Dynamic anti-plane sliding of dissimilar anisotropic linear elastic solids

The stability of steady, dynamic, anti-plane slipping at a planar interface between two dissimilar anisotropic linear elastic solids is studied. The solids are assumed to possess a plane of symmetry normal to the slip direction, so that in-plane displacements and normal stress changes on the slip plane do not occur. Friction at the interface is assumed to follow a rate and state-dependent law with velocity weakening behavior in the steady state. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along the interface is studied. The critical wavenumber magnitude, |k|_cr, above which there is stability and the corresponding phase velocity, c, of the neutrally stable mode are obtained from the stability analysis. Numerical plots showing the dependence of |k|_cr and c on the unperturbed sliding velocity, V_o, are provided for various bi-material combinations of practical interest.     

Studies of dual and tandem rigid wheel performance in sand: by G. D. Swanson

Soil parameters c, ?, k_c, k_? and n were determined by use of a Penetro-Shear apparatus. This device utilizes the rotating and penetrating motion of a circular plate. The performance of a full scale M113 track is predicted using soil parameters calculated from the Penetro-Shear Apparatus data and compared to experimental determination of the drawbar pull-weight ratio vs.per cent of track slip curves.     

Mode II stress intensity factors for a crack parallel to the surface of an elastic half-space subjected to a moving point load

The direction of propagation of rolling contact fatigue cracks is observed to depend upon the direction of motion of the load. In this paper approximate calculations are described of the variation of Mode II stress intensity factors at each tip of a subsurface crack, which lies parallel to the surface of an elastic half-space, due to a load moving over the surface. In particular the effect of frictional locking of the crack faces under the load is investigated. In consequence of frictional locking the range of SIF at the trailing tip ΔK_T is found to be about 30% greater than that of the leading tip ΔK_L, which is consistent with observations that subsurface cracks propagate predominantly in the direction of motion of the load over the surface. The effects on k_t and k_lof crack length, crack face friction, traction forces at the surface and residual shear stresses are also investigated.     

Limit point buckling of a finite beam on a nonlinear foundation

In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from 1 to k as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter k through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics.     

Shakedown limits for an oscillating,elastic rolling contact with Coulomb friction

We examine experimentally and theoretically the effect of frictional shakedown of a three-dimensional elastic rolling contact. Small oscillations of the local normal forces lead to incremental sliding processes within the area of contact. Consequently, this causes a macroscopic slip motion of the two contacting bodies. If the oscillation amplitude is sufficiently small, the frictional slip ceases after the first few loading periods and a safe shakedown occurs. Otherwise the slip motion is continued and the contact fails.     

Steady and transient sliding under rate-and-state friction

The physics of dry friction is often modelled by assuming that static and kinetic frictional forces can be represented by a pair of coefficients usually referred to as μ_s and μ_k, respectively. In this paper we re-examine this discontinuous dichotomy and relate it quantitatively to the more general, and smooth, framework of rate-and-state friction. This is important because it enables us to link the ideas behind the widely used static and dynamic coefficients to the more complex concepts that lie behind the rate-and-state framework. Further, we introduce a generic framework for rate-and-state friction that unifies different approaches found in the literature.We consider specific dynamical models for the motion of a rigid block sliding on an inclined surface. In the Coulomb model with constant dynamic friction coefficient, sliding at constant velocity is not possible. In the rate-and-state formalism steady sliding states exist, and analysing their existence and stability enables us to show that the static friction coefficient μ_s should be interpreted as the local maximum at very small slip rates of the steady state rate-and-state friction law.Next, we revisit the often-cited experiments of Rabinowicz (J. Appl. Phys., 22:1373–1379, 1951). Rabinowicz further developed the idea of static and kinetic friction by proposing that the friction coefficient maintains its higher and static value μ_s over a persistence length before dropping to the value μ_k. We show that there is a natural identification of the persistence length with the distance that the block slips as measured along the stable manifold of the saddle point equilibrium in the phase space of the rate-and-state dynamics. This enables us explicitly to define μ_s in terms of the rate-and-state variables and hence link Rabinowicz's ideas to rate-and-state friction laws.This stable manifold naturally separates two basins of attraction in the phase space: initial conditions in the first one lead to the block eventually stopping, while in the second basin of attraction the sliding motion continues indefinitely. We show that a second definition of μ_s is possible, compatible with the first one, as the weighted average of the rate-and-state friction coefficient over the time the block is in motion.     

