Instability of vibrations of a moving two-mass oscillator on a flexibly supported Timoshenko beam

Summary  Transversal vibrations of a uniformly moving two-mass oscillator on a Timoshenko beam of infinite length supported by a viscoelastic foundation are studied. By using integral transforms, the characteristic equation for the oscillator's vibrations is obtained. It is shown that the equation may have a root with a positive real part. The existence of such a root leads to the exponential increase of the amplitude of the oscillator vibrations, i.e. to instability. The reasons for the instability to occur are discussed. By employing the method of D-decomposition, the instability domains are found in the space of the system parameters. Received 30 October 2000; accepted for publication 28 March 2001     

A study of the aerodynamics of a generic container freight wagon using Large-Eddy Simulation

In this work simulations using the Large Eddy Simulation technique have been made of the flow around a generic container freight wagon model. The model consists of one 11.8 m standard length container placed on a wagon. Details of the undercarriage such as wheels are included, but the container is generic and smoothed in comparison to a real freight wagon. The Reynolds number of the flow is 10⁵ based on the container width of 2.354 m. Two cases have been considered in the study, one case where the wagon is standing alone and one case where it is submerged into a train set with wagons ahead and behind the wagon. The latter case is simulated using periodic boundary condition. Both the time-averaged and the instantaneous flow around the wagon for the two cases are described. For the single wagon case, it is found that the separation bubble formed on the roof of the container oscillates back and forth in the streamwise direction and that this oscillation is in phase with oscillations found in the upper shear layer of the ring vortex in the wake. The mechanism that is causing the synchronization of the oscillations of the separation bubble at the front and the upper shear layers in the wake is found to be waves of vorticity being shed from the separation bubble. The time-averaged ring vortex in the near wake of the single wagon is found to be inclined due to the disturbance of the undercarriage details on flow in the lower shear layer. The lower center of the ring vortex is located closer to the base face than the upper center. The drag coefficient of the wagon in the periodic case was found to be only 10% of that of the single wagon case. This is due to two symmetrical counter-rotating vortices found in the gaps which make the train set appear as a single body to the oncoming flow and shielding the wagon from any direct impingement of the flow. The counter-rotating vortices in the gap are found to inhibit periodic oscillations in the lateral direction. These oscillations cause vortical structures to form by the air that is pushed out from the gap and these flow structures cause a dominating oscillation of non-dimensional frequency St=0.12 in the side force signal.     

Vibration of a continuous beam with multiple elastic supports excited by a moving two-axle system with separation

This paper studies the dynamics of a two-axle system travelling along a continuous elastic beam resting on elastic supports modelled as linear springs. During its travel along the vibrating beam, the vibrating two-axle system may separate from the beam for certain time intervals when there is no interaction between the two-axle system and the beam. As a result, the dynamics of the whole system can be very different from the case when separation is ignored. The study is focused on conditions that facilitate the onset of separation between the moving and supporting structures and subsequent possible reattachment. Numerical results show that the occurrence of separation may be influenced to a large extent by the presence of intermediate elastic supports.     

Vibration of beams with general boundary conditions due to a moving random load

Summary  The transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed. The boundary conditions considered are: pinned–pinned, fixed–fixed, pinned–fixed, and fixed–free. Based on the Bernoulli beam theory, the problem is described by means of a partial differential equation. Closed-form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion: accelerated, decelerated and uniform. The effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail. Received 3 December 2001; accepted for publication 30 April 2002     

Nonlinear isochronous oscillations of a fluid in a paraboloid: theory and experiment

Within the framework of the shallow-water model, the nonlinear axisymmetric oscillations of a fluid in a paraboloid of revolution and in an unbounded parabolic channel are investigated. It is established that in the paraboloid of revolution the oscillation period does not depend on the amplitude, that is, the oscillations are isochronous. Experimental investigations of free fluid oscillations in a paraboloid confirm this theoretical result.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 131–142. Original Russian Text Copyright © 2004 by Kalashnik, Kakhiani, Lominadze, Patarashvili, Svirkunov, and Tsakadze.     

