Dynamics Of Rods Research Articles

Relaxation dynamics of a flexible rod in a fluid

The relaxation dynamics of a bent rod immersed in a fluid is strongly affected by its bending stiffness and the fluid viscosity. For large bending stiffness or low viscosity, the dynamics is underdamped and the rod oscillates around its equilibrium position with a well-defined frequency and a damped amplitude. For low bending stiffness or large viscosity, the dynamics is overdamped and the rod relaxes to its equilibrium position without oscillating. We show the existence of two overdamped regimes where the relaxation dynamics is characterized by two different polynomial decays showing that the viscous force is not proportional to the rod velocity in our experiments.

A simple remark about the Love hypothesis in rod dynamics

This note discusses the validity of the Love hypothesis in the dynamics of rods, which consists of assuming a linear relationship among the radial and longitudinal strains. In the case of incompressible materials, we show that such an assumption can be correctly employed only within the linear theory of infinitesimal strains. The use of this hypothesis in the nonlinear setting would indeed lead to ‘spurious’ mechanical theories.

Multi-particle collision dynamics with a non-ideal equation of state. II. Collective dynamics of elongated squirmer rods.

Simulations of flow fields around microscopic objects typically require methods that both solve the Navier-Stokes equations and also include thermal fluctuations. One such method popular in the field of soft-matter physics is the particle-based simulation method of multi-particle collision dynamics (MPCD). However, in contrast to the typically incompressible real fluid, the fluid of the traditional MPCD methods obeys the ideal-gas equation of state. This can be problematic because most fluid properties strongly depend on the fluid density. In a recent article, we proposed an extended MPCD algorithm and derived its non-ideal equation of state and an expression for the viscosity. In the present work, we demonstrate its accuracy and efficiency for the simulations of the flow fields of single squirmers and of the collective dynamics of squirmer rods. We use two exemplary squirmer-rod systems for which we compare the outcome of the extended MPCD method to the well-established MPCD version with an Andersen thermostat. First, we explicitly demonstrate the reduced compressibility of the MPCD fluid in a cluster of squirmer rods. Second, for shorter rods, we show the interesting result that in simulations with the extended MPCD method, dynamic swarms are more pronounced and have a higher polar order. Finally, we present a thorough study of the state diagram of squirmer rods moving in the center plane of a Hele-Shaw geometry. From a small to large aspect ratio and density, we observe a disordered state, dynamic swarms, a single swarm, and a jammed cluster, which we characterize accordingly.

Flow coupling between active and passive fluids across water\u2013oil interfaces

Active fluid droplets surrounded by oil can spontaneously develop circulatory flows. However, the dynamics of the surrounding oil and their influence on the active fluid remain poorly understood. To investigate interactions between the active fluid and the passive oil across their interface, kinesin-driven microtubule-based active fluid droplets were immersed in oil and compressed into a cylinder-like shape. The droplet geometry supported intradroplet circulatory flows, but the circulation was suppressed when the thickness of the oil layer surrounding the droplet decreased. Experiments with tracers and network structure analyses and continuum models based on the dynamics of self-elongating rods demonstrated that the flow transition resulted from flow coupling across the interface between active fluid and oil, with a millimeter–scale coupling length. In addition, two novel millifluidic devices were developed that could trigger and suppress intradroplet circulatory flows in real time: one by changing the thickness of the surrounding oil layer and the other by locally deforming the droplet. This work highlights the role of interfacial dynamics in the active fluid droplet system and shows that circulatory flows within droplets can be affected by millimeter–scale flow coupling across the interface between the active fluid and the oil.

On the statics and dynamics of granular-microstructured rods with higher order effects

Granular-microstructured rods show strong dependence of grain-scale interactions in their mechanical behavior, and therefore, their proper description requires theories beyond the classical theory of continuum mechanics. Recently, the authors have derived a micromorphic continuum theory of degree n based upon the granular micromechanics approach (GMA). Here, the GMA is further specialized for a one-dimensional material with granular microstructure that can be described as a micromorphic medium of degree 1. To this end, the constitutive relationships, governing equations of motion and variationally consistent boundary conditions are derived. Furthermore, the static and dynamic length scales are linked to the second-gradient stiffness and micro-scale mass density distribution, respectively. The behavior of a one-dimensional granular structure for different boundary conditions is studied in both static and dynamic problems. The effects of material constants and the size effects on the response of the material are also investigated through parametric studies. In the static problem, the size-dependency of the system is observed in the width of the emergent boundary layers for certain imposed boundary conditions. In the dynamic problem, microstructural effects are always present and are manifested as deviations in the natural frequencies of the system from their classical counterparts.