Aerodynamics of laminar separation flutter at a transitional Reynolds number

Experimental observations of self-sustained pitch oscillations of a NACA 0012 airfoil at transitional Reynolds numbers were recently reported. The aeroelastic limit cycle oscillations, herein labelled as laminar separation flutter, occur in the range 5.0×10⁴≤Re_c≤1.3×10⁵. They are well behaved, have a small amplitude and oscillate about θ=0°. It has been speculated that laminar separation leading to the formation of a laminar separation bubble, occurring at these Reynolds numbers, plays an essential role in these oscillations. This paper focuses on the Re_c=7.7×10⁴ case, with the elastic axis located at 18.6% chord. Considering that the experimental rig acts as a dynamic balance, the aerodynamic moment is derived and is empirically modelled as a generalized Duffing–van-der-Pol nonlinearity. As expected, it behaves nonlinearly with pitch displacement and rate. It also indicates a dynamically unstable equilibrium point, i.e. negative aerodynamic damping. In addition, large eddy simulations of the flow around the airfoil undergoing prescribed simple harmonic motion, using the same amplitude and frequency as the aeroelastic oscillations, are performed. The comparison between the experiment and simulations is conclusive. Both approaches show that the work done by the airflow on the airfoil is positive and both have the same magnitude. The large eddy simulation (LES) computations indicate that at θ=0°, the pitching motion induces a lag in the separation point on both surfaces of the airfoil resulting in negative pitching moment when pitching down, and positive moment when pitching up, thus feeding the LCO.     

Interaction of thermal contact resistance and frictional heating in thermoelastic instability

Thermoelastic contact problems can posess non-unique and/or unstable steady-state solutions if there is frictional heating or if there is a pressure-dependent thermal contact resistance at the interface. These two effects have been extensively studied in isolation, but their possible interaction has never been investigated. In this paper, we consider an idealized problem in which a thermoelastic rod slides against a rigid plane with both frictional heating and a contact resistance. For sufficiently low sliding speeds, the results are qualitatively similar to those with no sliding. In particular, there is always an odd number of steady-state solutions; if the steady-state is unique it is stable and if it is non-unique, stable and unstable solutions alternate, with the outlying solutions being stable. However, we identify a sliding speed V₀ above which the number of steady states is always even (including zero, implying possible non-existence of a steady-state) and again stable and unstable states alternate. A parallel numerical study shows that for V>V₀ there are some initial conditions from which the contact pressure grows without limit in time, whereas for V<V₀ the system will always tend to one of the stable steady states.     

Instabilities in Dynamic Anti-plane Sliding of an Elastic Layer on a Dissimilar Elastic Half-Space

The stability of dynamic anti-plane sliding at an interface between an elastic layer and an elastic half-space with dissimilar elastic properties is studied. Friction at the interface is assumed to follow a rate- and state-dependent law, with a positive instantaneous dependence on slip velocity and a rate weakening behavior in the steady state. The perturbations at the interface are of the form exp(ikx ₁+pt), where k is the wavenumber, x ₁ is the coordinate along the interface, p is the time response to the perturbation and t is time. A key feature of the problem is that interfacial waves both in freely slipping contact as well as in bonded contact exist for the problem. Attention is focused on the role of the interfacial waves on slip stability. Instabilities are plotted in the $\operatorname{Re} (pL/V_{o})$ versus $\operatorname{Im} (p/|k|c_{s})$ plane, where L is a length scale in the friction law, V _o is the unperturbed slip velocity and c _s is the shear wave speed of the layer. Stability of both rapid and slow slip is studied. The results show one mechanism by which instabilities occur is the destabilization by friction of the interfacial waves in freely slipping contact/bonded contact. This occurs even in slow sliding, thus confirming that the quasi-static approximation is not valid for slow sliding. The effect of material properties and layer thickness on the stability results is studied.     

Reducing cross-flow vibrations of underflow gates: Experiments and numerical studies

An experimental study is combined with numerical modelling to investigate new ways to reduce cross-flow vibrations of hydraulic gates with underflow. A rectangular gate section placed in a flume was given freedom to vibrate in the vertical direction. Horizontal slots in the gate bottom enabled leakage flow through the gate to enter the area directly under the gate which is known to play a key role in most excitation mechanisms.For submerged discharge conditions with small gate openings the vertical dynamic support force was measured in the reduced velocity range 1.5<V_r<10.5 for a gate with and without ventilation slots. The leakage flow significantly reduced vibrations. This attenuation was most profound in the high stiffness region at 2<V_r<3.5.Two-dimensional numerical simulations were performed with the Finite Element Method to assess local velocities and pressures for both gate types. A moving mesh covering both solid and fluid domain allowed free gate movement and two-way fluid–structure interactions. Modelling assumptions and observed numerical effects are discussed and quantified. The simulated added mass in still water is shown to be close to experimental values. The spring stiffness and mass factor were varied to achieve similar response frequencies at the same dry natural frequencies as in the experiment. Although it was not possible to reproduce the vibrations dominated by impinging leading edge vortices (ILEV) at relatively low V_r, the simulations at high V_r showed strong vibrations with movement-induced excitation (MIE). For the latter case, the simulated response reduction of the ventilated gate agrees with the experimental results. The numerical modelling results suggest that the leakage flow diminishes pressure fluctuations close to the trailing edge associated with entrainment from the wake into the recirculation zone directly under the gate that most likely cause the growing oscillations of the ordinary rectangular gate.     

