Dynamics Of Rods Research Articles

Dynamics of hard rods with initial domain wall state

Inspired by recent results on the non-equilibrium dynamics of many-body quantum systems, we study the classical hard rod problem in one dimension with initial domain wall condition. Hard rods are an integrable system, in the sense that for each velocity the density of particles is locally conserved. It was proven by Boldrighini et al (1983 J. Stat. Phys. 31 577) that on the hydrodynamic space-time scale, the fluid of hard rods satisfies Euler-type equations which comprise all conservation laws. We provide the general solution to these equations on the line, with an initial condition where the left and right halves are, asymptotically, in different states. The solution is interpreted as being composed of a continuum of contact discontinuities, one for each velocity. This is a classical counterpart of the transport problem solved recently in quantum integrable systems. We also discuss the Navier–Stokes (viscous) corrections, and study its effect on the broadening of the contact discontinuity and on entropy production.

A rheological constitutive model for semiconcentrated rod suspensions in Bingham fluids

A rheological constitutive law is developed for a suspension of rigid rods in a Bingham fluid for volume fractions ranging up to the semiconcentrated regime. Based on a cell model approach, which allows expressing the shear stress on the particle surface, the particle stress contribution is derived and involves additional yield stress terms related to an ensemble average orientation distribution of the rods. As a first approach, a von Mises criterion is used to describe the composite flow threshold, which is found to be anisotropic in the sense that it depends on the rod orientation. A rod dynamics equation is also proposed and incorporates some diffusion/perturbation due to yielded regions encountered throughout the suspension. In parallel, an equivalent kinetic theory is also developed. The model provides good agreement with shear stress experiments for kaolin pastes filled with steel fibers of two different aspect ratios.

Real time simulation of inextensible surgical thread using a Kirchhoff rod model with force output for haptic feedback applications

We discuss the dynamics of an inextensible thin Kirchhoff rod used in the modeling of surgical threads, and demonstrate a very efficient scheme to not only simulate the motion of the thread in real-time (up to 1 ms per frame) but also obtain the constraint axial forces which can be fed back to a haptic system. The numerical scheme is based on a family of schemes called geometric or discrete variational integrators guaranteeing that the momentum and energy are exactly conserved over long periods of time for conservative systems. Besides, we report on an efficient numerical procedure to handle the inextensibility of the thread through physically based Lagrange multipliers, as well as the internal dissipation of the thread. We have performed simulations to verify the capabilities of our model to conserve momentum and energy, accurately calculate the axial constraint forces along the thread for haptic feedback, and capture bending-torsion coupling leading to the formation of plectonemes. While many of the ideas are well known in the computer graphics community (especially in hair modeling), we have implemented several improvements for the specific purpose of speeding up the computations for developing physically based haptic interfaces for knot tying and suturing.

Low-Dimensional Modeling of a Pumping Unit to Cope With Multiphase Flow

Pumping unit efficiency is highly disturbed by the presence of gas influx reducing the productivity and inducing unpredictable system response due to the change of its intrinsic properties such as the natural frequency. A poor estimation of those properties may affect the on-field crew and system safety as well as the production rate. The purpose of this paper is to construct a hydromechanical model describing the coupled multiphase flow-pumping unit system dynamics and to develop a procedure to control the pumping speed for safety assurance and oil production maximization. A coupled mechanical-multiphase flow model capturing the interplay between the gas void fraction (GVF) and the driving harmonic force of the pumping unit is developed. Specifically, the predicted downhole pressure is used to determine the sucker rod effective load. Consequently, a reduced-order model, capturing the dynamics of the sucker rod, is used to estimate the saddle bearings axial displacements which are function of polished rod loading. An error-driven adaptation using the difference between presumed bearing displacement with known GVF and the predicted bearing displacement from the proposed multiphysics model is employed to estimate the unknown downhole GVF. The obtained results prove that the adaptation allows an accurate evaluation of the pumped fluid's GVF, thereby circumventing the need for a costly and inaccurate measurement of the two-phase flow gas fraction. Based on this estimation, a control strategy is then proposed to regulate the pump speed while avoiding the resonance frequency of the sucker-rod system.

