A method to determine dynamic loads on spur gear teeth and on bearings

A method is described which can be used to calculate dynamic gear tooth force and bearing forces. The model includes elastic bearings. The gear mesh stiffness and the path of contact are determined using the deformations of the gears and the bearings. This gives contact outside the plane-of-action and a time-varying working pressure angle. In a numerical example it is found that the only important vibration mode for the gear contact is the one where the gear tooth deformation is dominant. The bearing force variation, however, will be much more affected by the other vibration modes. The influence of the friction force is also studied. The friction has no dynamic influence on the gear contact force or on the bearing force in the gear mesh line-of-action direction. On the other hand, the changing of sliding directions in the pitch point is a source for critical oscillations of the bearings in the gear tooth frictional direction. These bearing force oscillations in the frictional direction appear unaffected by the dynamic response along the gear mesh line-of-action direction.     

Nonlinear shear wave interaction at a frictional interface: energy dissipation and generation of harmonics

Analytical and numerical modeling of the nonlinear interaction of shear wave with a frictional interface is presented. The system studied is composed of two homogeneous and isotropic elastic solids, brought into frictional contact by remote normal compression. A shear wave, either time harmonic or a narrow band pulse, is incident normal to the interface and propagates through the contact. Two friction laws are considered and the influence on interface behavior is investigated: Coulomb's law with a constant friction coefficient and a slip-weakening friction law which involves static and dynamic friction coefficients. The relationship between the nonlinear harmonics and the dissipated energy, and the dependence on the contact dynamics (friction law, sliding, and tangential stress) and on the normal contact stress are examined in detail. The analytical and numerical results indicate universal type laws for the amplitude of the higher harmonics and for the dissipated energy, properly non-dimensionalized in terms of the pre-stress, the friction coefficient and the incident amplitude. The results suggest that measurements of higher harmonics can be used to quantify friction and dissipation effects of a sliding interface.     

Dynamics of precursors to frictional sliding

We measure the spatial and temporal behavior of the true contact area A along a rough spatially extended interface between two blocks in frictional contact. Upon the application of shear the onset of motion is preceded by a discrete sequence of cracklike precursors, which are initiated at shear levels that are well below the threshold for static friction. These precursors arrest well before traversing the entire interface. They systematically increase in length with the applied shear force and significantly redistribute the true contact area along the interface. Thus, when frictional sliding occurs, the initially uniform contact area along the interface has already evolved to one that is highly nonuniform in space.     

Identification of Iwan distribution density function in frictional contacts

Mechanical contacts affect structural responses, causing localized nonlinear variations in the stiffness and damping. The physical behaviors of contact interfaces are quite complicated and almost impossible to model at the micro-scale. In order to establish a meaningful understanding of the friction effects and to predict the contact behavior, a robust parametric friction model is usually employed. This paper employs an Iwan-type model to predict the nonlinear effects of a frictional contact interface. The Iwan model is characterized by its distribution density function which is commonly identified by double differentiation of the experimentally obtained joint interface restoring force. Direct measurement of restoring forces at the contact interface is impractical and estimating it using an inverse approach introduces considerable uncertainties in identification of the density function. This paper develops a more reliable procedure in identification of the Iwan model by relating the density function to the joint interface dissipated energy. The energy dissipated in a contact interface is easily obtained from measurement and it is shown that the dissipation is uniquely defined using the density function and the vibration amplitude. In an experimental case study Iwan distribution density function in a frictional contact is obtained using measured dissipations at different vibration levels.     

Random motion with interfacial contact: Driven diffusion vis-à-vis mechanical activation

Rolling of a small sphere on a patterned support of an elastomer is governed by a non-linear friction. No motion occurs when the external field is weaker than the frictional resistance. However, with the intervention of an external noise, a viscous friction like behavior emerges; thus the sphere rolls with a uniform drift velocity that is proportional to the applied field. At a very low noise strength, the sphere exhibits a stick-slip behavior with motion occurring always along the bias. With the increase in the noise strength, the sphere exhibits a diffusive drift accompanied with forward and backward displacements. During this stage of driven diffusive motion, the ratio of the integrated probabilities of the negative-to-positive work fluctuations decreases monotonically with the time of observation, from which a temperature like intensive parameter can be estimated. This parameter conforms to Einstein's ratio of diffusivity and mobility that increases almost linearly, even though the diffusivity increases super-linearly, with the strength of the noise. A new barrier crossing experiment is introduced that can be performed either with a hard (e.g. a steel ball) or with a soft (e.g. a water drop) sphere in contact with a periodically undulated substrate. The frequency of barrier crossing follows a transition state equation allowing a direct estimation of the effective temperature. These experiments as well as certain numerical simulations suggest that the effective temperature of a system controlled by a non-linear friction may not have a unique value.     

