Non-linear analysis of stick/slip motion

The steady state relative motion of two masses with dry (Coulomb) friction contact is investigated. The bodies are assumed to have the same mass and stiffness and are subjected to harmonic excitation. By means of a combined analytical—numerical procedure, results are obtained for arbitrary values of Coulomb friction, excitation frequency, and natural frequencies of the bodies. For certain values of these parameters, multiple lockups per cycle are possible. In this respect, the problem investigated here is a natural extension of the one considered by Den Hartog, who in obtaining his closed form solution assumed a maximum of two lockups per cycle.     

FRICTIONAL BEHAVIOUR OF A BELT-DRIVEN AND PERIODICALLY EXCITED OSCILLATOR

A single-degree-of-freedom system with the parallel presence of a linear spring, a viscous damper and a contact dry friction device is studied here. The mass may slide or stick on the belt when the driver moves periodically or at a constant speed. We derive closed-form solutions according to a more complete two-phase formulation, and some interesting behaviours of the considered system are displayed. For the non-damping oscillator belt with fixed, we offer closed-form formulae for estimating the maximum displacement and the minimum driving speed amplitude needed to prevent sticking. Two friction laws are considered. For the Coulomb friction system, the positive damping term suffices to avoid the climb motion of the mass slider. We also investigate the friction behaviour of the mass slider under the influence of the friction force bound on mass speed, whose curve has negative slope when the mass speed is less than a certain value v_min. For the speed-dependent friction system we identify a critical speed denoted by v^*. According to the qualitative analysis in the phase plane we give simple criteria of the parameter values for stable equilibrium point as well as for stable limit cycle. When v varies from vv^* to v<v^*, subcritical Hopf bifurcation occurs. For the latter case the mass slider undergoes a slide-stick motion, but by increasing the driving speed the slide-stick motion can be avoided.     

A generic mechanism of sliding friction between charged soft surfaces

A universal sliding friction mechanism between soft, electrically charged surfaces (e.g., those carrying polyelectrolytes) is proposed. The Coulomb field of the randomly distributed charges induces a damped motion of the charges of the opposite surface, which gives rise to dissipation, and thus to friction. It is predicted that this friction force, which diverges at zero separation, may overcome viscous friction at distances significantly exceeding the Bjerrum length. This is not only of fundamental interest, but also of potential importance in life science and many practical situations.     

Pole assignment of friction-induced vibration for stabilisation through state-feedback control

Friction-induced self-excited linear vibration is often governed by a second-order matrix differential equation of motion with an asymmetric stiffness matrix. The asymmetric terms are product of friction coefficient and the normal stiffness at the contact interface. When the friction coefficient becomes high enough, the resultant vibration becomes unstable as frequencies of two conjugate pairs of complex eigenvalues (poles) coalesce (when viscous damping is low).This short paper presents a receptance-based inverse method for assigning complex poles to second-order asymmetric systems through (active) state-feedback control of a combination of active stiffness, active damping and active mass, which is capable of assigning negative real parts to stabilise an unstable system.     

Collective modes in some Fokker-Planck Coulomb systems: One-dimensional case

The collective modes in two Fokker-Planck Coulomb systems are studied in one dimension and in the long wavelength limit. The Fokker-Planck “friction coefficient”, which can be related to the viscosity of the medium through the Stokes' law, is treated as a parameter. In a single component system it is shown under what circumstances the propagative plasma mode can become a diffusive mode. Depending on an inequality involving the friction coefficient and the plasmon frequency, the collective mode can illustrate certain properties of the so-called central peak. A two-component Vlasov-Fokker-Planck system exhibits three modes analogous to the Rayleigh scattering in a viscous fluid.     

Two particle model for the diffusion of interacting particles in periodic potentials

The random motion of two interacting particles in a periodic potential with a finite number of sites is investigated as a model that may be applied to superionic conductivity. Starting from the Fokker-Planck-equation for the model and using an appropriate series expansion for the probability density, solutions for the frequency dependent conductivity are given. Explicit numerical results are shown for hard-core and Coulomb interaction in the range of intermediate and high friction constants.     

Separation and determination of combined dampings from free vibrations

The effects of combined viscous with Coulomb damping, Reid damping and bilinear hysteretic damping on the free motion of an oscillator are studied directly by integration of the differential equations of motion and by the use of first integrals for the first time. The amplitude decay during the free motion is considered in order to separate and determine the various types of damping.     

TIME-DEPENDENT SOLUTION OF FOKKER-PLANCK EQUATION FOR N DEGREES OF FREEDOM AND DYNAMICAL THRESHOLD FOR COMPOUND NUCLEUS FORMATION IN SYMMETRIC HEAVY ION REACTION

We generalized the time-dependent solution of the Fokker-Planck equation to n degrees of freedom.Using the solution we calculated the extra push energy ΔE from the view of Brownian motion as a diffusion process over a saddle point, which is caused by overcoming both repulsive Coulomb and dissipative forces to form a compound nucleus.It depends on the initial condition and the friction, other degrees of freedom acting as a heat bath of temperature T and causing thermal fluctuations.     

