moving along coupled potential energy surfaces. In solving these equations of motion, we obtain agreement with exact quantum results computed by solving the time-dependent Schrodinger equation on a space-fixed grid. On each of the coupled potential energy surfaces, the dynamics of the trajectories is fully quantum mechanical, i.e., there are no “surface–hopping transitions.” We present a detailed analysis of the QTM results including density changes, complex oscillations of the wave functions, phase space analysis, and a detailed discussion of the forces that contribute to the evolution the trajectories.

Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash

Abstract: Dynamics of gear-pair systems driven by motors and presenting speed-dependent moment resistance is investigated. First, a suitable mechanical model is developed, taking into account the gear mesh backlash and static transmission error as well as the essential non-linearities due to the bearing clearance and contact characteristics. In comparison with earlier related studies, the new element of the present work is that the model developed determines the system response by simply specifying the external loads, rather than by assuming an a priori value for the constant mean angular velocity of the gear shafts. In fact, models employed in earlier research studies are shown to be obtained as special cases of the present model. The study is completed by numerical results. First, classical response diagrams are presented, illustrating the effect of the most important parameters on the system response. Finally, direct integration of the equations of motion is also performed, demonstrating the existence of quasi-periodic and chaotic long time response for selected combinations of the system parameters.

On a class of exact solutions of the equations of motion of a second grade fluid

Abstract: By means of Fourier sine transform, the velocity field corresponding to a flow of a suddenly moved flat plate in a second grade fluid is determined. The adequate solution for the Rayleigh-Stokes problem for the edge is also presented.

Nonlinear Supersonic Flutter of Circular Cylindrical Shells

Abstract: The aeroelastic stability of simply supported, circular cylindrical shells in supersonic flow is investigated. Nonlinearities caused by large-amplitude shell motion are considered by using the Donnell nonlinear shallow-shell theory, and the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied to the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear piston theory is applied to describe the fluid-structure interaction by using two different formulations, taking into account or neglecting the curvature correction term. The system is discretized by Galerkin projections and is investigated by using a model involving seven degrees of freedom, allowing for traveling-wave flutter of the shell and shell axisymmetric contraction. Results show that the system loses stability by standing-wave flutter through supercritical bifurcation; however, traveling-wave flutter appears with a very small increment of the freestream static pressure that is used as the bifurcation parameter. A very good agreement between theoretical and existing experimental data has been found for flutter amplitudes. The influence of internal static pressure has also been studied.

On the dynamics of parallel manipulators

Abstract: Studies the dynamics of parallel manipulators. We first have a brief review and discussion on different dynamics formulations in the literature (Newton-Euler, direct Lagrangian, and Lagrange-D'Alembert formulation on the reduced system). Then we show the equivalence of these methods. Based on the concepts from differential manifolds, we prove that away from configuration singularity, there exists a projection from the joint space to parameterize the configuration space. The fact that the dynamics is well defined even at actuators singularity, end-effector singularity and other kinds of parameterization singularity is highlighted. For the method of reduced systems, there are two main drawbacks. Firstly the joints being cut for forming the tree system are presumed to have no external torque. Secondly the force and torque applied to other links of the manipulator is not considered. We propose two methods to remedy the situation. Firstly by cutting a link instead of a joint, all the joints torque can be incorporated into our equations of motion. This is useful not only for the case of actuating all the joints, but also if we consider compensating the joints friction. Secondly we propose a concept of transforming force to the generalized force space so that all the other forces and torque can be considered.

Determining stress parameters of formations from multi-mode velocity data

Abstract: A method for determining unknown stress parameters in earth formation measures velocities in four sonic transmissions modes (compression, fast shear, slow shear and Stoneley) at a series of depths. Relationships between measured velocities and other measured values, two independent linear constants, and three nonlinear constant associated with equations of motion for pre-stressed isotropic materials are expressed in a set of four velocity difference equations derived from non-linear continuum mechanics. The velocity difference equations are solved using inversion for useful stress parameters, including maximum horizontal stress, minimum horizontal stress, pore pressure, and change in pore pressure over time.

Collective variable theory for optical solitons in fibers.

Abstract: We present a projection-operator method to express the generalized nonlinear Schrodinger equation for pulse propagation in optical fibers, in terms of the pulse parameters, called collective variables, such as the pulse width, amplitude, chirp, and frequency. The collective variable (CV) equations of motion are derived by imposing a set of constraints on the CVs to minimize the soliton dressing during its propagation. The lowest-order approximation of this CV approach is shown to be equivalent to the variational Lagrangian method. Finally, we demonstrate the application of this CV theory for pulse propagation in dispersion-managed optical fiber links.