Transport of neutral and charged nanorods across varying-section channels

We study the dynamics of neutral and charged rods embedded in varying-section channels. By means of systematic approximations, we derive the dependence of the local diffusion coefficient on both the geometry and charge of the rods. This microscopic insight allows us to provide predictions for the permeability of varying-section channels to rods with diverse lengths, aspect ratios and charge. Our analysis shows that the dynamics of charged rods is sensitive to the geometry of the channel and that their transport can be controlled by tuning both the shape of the confining walls and the charge of the rod. Interestingly, we find that the channel permeability does not depend monotonically on the charge of the rod. This opens the possibility of a novel mechanism to separate charged rods.

Rotational dynamics of bottom-heavy rods in turbulence from experiments and numerical simulations

We successfully perform the three-dimensional tracking in a turbulent fluid flow of small axisymmetrical particles that are neutrally-buoyant and bottom-heavy, i.e., they have a non-homogeneous mass distribution along their symmetry axis. We experimentally show how a tiny mass inhomogeneity can affect the particle orientation along the preferred vertical direction and modify its tumbling rate. The experiment is complemented by a series of simulations based on realistic Navier–Stokes turbulence and on a point-like particle model that is capable to explore the full range of parameter space characterized by the gravitational torque stability number and by the particle aspect ratio. We propose a theoretical perturbative prediction valid in the high bottom-heaviness regime that agrees well with the observed preferential orientation and tumbling rate of the particles. We also show that the heavy-tail shape of the probability distribution function of the tumbling rate is weakly affected by the bottom-heaviness of the particles.

GENERALIZATION OF KIRCHHOFF KINETIC ANALOGY TO THIN ELASTIC SHELLS

The Kirchhoff kinetic analogy is generalized from thin elastic rods to thin elastic shells. The generalization makes thin shell deformations physically correspond and mathematically equivalent to rigid body motions. Hence theories and methods of rigid body dynamics can be applied to investigate deformations of thin elastic shell, and also provide a novel discretization for continuous thin elastic shells. An orthogonal spatial axis system is established along the coordinate lines of the middle surface under the straight normal assumption. The moving of the axis system along the coordinate lines in unit velocity forms its angular velocity, which is the curvature-twist vector with two independent variables. The curvature-twist vector along two coordinate lines expresses the deformation and the configuration of a thin elastic shell. It is demonstrated that curvature-twist vectors are compatible, and curvature-twist vectors and tangential vectors of middle surface are compatible. Nonholonomic constraints and differential equations of middle surfaces are established in the Euler angles and the Lam$\acute{e}$ coefficient form. The strain, the stress and the internal forces are formulated in the curvature-twist vectors and the Lam$\acute{e}$ coefficients. The equilibrium partial differential equations are presented with distributed internal forces intensity of thin elastic shells. The forms of the equations are similar to the Euler equations of rigid body dynamics and Kirchhoff equations of thin elastic rods. The fact means that the Kirchhoff kinetic analogy of thin elastic rods is generalized to thin elastic shells. The analogy relations between thin elastic shells and dynamics of rigid body or thin elastic rods are concluded. Finally, an example is given to show the application of this method. The proposed analogy leads to novel views and approaches to model and to analyze deformation of thin elastic shells. It is possible to generalize further the analogy for dynamics of thin elastic shells.

Dynamics of Nonlocal Rod by Means of Fractional Laplacian

The use of fractional models to analyse nonlocal behaviour of solids has acquired great importance in recent years. The aim of this paper is to propose a model that uses the fractional Laplacian in order to obtain the equation ruling the dynamics of nonlocal rods. The solution is found by means of numerical techniques with a discretisation in the space domain. At first, the proposed model is compared to a model that uses Eringen’s classical approach to derive the differential equation ruling the problem, showing how the parameters used in the proposed fractional model can be estimated. Moreover, the physical meaning of the model parameters is assessed. The model is then extended in dynamics by means of a discretisation in the time domain using Newmark’s method, and the responses to different dynamic conditions, such as an external load varying with time and free vibrations due to an initial deformation, are estimated, showing the difference of behaviour between the local response and the nonlocal response. The obtained results show that the proposed model can be used efficiently to estimate the response of the nonlocal rod both to static and dynamic loads.