Random motion with interfacial contact: Driven diffusion vis-à-vis mechanical activation

Rolling of a small sphere on a patterned support of an elastomer is governed by a non-linear friction. No motion occurs when the external field is weaker than the frictional resistance. However, with the intervention of an external noise, a viscous friction like behavior emerges; thus the sphere rolls with a uniform drift velocity that is proportional to the applied field. At a very low noise strength, the sphere exhibits a stick-slip behavior with motion occurring always along the bias. With the increase in the noise strength, the sphere exhibits a diffusive drift accompanied with forward and backward displacements. During this stage of driven diffusive motion, the ratio of the integrated probabilities of the negative-to-positive work fluctuations decreases monotonically with the time of observation, from which a temperature like intensive parameter can be estimated. This parameter conforms to Einstein??s ratio of diffusivity and mobility that increases almost linearly, even though the diffusivity increases super-linearly, with the strength of the noise. A new barrier crossing experiment is introduced that can be performed either with a hard (e.g. a steel ball) or with a soft (e.g. a water drop) sphere in contact with a periodically undulated substrate. The frequency of barrier crossing follows a transition state equation allowing a direct estimation of the effective temperature. These experiments as well as certain numerical simulations suggest that the effective temperature of a system controlled by a non-linear friction may not have a unique value.     

Relationship between the real contact area and contact force in pre-sliding regime

The pre-sliding regime is typically neglected in the dynamic modelling of mechanical systems. However, the change in contact state caused by static friction may decrease positional accuracy and control precision. To investigate the relationship between contact status and contact force in pre-sliding friction, an optical experimental method is presented in this paper.With this method, the real contact state at the interface of a transparent material can be observed based on the total reflection principle of light by using an image processing technique. A novel setup, which includes a pair of rectangular trapezoidal blocks, is proposed to solve the challenging issue of accurately applying different tangential and normal forces to the contact interface. The improved Otsu's method is used for measurement. Through an experimental study performed on polymethyl methacrylate(PMMA), the quantity of contact asperities is proven to be the dominant factor that affects the real contact area. The relationship between the real contact area and the contact force in the pre-sliding regime is studied, and the distribution of static friction at the contact interface is qualitatively discussed. New phenomena in which the real contact area expands along with increasing static friction are identified. The aforementioned relationship is approximately linear at the contact interface under a constant normal pressure, and the distribution of friction stress decreases from the leading edge to the trailing edge.     

Superlubricity of graphite

Using a home-built frictional force microscope that is able to detect forces in three dimensions with a lateral force resolution down to 15 pN, we have studied the energy dissipation between a tungsten tip sliding over a graphite surface in dry contact. By measuring atomic-scale friction as a function of the rotational angle between two contacting bodies, we show that the origin of the ultralow friction of graphite lies in the incommensurability between rotated graphite layers, an effect proposed under the name of "superlubricity" [Phys. Rev. B 41, 11 837 (1990)]].     

A model for the characterization of friction contacts in turbine blades

Stresses produced by the forced vibrations can lead to a significant reduction of the life of turbo engine blades. To predict the vibration amplitudes of this components an accurate dynamic analysis is necessary. The forced response calculation of these dynamic systems is strongly affected by the presence of the contact interfaces (i.e., underplatform dampers, shrouds, root joints). Different contact models are available in literature. These models make use of contact parameters, contact stiffness and friction coefficient to evaluate the damping and stiffness related to the contact interfaces. In this paper a model is proposed to characterize friction contact of non-spherical contact geometries obeying the Coulomb friction law with constant friction coefficient and constant normal load. The hysteresis curves of the oscillating tangential contact forces vs. relative tangential displacements and the dissipated energy at the contact are obtained for different contact geometries. The developed model is suitable to be implemented in numerical solvers for the calculation of the forced response of turbine blades with embedded friction contacts.     

Observed transition from linear to non-linear friction-load behavior using a lateral force microscope

The most commonly observed friction behavior for sliding systems is that described by Amontons laws of friction. In this case, sliding friction is independent of the gross or apparent area of contact between the materials and a linear function of the applied normal load, where the constant of proportionality is called the friction coefficient. However, for dry sliding solids in contact via a single-asperity junction, Amontons (linear) friction-load behavior is not strictly relevant. In experiments measuring sliding friction between a silicon tip and a quartz surface using an atomic force microscope (AFM), a transition from linear to non-linear friction-load behavior has been observed. This is proposed to result from a nanoscale ‘conditioning’ of a multiple-contact tip-surface interface to form a single-asperity contact.     