On the nonlinear slewing dynamics and control of the Space Station based Mobile Servicing System

A relatively general Lagrangian formulation for studying the nonlinear dynamics and control of space-craft with interconnected flexible members in a tree-type topology is developed. Versatility of the formulation is illustrated through a dynamical study of the Space Station based two-link Mobile Servicing System (MSS). The performance of the MSS undergoing inplane and out-of-plane slewing maneuvers is compared. Results indicate that, in absence of control, the maneuvers induce undesirable librational motion of the Space Station as well as vibration of the links. Nonlinear control, based on the Feedback Linearization Technique (FLT), appears promising. Quasi-Closed Loop Control (QCLC), a variation of the FLT, is applied to control the libration of the Space Station. Once the attitude of the Space Station is controlled, the performance of the MSS improves significantly. For a 5-minute maneuver of the MSS, the maximum control torque required is only 34.5 Nm.Nomenclature f _i ¹ , f _i,j ¹ fundamental frequency of bodies B _i and B _i,j, respectively - l _c, l _i, l _i,j length of bodies B _c, B _i, and B _i,j, respectively - m _c, m _i, m _i,j mass of bodies B _c, B _i, and B _i,j, respectively - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbhv2BYDwAHbacfiGab8xCayaaraqefavySfgDP52BGWuAU9gD% 5bxzaGGbciaa+zgacaWFSaGaa8hiaiqa-fhagaqeaiaa-jhaaaa!4B1F!\[\bar qf, \bar qr\] vector representing flexible and rigid generalized coordinates - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbhv2BYDwAHbacfiGaa8hkaiqa-fhagaqeaiaa-jhacaWFPaqe% favySfgDP52BGWuAU9gD5bxzaGGbciaa+rgaaaa!4A18!\[(\bar qr)d\] vector representing the desired rigid generalized coordinates - (I _xx)k, (I _yy)k, (I _zz)k principal inertia of body B _k about X _k, Y _k, and Z _k axes, respectively; ksc, i or i, j - K _p, K _v displacement and velocity gain matrices - N _q total number of generalized coordinates - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbwvMCKfMBHbacfiGab8xuayaaraqefavySfgDP52BGWuAU9gD% 5bxzaGGbciaa+zgaieaacaqFSaGaa0hiaiqa-ffagaqeaGqaciaa8j% haaaa!4AEF!\[\bar Qf, \bar Qr\] control effort vectors for flexible and rigid coordinates, respectively - Q , Q , Q control effort for pitch, roll and yaw degree of freedom, respectively - _k ^y , _k ^z tip deflection of a beam type appendage (B _k) in the Y _k and Z _k directions, respectively.     

Stability of frictional slipping at an anisotropic/isotropic interface

The stability of steady, quasi-static slip at a planar interface between an anisotropic elastic solid and an isotropic elastic solid is studied. The paper begins with an analysis of anti-plane sliding at an orthotropic/isotropic interface. Friction at the interface is assumed to follow a rate- and state-dependent law. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along which the interface is studied. An expression is derived for the critical wavenumber ∣k∣_cr above which there is stability. In-plane sliding at an anisotropic/isotropic interface is subsequently studied. In this case, slip couples with normal stress changes and a constitutive law for dynamic normal stress changes is adopted. Again a formula for ∣k∣_cr is derived and the results are specialized to the case of an orthotropic/isotropic interface. Numerical plots of the dependence of ∣k∣_cr on the orientation of the orthotropic solid, as well as on the material parameters are provided.     

Ergodicity of the Infinite Dimensional Fractional Brownian Motion

In this short note we prove that an infinite dimensional fractional Brownian motion B _H of any Hurst parameter ${H \in (0, 1)}In this short note we prove that an infinite dimensional fractional Brownian motion B _H of any Hurst parameter H ? (0, 1){H \in (0, 1)} forms an ergodic metric dynamical system. For the proof we mainly use the fundamental theorems of Kolmogorov.     