Method of generalized functions in boundary value problems of thermoelastic rod dynamics

The method of generalized functions (GFM) has been developed to solve transientand vibrational boundary value problems of thermoelastic rod dynamics using a model of coupled thermoelasticity. Thermoelastic shock waves arising in such structures under the influence ofshock loads and heat flows are considered. Conditions on their fronts were obtained. Thesingularity of the assigned boundary tasks taking into account shock waves has been proved.On the basis of GFM, a system of algebraic resolving equations is built for a wide class ofboundary problems to determine their analytical solutions. Dynamics of the rod under the actionof forces and heat sources of various types, including those described by singular generalizedfunctions, which allow modeling the effect of pulsed concentrated sources, are studied. Computerimplementation of solutions of one edge problem at stationary oscillations was carried out, resultsof numerical experiments of calculation of rod thermodynamics at low and high frequenciesare presented. These solutions and algorithms can be used for engineering calculations of rodstructures to evaluate their strength properties.

Bifurcations in the dynamics of a dipolar spheroid in a shear flow subjected to an external field

The dynamical phase behavior of a spheroid with a magnetic dipole rotating in a shear flow and external field is investigated theoretically. Depending on the ratio of the torques due to the shear flow, a sequence of bifurcations leads to a rich complexity in the phase portraits in orientation space. We also find that the dynamics of an ideal thin rod could be very different from that of a high aspect ratio spheroid.

Speeding up Dynamics by Tuning the Noncommensurate Size of Rodlike Particles in a Smectic Phase.

Using simulations, we study the diffusion of rodlike guest particles in a smectic environment of rodlike host particles. We find that the dynamics of guest rods across smectic layers changes from a fast nematiclike diffusion to a slow hopping-type dynamics via an intermediate switching regime by varying the length of the guest rods with respect to the smectic layer spacing. We determine the optimal rod length that yields the fastest and the slowest diffusion in a lamellar environment. We show that this behavior can be rationalized by a complex 1D effective periodic potential exhibiting two energy barriers, resulting in a varying preferred mean position of the guest particle in the smectic layer. The interplay of these two barriers controls the dynamics of the guest particles yielding a slow, an intermediate, and a fast diffusion regime depending on the particle length.

Обобщенные решения краевых задач динамики термоупругих стержней

The non-stationary boundary-value problems of the dynamics of thermoelastic rods are considered. Based on the method of generalized functions, generalized solutions of boundary value problems are constructed using a model of unbound thermoelasticity under the influence of non-stationary power and thermal loads of various types. The integral representations of the solution of the boundary-value problem are given, which make it possible to determine the thermally stressed state of the rod under the influence of various power and thermal influences from the known boundary displacements, stresses, temperature, and heat fluxes at the ends of the rod.

Dynamics of mandrel holding mechanism of pipe-rolling line automatic mill

The different walled parameter of shells is one of the essential indices, determining quality of hot-rolled seamless pipes, produced at pipe-rolling line (PRL). Thickness variation of shell walls makes a considerable share of the longitudinal and cross-section different walled parameter of finished pipes. The phenomena is stipulated mainly by the rod dynamics with the mandrel in the mill deformation hearth. Based on a study of dynamic model of the mandrel holding mechanism rod, the character of dynamic processes progress at the exit part of PRL automatic mill determined, accounting the shell rolling parameters. A system of differential equations composed for the mandrel holding mechanism rod dynamics, accounting technological parameters and inertia parameters of the rolled shell. The solution of the differential equations system with variable coefficients until the forth approximation realized digitally by application of Runge-Kutt method for correspondent forms of the mandrel rod oscillation. The solution obtained taking the example of calculation of the PRL 350 automatic mill. The picture of dynamic processes of the mandrel holding mechanism rod system at realization of the whole technological process of seamless pipes production at the PRL 350 automatic mill specified. At that, the variable impact of the deformation hearth and change of the rolled shell mass were taken into account. A wide range of mathematic experiment accomplished, that enabled to select necessary technological and dynamic system parameters, determine the stable pipe rolling regimes. Analytical solution of the task obtained, that considerably differs from the earlier known mathematic models. The obtained solution reflects much better the dynamic processes in the mandrel holding mechanism rod system of the PRL 350 automatic mill and specifies the peculiarities of pipe longitudinal different-walled phenomena formation mechanism.

Dynamic Effect Research of Cable-Lead-In Rod on Towed System

This paper aims to research the cable-lead-in rod effect on a towed system through mathematical modeling and numerical simulations. The rod dynamics, as a key part of this study, is modeled using the combination of cable node governing equations and kinematic constraint conditions. As the first attempt to analyze such a problem, the rod is simply treated as an elastic cable segment so as to be incorporated into the dynamics of the cable, and a set of algorithm is then proposed based on the kinematic constraint conditions to fully describe its motions. Meanwhile, the cable and the underwater vehicle are modeled by the traditional lumped mass method and the 6 degree-of-freedom maneuverability equations for submarines respectively; the coupling boundary conditions besides the rod dynamics are also given to form the whole system’s model. Several numerical cases are performed to investigate the rod effect on the system in different maneuver situations. Some meaningful conclusions are drawn through comparative analysis.