Dynamics of rod under axial impact by a body

Dynamics of rod under axial impact by a body

Quasi-resonance Behaviour of Viscoelastic Rods Subjected to Free and Forced Longitudinal Vibrations

A classical viscoelastic rod subjected to longitudinal vibrations is considered. The boundary conditions are so that the left end of the rod is fixed and the right end is free. Moreover it is assumed that the rod is growing and hence, its length is changing in time. The function of growth is assumed to be twice continuously differentiable with respect to time. A particular case of rod's growth proportional to time is of special interest. The change of variables is introduced so that in new variables the rod length becomes constant. The new partial differential equation describing the rod's dynamics is derived in these variables. This equation is simplified using assumptions on slow rate growth constant and small viscoelastic damping factor. A special representation of solution is introduced which uses eigenfunctions of the generating problem, when growth and damping are neglected, and satisfy the boundary conditions. By means of this representation the governing partial differential equation is converted in infinite system of ordinary differential equations. It is shown that solutions of truncated systems converge to solution of the original system of equations. Three major problems of the growing rod vibrations are formulated and solved. In the first problem free undamped vibrations are considered. It is shown that at linear growth of rod length amplitudes of all its modes are also growing linearly in time. The simplified model neglecting the modes cross-coupling is composed for the explanation of this effect. The corresponding differential equation is solved exactly in elementary functions and it is shown that amplitudes of vibration of any modes grow linearly and almost-periods of vibrations have logarithmic dependence on time. In the second problem free damped vibrations of linearly growing rod are considered. It is shown that time behaviour of the rod has two characteristic domains: in the first the vibration amplitudes decay exponentially due to domination of the viscoelastic damping effects; in the second domain these amplitudes start to grow linearly in time due to domination of the effects considered in the first problem. The simplified model describing this effect and neglecting the modal cross-coupling is developed. The exact solution of the corresponding differential equation is obtained in the confluent hypergeometric functions, which qualitatively explain the abovementioned behaviour of the rod. In the third problem the forced damped vibrations of the rod are considered. It is shown that at fixed frequency of excitation the resonant effects are manifested subsequently in all modes of the rod in the process of its growing.

ミニマルな介入によるヒトに近い高速走行

In this study, we aim to the human-like high-speed running that had high energy efficiency based on bouncing rod dynamics. Humans are intervening in the robot’s running and are trying to improve while feeling running conditions. It is significantly different from the normal development process on robotics. We propose the remote manual operation system with wires and realize the human-like high-speed running by a minimal intervention.

Dynamic instability of super-long elastic rod in viscous fluid

The external environment affects the structural form of biological system. Many biological systems are surrounded by cell solutions, such as DNA and bacteria. The solution will offer a viscous resistance as the biological system moves in the viscous fluid. How does the viscous resistance affect the stability of biological system and what mode will be selected after instability? In this paper, we establish a super-long elastic rod model which contains the viscous resistance to model this phenomenon. The stability and instability of the super-long elastic rod in the viscous fluid are studied. The dynamic equations of motion of the super-long elastic rod in viscous fluid are given based on the Kirchhoff dynamic analogy. Then a coordinate basis vector perturbation scheme is reviewed. According to the new perturbation method, we obtain the first order perturbation representation of super-long elastic rod dynamic equation in the viscous fluid, which is a group of the second order linear partial differential equations. The stability of the super-long elastic rod can be determined by analyzing the solutions of the second order linear partial differential equations. The results are applied to a twisted planar DNA ring. The stability criterion of the twisted planar DNA ring and its critical region are obtained. The results show that the viscous resistance has no effect on the stability of super-long elastic rod dynamics, but affects its instability. The mode selection and the influence of the viscous resistance on the instability of DNA ring are discussed. The amplitude of the elastic loop becomes smaller under the influence of the viscous resistance, and a bifurcation occurs. The mode number of instability of DNA loop becomes bigger with the increase of viscous resistance.

Diffusion of a nanowire rod through an obstacle field.

We report the experimental realization of a rod diffusing in a two-dimensional obstacle field following the single rod dynamics. We use a silver nanowire as our rod and two types of obstacles: repelling light beams and polymer pillars. We study the effect of hydrodynamic interactions on the transport of the rod, comparing both experimental realizations and recent simulations. We propose a framework for analyzing the transport through such systems, and we predict a new superdiffusive regime of rod transport at high obstacle concentration and short times.