Nonlinear behaviour of different flexible size-dependent beams models based on the modified couple stress theory. Part 1: Governing equations and static analysis of flexible beams

In the first part of the paper we employ the Sheremetev-Pelekh-Reddy-Levinson hypotheses, which yield a non-linear mathematical model of a beam taking into account geometric and physical non-linearity as well as transverse shear based on the modified couple stress theory. The general model includes both Bernoulli-Euler and Timoshenko models with/without geometric/physical non-linearity, and the size-dependent beam behaviour.In addition, we present results of the development of the relaxation method for solution to numerous static problems. The influence of the size-dependent coefficient on the load-deflection and stress-strain states of the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson mathematical models has been also studied.     

Tensionless contact of a finite beam: Concentrated load inside and outside the contact zone

For a finite beam with a nonzero gap distance, an asymmetric concentrated load can be either inside or outside of the contact zone. A new governing equation is given for the case of a concentrated load outside the contact zone. By numerically solving the left-side and right-side contact lengths of the beam, a criterion is established to determine whether the concentrated load is inside or outside the contact zone. A more general approach on the tensionless contact of a beam is thus presented.     

Mechanical and acoustical study of a structural bond: comparison theory/numerical simulations/experiment

To carry out numerical simulations able to describe the behavior of a structural bond during a mechanical test, it is common to use a theory of damage. This approach consists in modeling the bonded zone by a surface distribution of springs with or without mass. The elasto-plastic with damage model and the numerical simulations are carried out with a finite element code. The use of this model requires two types of data: the critical energies in modes I and II measured during mechanical tests and the stiffnesses of the springs. These ones cannot be identified by mechanical measurements and, in this paper we propose an ultrasonic method to measure them. The ultrasonic approach and its experimental validation are first presented. Then, the mechanical model is detailed. The whole identification strategy is applied on aluminum/epoxy/aluminum samples.     

The self-force and effective mass of a generally accelerating dislocation I: Screw dislocation

The self-force and effective mass of a moving dislocation in a generally accelerating motion are explicitly obtained on the basis of a surface-independent dynamic J-integral. Logarithmic singularities due to non-zero acceleration result in divergent integrals in the dynamic J-integral, which are treated by smearing out the dislocation core (ramp-core) and by regularizing in the sense of distributions, both coinciding in the leading terms.     

Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1：2 resonance case

A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed. The project supported by the National Natural Science Foundation of China (10572055, 50475109) and the Natural Science Foundation of Gansu Province Government of China (3ZS051-A25-030(key item)) The English text was polished by Keren Wang.     

Coupling between countercurrent gas and solid flows in a moving granular bed: The role of shear bands at the walls

We extended the standard approach to countercurrent gas–solid flow in vertical vessels by explicitly coupling the gas flow and the rheology of the moving bed of granular solids, modelled as a continuum, pseudo-fluid. The method aims at quantitatively accounting for the presence of shear in the granular material that induces changes in local porosity, affecting the gas flow pattern through the solids. Results are presented for the vertical channel configuration, discussing the gas maldistribution both through global and specific indexes, highlighting the effect of the relevant parameters such as solids and gas flowrate, channel width, and wall friction. Non-uniform gas flow distribution resulting from uneven bed porosity is also discussed in terms of gas residence time distribution (RTD). The theoretical RTD in a vessel of constant porosity and Literature data obtained in actual moving beds are qualitatively compared to our results, supporting the relevance under given circumstances of the coupling between gas and solids flow.     

Shocks and slip systems: Predictions from a mesoscale theory of continuum dislocation dynamics

Exploring a recently developed mesoscale continuum theory of dislocation dynamics, we derive three predictions about plasticity and grain boundary formation in crystals. (1) There is a residual stress jump across grain boundaries and plasticity-induced cell walls as they form, which self-consistently acts to attract neighboring dislocations; residual stress in this theory appears as a remnant of the driving force behind wall formation under both polygonization and plastic deformation. We derive the predicted asymptotic late-time dynamics of the grain-boundary formation process. (2) During grain boundary formation at high temperatures, there is a predicted cusp in the elastic energy density. (3) In early stages of plasticity, when only one type of dislocation is active (single-slip), cell walls do not form in the theory; instead we predict the formation of a hitherto unrecognized jump singularity in the dislocation density.     