A longitudinal magnetoelastic wave in a rod with account of the damage of its material

For an electrically conducting rod that performs longitudinal oscillations, a self-consistent system is formulated that includes the equation of rod dynamics, the equation for the variation of the strength of the external magnetic field, and the kinetic equation for the accumulation of damages in the material of the rod. The linearized system and the system of equations, including geometrical and physical elastic nonlinearities, are consecutively considered. It is shown that the waves described by linearized system have dispersion and frequency-dependent damping due to the presence of two types of dissipation, one of which is caused by the damage of the material and the other by a magnetic field. In nonlinear case, an evolution equation is derived that generalizes the Burgers equation, known in nonlinear wave dynamics. Its solutions are found and analyzed.

Slender body theory for particles with non-circular cross-sections with application to particle dynamics in shear flows

This paper presents a theory to obtain the force per unit length acting on a slender filament with a non-circular cross-section moving in a fluid at low Reynolds number. Using a regular perturbation of the inner solution, we show that the force per unit length has$O(1/\ln (2A))+O(\unicode[STIX]{x1D6FC}/\ln ^{2}(2A))$contributions driven by the relative motion of the particle and the local fluid velocity and an$O(\unicode[STIX]{x1D6FC}/(\ln (2A)A))$contribution driven by the gradient in the imposed fluid velocity. Here, the aspect ratio ($A=l/a_{0}$) is defined as the ratio of the particle size ($l$) to the cross-sectional dimension ($a_{0}$) and$\unicode[STIX]{x1D6FC}$is the amplitude of the non-circular perturbation. Using thought experiments, we show that two-lobed and three-lobed cross-sections affect the response to relative motion and velocity gradients, respectively. A two-dimensional Stokes flow calculation is used to extend the perturbation analysis to cross-sections that deviate significantly from a circle (i.e.$\unicode[STIX]{x1D6FC}\sim O(1)$). We demonstrate the ability of our method to accurately compute the resistance to translation and rotation of a slender triaxial ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a simple shear flow that translate and rotate quasi-periodically if they have two-lobed cross-section, and rotate chaotically and translate diffusively if they have a combination of two- and three-lobed cross-sections. Finally, we show the remarkable ability of our theory to accurately predict the motion of rings, retaining great accuracy for moderate aspect ratios (${\sim}10$) and cross-sections that deviate significantly from a circle, thereby making our theory a computationally inexpensive alternative to other Stokes flow solvers.

Discrete formulation for the dynamics of rods deforming in space

The movement of rods in an Euclidean space can be described as a field theory on a principal bundle. The dynamics of a rod is governed by partial differential equations that may have a variational origin. If the corresponding smooth Lagrangian density is invariant by some group of transformations, there exist the corresponding conserved Noether currents. Generally, numerical schemes dealing with PDEs fail to reflect these conservation properties. We describe the main ingredients needed to create, from the smooth Lagrangian density, a variational principle for discrete motions of a discrete rod, with the corresponding conserved Noether currents. We describe all geometrical objects in terms of elements on the linear Atiyah bundle using a reduced forward difference operator. We show how this introduces a discrete Lagrangian density that models the discrete dynamics of a discrete rod. The presented tools are general enough to represent a discretization of any variational theory in principal bundles, and its simplicity allows us to perform an iterative integration algorithm to compute the discrete rod evolution in time, starting from any predefined configurations of all discrete rod elements at initial times.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

Configurational forces and nonlinear structural dynamics

Configurational, or Eshelby-like, forces are shown to strongly influence the nonlinear dynamics of an elastic rod constrained with a frictionless sliding sleeve at one end and with an attached mass at the other end. The configurational force, generated at the sliding sleeve constraint and proportional to the square of the bending moment realized there, has been so far investigated only under quasi-static setting and is now confirmed (through a variational argument) to be present within a dynamic framework. The deep influence of configurational forces on the dynamics is shown both theoretically (through the development of a dynamic nonlinear model in which the rod is treated as a nonlinear spring, obeying the Euler elastica, with negligible inertia) and experimentally (through a specifically designed experimental set-up). During the nonlinear dynamics, the elastic rod may slip alternatively in and out from the sliding sleeve, becoming a sort of nonlinear oscillator displaying a motion eventually ending with the rod completely injected into or completely ejected from the sleeve. The present results may find applications in the dynamics of compliant and extensible devices, for instance, to guide the movement of a retractable and flexible robot arm.