Mathematical Models for the Propagation of Stress Waves in Elastic Rods: Exact Solutions and Numerical Simulation

AbstractIn this work, the Bishop and Love models for longitudinal vibrations are adopted to study the dynamics of isotropic rods with conical and exponential cross-sections. Exact solutions of both models are derived, using appropriate transformations. The analytical solutions of these two models are obtained in terms of generalised hypergeometric functions and Legendre spherical functions respectively. The exact solution of Love model for a rod with exponential cross-section is expressed as a sum of Gauss hypergeometric functions. The models are solved numerically by using the method of lines to reduce the original PDE to a system of ODEs. The accuracy of the numerical approximations is studied in the case of special solutions.

НЕЛИНЕЙНЫЕ СТАЦИОНАРНЫЕ ИЗГИБНЫЕ ВОЛНЫ В СТЕРЖНЕ ИЗ УПРОЧНЯЮЩЕГОСЯ МАТЕРИАЛА

In this paper nonlinear bending waves in an infinite straight rod made of material whose hardening is described by Hollomon - Ludwik power-law are considered. The process of bending is described by Bernoulli - Euler theory. The equations of rod's dynamics were obtained using the Hamilton - Ostrogradsky principle based on expressions of potential and kinetic energy for the elementary portion of the rod. The search of solutions was performed in the form of travelling stationary waves, which allowed reducing the partial differential equation to ordinary one. Investigation of phase portraits of obtained equation was performed using the MAPLE software. Possibility of existence of nonlinear travelling stationary waves, which propagate with constant velocity and maintain its initial shape, were determined. Also the character of dependences between the length of stationary wave and its amplitude, hardening index and strength index is established. Keywords: bending waves, nonlinearity, work hardening, Hamilton − Ostrogradsky principle, phase portraits.

自由度凍結の解除とワイヤ拘束によるヒトに近い走行

In this study, we focus on the bouncing rod dynamics and its mechanism. We aim to realize the passive running that had high energy efficiency. We tried a passive running with high self-subsistence by using wires and levers, but the running was unstable. We developed new mechanism based on releasing fixed degree of freedom and wire-restraint, resulting in more stable human-like running.

Electric field–induced criticality and frequency responsive dynamics of charged rods

We use the suspensions of charged fibrous virus (fd) as a model system for charged colloidal rods, where thick electric double layers are present. In particular, the system has shown the electric field–induced phase/state transitions under a low electric field strength. We briefly present the frequency responsive dynamics in the phase/state diagram and field-induced criticality in the dynamical states by approaching the field-induced critical point. In addition, the possible mechanism of low-frequency responsive dynamical states is briefly explained by the local charge redistribution of dissociation and association of condensed ions on the charged rods. The found phases/states are unique in the bulk phase diagram, away from the electrode polarization that is dominant near to the surface of the indium tin oxide glasses, in the sense that they are observed independent from the applied field amplitude and frequency.

Periodicity effects of axial waves in elastic compound rods

Floquet analysis is applied to the Bernoulli–Euler model for axial waves in a periodic rod. Explicit asymptotic formulae for the stop band borders are given and the topology of the stop band pattern is explained. Eigenfrequencies of the symmetric unit cell are determined by the Phase-closure Principle, and their correspondence with stop band formation is shown. Steady-state and transient dynamics of a periodic rod of finite length are analysed numerically and the difference in structural response when excitation is done in either stop- or pass bands is demonstrated. A physical interpretation of the underlying mechanisms of stop bands is proposed.

Stochastic dynamics of penetrable rods in one dimension: Entangled dynamics and transport properties.

The dynamical properties of a system of soft rods governed by stochastic hard collisions (SHCs) have been determined over a varying range of softness using molecular dynamics simulations in one dimension and analytic theory. The SHC model allows for interpenetration of the system's constituent particles in the simulations, generating overlapping clustering behavior analogous to the spatial structures observed in systems governed by deterministic bounded potentials. Through variation of an assigned softness parameter δ, the limiting ranges of intermolecular softness are bridged, connecting the limiting ensemble behavior from hard to ideal (completely soft). Various dynamical and structural observables are measured from simulation and compared to developed theoretical values. The spatial properties are found to be well predicted by theories developed for the deterministic penetrable-sphere model with a transformation from energetic to probabilistic arguments. While the overlapping spatial structures are complex, the dynamical properties can be adequately approximated through a theory built on impulsive interactions with Enskog corrections. Our theory suggests that as the softness of interaction is varied toward the ideal limit, correlated collision processes are less important to the energy transfer mechanism, and Markovian processes dominate the evolution of the configuration space ensemble. For interaction softness close to hard limit, collision processes are highly correlated and overlapping spatial configurations give rise to entanglement of single-particle trajectories.