Impact energy absorption by a rate-dependent locking target

The impact by an elastic cylindrical piston on a thin plate-like target resting on a rigid foundation is considered. The relationship between force F acting on the target and displacement x is given by F=kx+q dx/dt provided dx/dt≥0 and 0≤x<d (k, q and d≥0). When x=d locking occurs, and F can assume any value ≥kd without increase in x. The displacement is assumed to be completely irreversible. The motion of the impactor is assumed to be governed by the elementary wave equation and, since the target is thin, wave motion in the target is neglected. The energy W=εFdx and its components W _k=kεx dx (the energy absorbed in a corresponding quasistatic process) and W _q=qε(dx/dt)² dt (the excessive energy because of the rate-dependence) are determined in terms of the impact energy as functions of non-dimensional parameters representing k, q and d. With the aid of diagrams, it is shown under what circumstances locking occurs, and under what circumstances W _k or W _q, or both, are large.     

Winkler partial slip solution for harmonic oscillations in steady rolling contact problems

A Winkler model (Kalker’s simplified theory) is adopted for solving analytically partial slip rolling contact problem in the first order perturbation form of small periodic oscillations of generally both normal and tangential load about a steady state. At present, only numerical investigations exist for this problem, with various approximations to deal with the transient effects (often, simply neglected), and particularly the effect of varying normal load and hence contact area, has not been investigated in detail, despite the problem of corrugation is essentially driven by the change of normal load.The linear perturbation analysis is used to obtain closed form expressions for the receptances of the tangential load. Also, similar expressions are obtained for the energy dissipation, which is correlated with the local wear.     

A note on the characteristics of some slider bearings

Summary The elementary theory of lubrication is applied to some slider bearings having different film shapes and the minimisation of the friction coefficient is particularly considered. The inadequacy of the mathematical model is shown since it involves, in the usually dealt with situations, discontinuities not present in physics, and the characteristics are given of a particular profile for which this difficulty is overcome.

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Sommario Si applica la teoria elementare della lubrificazione a diversi tipi di accoppiamenti prismatici e si considera, in particolare, la minimizzazione del coefficiente di attrito. Si mostra l'inadeguatezza del modello matematico che comporta, nei casi usualmente trattati, discontinuità non presenti nella realtà fisica e si danno le caratteristiche di una particolare coppia in cui questa difficoltà è superata.

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Nomenclature f reduced friction coefficient,f/(h ₁ /L) - f friction coefficient - F reduced load capacity,F _N h ₁ ² /VL ² - F _N load capaticy - h film thickness - L length of the slider - x longitudinal coordinate, dimensionless with respect toL - V velocity of the slider - inclination of the plane slider - viscosity of the lubricant This research was partially supported by the Italian Ministry for Education (M.P.I.).     

Non-linear dynamics of wind turbine wings

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli-Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω. Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω₂/ω₁?2, internal resonances between these modes have been studied. It is demonstrated that for ω/ω₁?0.66,1.33,1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω/ω₁?1.0 and ω/ω₂?1.0 partly caused by the additive load term.     

Analytical and experimental evaluations of the influence of fracture surface roughness,its sliding actions and the associated plasticity on fatigue crack propagation

An investigation of fatigue crack propagation in rectangular AM60B magnesium alloy plates containing an inclined through crack is presented in this paper. The behavior of fatigue crack growth in the alloy is influenced by the fracture surface roughness. Therefore, in the present investigation, a new model is developed for estimating the magnitude of the frictional stress intensity factor, k_f, arising from the mismatch of fracture surface roughness during in-plane shear. Based on the concept of k_f, the rate of fatigue crack propagation, db/dN, is postulated to be a function of the effective stress intensity factor range, Δk_eff. Subsequently, the proposed model is applied to predict crack growth due to fatigue loads. Experiments for verifying the theoretical predictions were also conducted. The results obtained are compared with those predicted using other employed mixed mode fracture criteria and the experimental data.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

Self-excited stick–slip oscillations of drill bitsOscillations stick–slip auto-entretenues d'un outil de forage

This Note studies the self-excited stick–slip oscillations of a rotary drilling system with a drag bit, using a discrete model which takes into consideration the axial and torsional vibration modes of the bit. Coupling between these two vibration modes takes place through a bit-rock interaction law which accounts for both frictional contact and cutting processes at the bit-rock interface. The cutting process introduces a delay in the equations of motion which is ultimately responsible for the existence of self-excited vibrations, exhibiting stick–slip oscillations under certain conditions. To cite this article: T. Richard et al., C. R. Mecanique 332 (2004).