Behavior of a net of fibers linked by viscous interactions: theory and mechanical properties

This paper presents an investigation of the macroscopic mechanical behavior of highly concentrated fiber suspensions for which the mechanical behavior is governed by local fiber-fiber interactions.The problem is approached by considering the case of a net of rigid fibers of uniform length, linked by viscous point interactions of power-law type. Those interactions may result in local forces and moments located at the contacting point between two fibers, and respectively power-law functions of the local linear and angular velocity at this point.Assuming the existence of an elementary representative volume which size is small compared to the size of the whole structure, the fiber net is regarded as a periodic assembly of identical cells. Macroscopic equilibrium and constitutive equations of the equivalent continuum are then obtained by the discrete and periodic media homogenization method, based on the use of asymptotic expansions.Depending on the order of magnitude of local translational viscosities and rotational viscosities, three types of the equivalent continua are proved to be possible. One of them leads to an effective Cosserat medium, the other ones being usual Cauchy media. Lastly, formulations that enable an effective computation of constitutive equations are detailed. They show that the equivalent continuum behaves like an anisotropic power-law fluid.     

Simulation of self-coordination in a row of beating flexible flaplets for micro-swimmer applications: Model and experiment study

In this study, we present a model that simulates hydrodynamic self-coordination in a row of flexible flaplets. We control the flaplets in order that their tips follow a fixed-amplitude oscillatory motion profile. When brought together at a low Reynolds-number environment, the flaplets interact with each other in the form of bending deflections at their tips, which causes the frequency of the individual oscillations to vary until a coordinated steady state is reached. The model design steps are experimentally verified and the coordination results of both the experiment and the model are compared. The model’s internal states are then analysed for a better understanding of the synchronization collective effect. The coordination of the flaplets is found to settle in the direction of propulsion forces ascent. The stability of the resulted synchronization and propulsion forces are examined over long periods. The model is meant to be simplified and mostly linear so that it can be utilized for state forecasting in a real-time control application of a swimmer robot. Finally, we experimentally study the propulsion performance of five beating flaplets that follow prescribed oscillation profiles forming a metachronal wave. The flow results show that the flaplets, that beat in coordination, are efficient at generating a uni-directional steady-streaming transport of the fluid at their surface.     

Slip-line modeling of machining with a rounded-edge tool—Part I: new model and theory

The effect of tool edge roundness attracts growing attention from the international machining research community due to ever accelerating applications of precision, super-precision, micro-, and nano-machining technologies in a wide variety of modern industries. A new slip-line model for machining with a rounded-edge tool and its associated hodograph are proposed in this paper. The model consists of 27 slip-line sub-regions, each sub-region having its own physical meaning. It is demonstrated that the model simultaneously takes into account nine effects, such as the shear-zone effect and the size effect, which commonly occur in machining. Eight groups of machining parameters, such as the ploughing (parasitic or non-cutting) force and the chip up-curl radius, can be simultaneously predicted from the model. Furthermore, the model incorporates eight slip-line models previously developed for machining during the last six decades as special cases. An additional special case that involves a parallel-sided shear zone can also be derived from the new model. A mathematical formulation of the model is established based on Dewhurst and Collins's (1973) matrix technique for numerically solving slip-line problems. A purely analytical equation is proposed to predict the thickness of the primary shear zone. This equation is also employed to predict the shear strain-rate in the primary shear zone.     

Évolution quasi-statique d'un fil souple inextensible sur un plan avec frottement sec : cas discretQuasi-static motion of a supple,homogenous and inextensible string on a horizontal plane with dry friction: discret case

The differential inclusion describing the quasi-static motion of a supple, homogeneous and inextensible string on a horizontal plane with dry friction (Coulomb's law) is a one dimensional evolution model of a continuous medium, with non-linear geometry, obeying a “plastic-rigid” law. With a view to numerical simulation, we treat the discrete case: the string is assimilated to a chain constituted by rigid rods perfectly articulated around ball-joints. We give variational formulation of the problem and prove existence and uniqueness of solutions. We construct an algorithm that describes the instantaneous solutions when the initial configuration of the string is given. Then, some examples are treated. To cite this article: H. Sayah, C. R. Mecanique 332 (2004).