Study of the double rebuilding effect of the nonstationary longitudinal wave in rod with rectangular cross section

The article investigates obtained by the author new effect of a dual modification of non-stationary wave in a rod with rectangular cross section. The dynamics of rod is described by derived earlier improved one-dimensional dynamic equations of the longitudinal deformation. This equations were got basis on strict mathematical algorithm (so-called non-minimal simplification of the three-dimensional dynamic equations of the theory of elasticity) – “asymptotic-group” analysi. On the solutions of mentioned equations were built graphs of the longitudinal wave distribution in the rod. Analysis of them allowed to describe tridimentionality of the process. At the same time the transfer of three-dimensional state of strain to one-dimensional classic case followed by arising of the appropriate quasifronts is studied. Also must pay attention to the fact that all wave fronts velocities are the same as in the problems of theory of elasticity.

ON MOTION OF AN ELASTIC WIRE IN A RIEMANNIAN MANIFOLD AND SINGULAR PERTURBATION

R.E. Caflish and J.H. Maddocks analyzed the dynamics of a planar slender elastic rod. We consider a thin elastic rod $\gamma$ in an $N$-dimensional riemannian manifold. The former model represents an elastic rod with positive thickness, and the equation becomes a semilinear wave equation. Our model represents an infinitely thin elastic rod, and the equation becomes a 1-dimensional semilinear plate equation. We prove the short time existence of solutions. We also discuss the behaviour of the solution when the resistance goes to infinity, and find that the solution converges to a solution of a gradient flow equation.

Asymptotic analysis of Painlevé’s paradox

Painlevé’s paradox is a well-known problem in rigid-body dynamics, of which the forward dynamics equations could have no solution. To handle this situation, an assumption of tangential impact is often introduced. Although the assumption seems to provide a good fix, it still needs to be mathematically examined via analyzing the asymptotic property of a compliance-based model at the limit of rigidity. In this paper, we revisit the paradox using the typical Painlevé’s example of a rod sliding on a rough surface. For convenience, the interaction at the contact point of the rod is represented by a linear spring to scale the local normal compliance, coupled with Coulomb’s law to reflect friction. We perform an asymptotic analysis using the spring stiffness as a perturbation parameter. The rod dynamics in the Painlevé’s paradox, accompanying the variation of friction status, consists of three distinct phases as follows: An initial period of sliding which allows contact force to diverge with the increase of the spring stiffness, a period of sticking which ends at the occurrence of a reverse slip motion, and a reverse slip phase which causes the rod to be detached from the contact surface. As the stiffness goes to infinity, all the time intervals of the three phases converge to zero. This analysis theoretically confirms the assumption of the tangential impact in the paradox of sliding rod dynamics.

An energy‐momentum method for in‐plane geometrically exact Euler–Bernoulli beam dynamics

SummaryNonlinear geometrically exact rod dynamics is of great interest in many areas of engineering. In recent years, the research was focused towards Timoshenko‐type rod theories where shearing is of importance. However, in many general model of mechanisms and spatial deformations, it is desirable to have a displacement‐only formulation, which brings us back to the classical Bernoulli beam. While it is well known for linear analysis, the Bernoulli beam is not as common in geometrically exact models of dynamics, especially when we want to incorporate the rotational inertia into the model. This paper is about the development of an energy‐momentum integration scheme for the geometrically exact Bernoulli‐type rod. We will show that the task is achievable and devise a general framework to do so. Copyright © 2014 John Wiley & Sons, Ltd.

Dynamics of a rigid rod in a disordered medium with long-range spatial correlation

We investigate the diffusion of a rigid rodlike object in a two-dimensional disordered host medium, which consists of static pointlike sources of force. The points are distributed with long-range spatial correlation and interact with the rod via a repulsive potential. The time dependence of the rod's center-of-mass mean-squared displacement and its rotational mean-squared displacement are obtained for various degrees of long-range spatial correlation and rod's lengths. These transport characteristics are compared to those obtained in previous studies for the case of homogeneous distribution of force points. It is shown that existence of long-range correlation among force points makes the center of mass diffusion anomalous.