Classical dynamical theory of heavy ion fusion and scattering

A classical dynamical model which includes dissipative forces is suggested for heavy ion reactions high above the Coulomb barrier. Internal degrees of freedom corresponding to rotation of the ions are included. The reactions divide into three parts: (1) quasi-elastic scattering, with relatively small energy loss, associated with higher angular momenta, (2) deep inelastic scattering, with larger energy loss and considerable transfer of mass, associated with intermediate angular momenta, and (3) complete fusion where a highly excited compound state is formed, generally associated with the lowest angular momenta. One can predict a fusion cross section for two values of the friction coefficient, “weak” and “strong” friction cases. Reasonable values for fusion can be obtained in the weak friction case, but scattering angular distributions are not consistent with available experimental data. In the strong friction case it is more difficult to fit all the fusion cross sections with a single friction parameter. But the predicted angular distributions and energy losses are in better agreement with experiment than for the weak friction case.     

Secular Motion Frequencies of ~9Be~+ Ions and ~(40)Ca~+ Ions in Bi-component Coulomb Crystals

We obtain bi-component Coulomb crystals using laser-cooled ~(40)Ca~+ ions to sympathetically cool ~9Be~+ ions in a linear Paul trap. The shell structures of the bi-component Coulomb crystals are investigated. The secular motion frequencies of the two different ions are determined and compared with those in the single-component Coulomb crystals. In the radial direction, the resonant motion frequencies of the two ionic species shift toward each other due to the strong motion coupling in the ion trap. In the axial direction, the motion frequency of the laser-cooled~(40)Ca~+is impervious to the sympathetically cooled ~9Be~+ ions because the spatially separation of the two different ionic species leads to the weak motion coupling in the axial direction.     

Computer simulation of amplitude-dependent internal friction

The internal friction in crystalline materials has been studied in the model of viscous motion of dislocations through a system of randomly arranged defects. The results of the calculation of the amplitude dependence of the internal friction for different frequencies and defects of different powers are presented. Three characteristic portions have been revealed in the amplitude dependences of the internal friction, which correspond to different modes of dislocation motion.     

Frictional vibration transmission from a laterally moving surface to a traveling beam

As the density of information stored in automated magnetic tape libraries continues to increase, greater requirements are placed on the precision of mechanical positioning in order to successfully read and write data bits. The location of the read/write head in the direction across the tape's width (termed the lateral direction) is actively controlled in order to maintain alignment between the head and data tracks, even in the presence of the tape's lateral vibration. However, during repositioning, vibration is undesirably transmitted from the laterally moving head structure to the axially moving tape because of frictional contact between the two adjacent surfaces. As an analog of that interaction, a model is developed here to describe frictional vibration transmission from a surface having prescribed lateral motion to a tensioned beam that travels and slides over it. For a transport speed that is high when compared to the lateral vibration velocity, Coulomb friction between the surface and the beam can be well-approximated by an equivalent form of viscous damping. The beam is divided into contiguous regions corresponding to free spans and the beam's portion that contacts the surface. A critical engagement length between the beam and the surface exists for which vibration transmission at a particular natural frequency can be substantially reduced, and for a given mode, that length depends weakly on the surface's position along the beam's span. By contouring the surface to have portions of differing radii of curvature, the extent of vibration transmission can be reduced over a broad range of frequency.     

Control of hysteretic instability in rotating machinery by elastic suspension systems subject to dry and viscous friction

Most of the undesired whirling motions of rotating machines can be efficiently reduced by supporting journal boxes elastically and controlling their movement by viscous dampers or by dry friction surfaces normal to the shaft axis, which rub against the frame. In the case of dry dampers, resonance ranges of the floating support configuration can be easily cut off by planning a motionless adhesive state of the friction surfaces. On the contrary, the dry friction contact must change automatically into sliding conditions when the fixed support resonances are to be feared. Moreover, the whirl amplitude can be restrained throughout the speed range by a proper choice of the suspension-to-shaft stiffness ratio and of the support-to-rotor mass ratio.This theoretical research deals firstly with the natural precession speeds and looks for Campbell plots in dependence on the shaft angular speed, for several rotor-suspension systems. Then, the steady response to unbalance is investigated, in terms of rotor and support orbits and of conical path of the rotor axis. In this search, the ranges of adhesive or sliding contact are identified in particular for system with dry friction damping. At last, the destabilizing influence of the shaft hysteresis in the supercritical regime is focalized and the counterbalancing effect of the other dissipative sources is verified. In the nonlinear case of dry friction dampers, the control of linear stability is fulfilled by a perturbation procedure, checking the magnitude of Floquet characteristic multipliers on the complex plane. Moreover, the nonlinear stability far from steady motion is tested by the direct numerical solution of the full motion equations. The comparison configuration of suspension systems with viscous dampers and no dry friction is examined through an analytical first approximation approach and closed-form results for stability thresholds are derived in particular for the symmetric case.     