Measurement of the frictional forces in fretting fatigue

This paper presents the results of a study of the frictional forces associated with the tangential harmonic displacements of a slider in the frequency range from 20 to 1000 Hz.On the basis of our experiments, we establish that the frictional interaction creates a normal periodic force which, in turn, creates normal vibrations leading to a change in the actual load and hence in the peak value of the force of friction.A change in the force of friction with the frequency of the tangential displacements has also been noted elsewhere [1]; however, the causes of this behavior have not previously been ascertained.We show that the force of friction depends significantly not only on the amplitude of the normal vibrations but also on the phase-frequency-relations between the vibrations in the normal and tangential directions.     

Experimental and numerical investigation of friction-induced vibration of a beam-on-beam in contact with friction

The vibrations generated by friction are responsible for various noises such as squealing, squeaking and chatter. Although these phenomena have been studied for a long time, it is not well-understood. In this study, an experimental and numerical study of friction-induced vibrations of a system composed of two beams in contact is proposed. The experimental system exhibits periodic steady state vibrations of different types. To model and understand this experimental vibratory phenomenon, complex eigenvalue and dynamic transient analyses are performed. In the linear complex eigenvalue analysis, flutter instability occurs via the coalescence of two eigenmodes of the system. This linear study provides an accurate value of the experimental frequency of vibration. To understand what happens physically during friction-induced instability, a dynamic transient analysis that takes account of the non-linear aspect of a frictional contact is performed. In this analysis, friction-induced instability is characterized by self-sustained vibrations and by stick, slip and separation zones occurring at the surface of the contact. The results stemming from this analysis show that good correlation between numerical and experimental vibrations can be obtained (in time and frequency domains). Moreover, time domain simulations permit understanding the physical phenomena involved in two different vibratory behaviours observed experimentally.     

Stochastic rolling of a rigid sphere in weak adhesive contact with a soft substrate

We study the rolling motion of a small solid sphere on a fibrillated rubber substrate in an external field in the presence of a Gaussian noise. From the nature of the drift and the evolution of the displacement fluctuation of the ball, it is evident that the rolling is controlled by a complex non-linear friction at a low velocity and a low noise strength (K), but by a linear kinematic friction at a high velocity and a high noise strength. This transition from a non-linear to a linear friction control of motion can be discerned from another experiment in which the ball is subjected to a periodic asymmetric vibration in conjunction with a random noise. Here, as opposed to that of a fixed external force, the rolling velocity decreases with the strength of the noise suggesting a progressive fluidization of the interface. A state (K) and rate (V) dependent friction model is able to explain both the evolution of the displacement fluctuation as well as the sigmoidal variation of the drift velocity with K. This research sets the stage for studying friction in a new way, in which it is submitted to a noise and then its dynamic response is studied using the tools of statistical mechanics. Although more works would be needed for a fuller realization of the above-stated goal, this approach has the potential to complement direct measurements of friction over several decades of velocities and other state variables. It is striking that the non-Gaussian displacement statistics as observed with the stochastic rolling is similar to that of a colloidal particle undergoing Brownian motion in contact with a soft microtubule.     

Effect of substrate dimensions on dynamics of ultrasonic consolidation

An FEM model is developed for a fundamental study of the time-dependent mechanical behavior of the substrate and its dimensions on ultrasonic consolidation. The simulation shows that for a given vibration condition, the amplitude of contact friction stress and displacement stabilizes to a saturated state after certain number of ultrasonic cycles. With the increased substrate height, the amplitude of contact frictional stress decreases, while that of contact interface displacement increases. The reason for the decrease in the frictional stress at the contact interface for certain substrate heights is the complicated wave interference occurring in the substrate. An analytical wave model has been built to validate the FEM model. A specific substrate geometry (height:width = 1.0) generates a minimum frictional strain state at the interface as a result of wave superposition. Such minimum strain state is believed to have produced the “lack of bonding” defect for the geometry. The energy density and transfer coefficient at the contact interface with different substrate heights is used as an indicator to correlate with the bond formation in ultrasonic bonding.     

A finite element study on rail corrugation based on saturated creep force-induced self-excited vibration of a wheelset-track system

The present work proposes friction coupling at the wheel-rail interface as the mechanism for formation of rail corrugation. Stability of a wheelset-track system is studied using the finite element complex eigenvalue method. Two models for a wheelset-track system on a tight curved track and on a straight track are established. In these two models, motion of the wheelset is coupled with that of the rail by friction. Creep force at the interface is assumed to become saturated and approximately equal to friction force, which is equal to the normal contact force multiplied by dynamic coefficient of friction. The rail is supported by vertical and lateral springs and dampers at the positions of sleepers. Numerical results show that there is a strong propensity of self-excited vibration of the wheelset-track system when the friction coefficient is larger than 0.21. Some unstable frequencies fall in the range 60-1200 Hz, which correspond to frequencies of rail corrugation. Parameter sensitivity analysis shows that the dynamic coefficient of friction, spring stiffness and damping of the sleeper supports all have important influences on the rail corrugation formation. Bringing the friction coefficient below a certain level can suppress or eliminate rail corrugation.     

Theory of nuclear friction

From the general many-body Schrödinger equation for colliding heavy ions a classical equation of motion including frictional forces is deduced. It is argued that one of the main origins of friction between two colliding heavy ions is the reflection of nucleons off the moving edges of the single-particle potentials. A simple classical analogue is given which illustrates how friction appears.     

Micro/macro-slip damping in beams with frictional contact interface

Friction in contact interfaces of assembled structures is the prime source of nonlinearity and energy dissipation. Determination of the dissipated energy in an assembled structure requires accurate modeling of joint interfaces in stick, micro-slip and macro-slip states. The present paper proposes an analytical model to evaluate frictional energy loss in surface-to-surface contacts. The goal is to develop a continuous contact model capable of predicting the dynamics of friction interface and dissipation energy due to partial slips. To achieve this goal, the governing equations of a frictional contact interface are derived for two distinct contact states of stick and partial slip. A solution procedure to determine stick–slip transition under single-harmonic excitations is derived. The analytical model is verified using experimental vibration test responses performed on a free-frictionally supported beam under lateral loading. The theoretical and experimental responses are compared and the results show good agreements between the two sets of responses.     

Local friction at a sliding interface between an elastomer and a rigid spherical probe

This paper reports on spatially resolved measurements of the shear stress distribution at a frictional interface between a flat rubber substrate and a glass lens. Silicone rubber specimens marked close to their surface by a colored pattern have been prepared in order to measure the surface displacement field induced by the steady-state friction of the spherical probe. The deconvolution of this displacement field then provides the actual shear stress distribution at the contact interface. When a smooth glass lens is used, a nearly constant shear stress is achieved within the contact. On the other hand, a bell-shaped shear stress distribution is obtained with rough lenses. These first results suggest that simple notions of real contact area and constant interface shear stress cannot account for the observed changes in local friction when roughness is varied.     

Rotor–stator contact dynamics using a non-ideal drive—Theoretical and experimental aspects

The possible contact between rotor and stator is considered a serious malfunction that may lead to catastrophic failure. Rotor rub is seen as a secondary phenomenon caused by a primary source, i.e. sudden mass unbalance, instabilities generated by aerodynamic and hydrodynamic forces in seals and bearings among others. The contact event gives rise to normal and friction forces exerted on the rotor at impact events. The friction force plays a significant role by transferring some rotational energy of the rotor to lateral motion. A mathematical model has been developed to capture this for a conventional backup annular guide setup. It is reasonable to superpose an impact condition to the rub, where the rotor spin energy can be fully transformed into rotor lateral movements. Using a nonideal drive, i.e. an electric motor without any kind of velocity feedback control, it is even possible to stop the rotor spin under rubbing conditions. All the rotational energy will be transformed in a kind of “self-excited” rotor lateral vibration with repeated impacts against the housing. This paper studies the impact motion of a rotor impacting a conventional backup annular guide for the case of dry and lubricated inner surface of the guide. For the dry surface case, the experimental and numerical analysis shows that the rotational energy is fully transformed into lateral motion and the rotor spin is stopped. Based on this study this paper proposes a new unconventional backup bearing design in order to reduce the rub related severity in friction and center the rotor at impact events. The analysis shows that the rotor at impacts is forced to the center of the backup bearing and the lateral motion is mitigated. As a result of this, the rotor spin is kept constant.     

Methods of calculation of a friction coefficient: application to nanotubes

In this Letter we develop theoretical and numerical methods to calculate the dynamic friction coefficient. The theoretical method is based on an adiabatic approximation which allows us to express the dynamic friction coefficient in terms of the time integral of the autocorrelation function of the force between both sliding objects. The motion of the objects and the autocorrelation function can be numerically calculated by molecular-dynamics simulations. We have successfully applied these methods to the evaluation of the dynamic friction coefficient of the relative motion of two concentric carbon nanotubes. The dynamic friction coefficient is shown to increase with the temperature.