Heavy-ion fusion: Comparison of experimental data with classical trajectory models

Currently available data on fusion excitation functions for heavy-ion induced reactions over a wide mass range are compared to results calculated with a classical dynamical model based on the proximity nuclear potential of Blocki et al., the Coulomb potential of Bondorf et al., and one-body nuclear friction in the proximity formalism of Randrup. With these conservative and dissipative forces and the radial parameters of Myers, overall good agreement is obtained between the theoretical excitation functions and most of the available data. Extensive calculations have been performed to test the sensitivity of the calculated fusion cross-sections to a number of parameters, including the radial dependence of the Coulomb and nuclear potentials, the radial and tangential friction form factors as well as the projectile and target radii. The theoretical excitation functions for the lighter heavy-ion systems are rather insensitive to changes in either the conservative or dissipative forces. The calculations show that tangential friction sufficient to produce the rolling condition is necessary to explain the magnitude of the fusion cross-sections at high energies, which are also sensitive to the magnitude of the radial friction component. This is in contrast to the fusion cross-sections at low energies which are determined by the nuclear potential at larger separations, and to a lesser extent by tangential friction. The low energy fusion data are most sensitive to the nuclear radii. The calculations reveal the importance of more experimental measurements of fusion cross-sections at high energies, especially for heavy systems where the magnitudes of the fusion cross-sections are the most sensitive to the assumed forces. However, even for these cases the effects of the conservative and dissipative forces are similar and difficult to separate. These studies indicate, however, that it is possible to construct a conservative potential that will give calculated fusion excitation functions which are in good agreement with all experimental data over the entire mass range. The maximum fusion cross-sections as defined here exceed considerably the liquid-drop limiting value for heavy systems.     

Magnetic order in the Shastry-Sutherland model

We investigate the influence of energetic disorder, viscous damping and an external field on the electron transfer (ET) in DNA. The double helix structure of the λ-form of DNA is modeled by a steric oscillator network. In the context of the base-pair picture two different kinds of modes representing twist motions of the base pairs and H-bond distortions are coupled to the electron amplitude. Through the nonlinear interaction between the electronic and the vibrational degrees of freedom localized stationary states in the form of standing electron-vibron breathers are produced which we derive with a stationary map method. We show that in the presence of additional energetic disorder the degree of localization of such breathers is enhanced. It is demonstrated how an applied electric field initiates the long-range coherent motion of breathers along the bases of a DNA strand. These moving electron-vibron breathers, absorbing energy from the applied field, sustain energetic losses due to the viscous friction caused by the aqueous solvent as well as the impact of a moderate amount of energetic disorder. Moreover, it is illustrated that with the choice of the amplitude and frequency of the external field, the breather can be steered to a desired lattice position achieving control of the ET. Received 5 July 2002 Published online 29 November 2002     

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

A new frequency-time domain procedure, the dynamic Lagrangian mixed frequency-time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.     

Torsional friction damper optimization

A new approach for the analysis of friction dampers is presented in this work. The exact form of the steady-state solution for a friction damper implemented on a primary system is developed and numerical solutions are used to determine the optimum friction in a friction damper applied to a specific primary system. When compared to classical results presented by earlier authors, the new approach provides a more optimal solution. In addition, viscous damping in the primary system may be included with the new analysis approach. The ability to optimize a friction damper when viscous damping is included in the primary system is a significant improvement over earlier methods and shows potential for serving as a guide to realizing a more accurate estimate of the optimal damping for friction dampers.     

基于Coulomb摩擦效应的一类非线性相对转动系统的混沌研究

研究一类非线性相对转动系统在负载Coulomb摩擦效应下的混沌运动行为. 根据Lagrange方程建立一类含非线性负载Coulomb摩擦阻尼的两个质量相对转动系统的动力学方程. 利用Cardano公式讨论自治系统的特征值, 在此基础上, 应用待定系数法给出系统同宿轨道的存在性, 并借助Silnikov定理研究了系统的混沌行为. 最后数值模拟了给定参数下系统的混沌运动, 并给出在Coulomb摩擦阻尼变化下系统由周期、倍周期通向混沌的途径, 验证了理论分析的正确性.     

Resonant charge exchange involving highly excited atoms

Transition of a classical electron between two Coulomb centers is analyzed on the basis of computer simulations. The contribution to the electron transfer cross section from a tunnel electron transition is evaluated taking into account the strong mixing of highly excited electron states due to motion of Coulomb centers. The rate of transition of a highly excited electron between two Coulomb cores with a fixed separation is evaluated together with the cross section of resonant charge exchange in slow collisions. Typical times of change of the electron momentum as a result of electron motion in the field of two Coulomb centers are determined. The text was submitted by authors in English.