A phase-plane analysis of localized frictional waves.

Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick-slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.

Asynchronous partial contact motion due to internal resonance in multiple degree-of-freedom rotordynamics.

Sudden onset of violent chattering or whirling rotor-stator contact motion in rotational machines can cause significant damage in many industrial applications. It is shown that internal resonance can lead to the onset of bouncing-type partial contact motion away from primary resonances. These partial contact limit cycles can involve any two modes of an arbitrarily high degree-of-freedom system, and can be seen as an extension of a synchronization condition previously reported for a single disc system. The synchronization formula predicts multiple drivespeeds, corresponding to different forms of mode-locked bouncing orbits. These results are backed up by a brute-force bifurcation analysis which reveals numerical existence of the corresponding family of bouncing orbits at supercritical drivespeeds, provided the damping is sufficiently low. The numerics reveal many overlapping families of solutions, which leads to significant multi-stability of the response at given drive speeds. Further, secondary bifurcations can also occur within each family, altering the nature of the response and ultimately leading to chaos. It is illustrated how stiffness and damping of the stator have a large effect on the number and nature of the partial contact solutions, illustrating the extreme sensitivity that would be observed in practice.

The landscape of nonlinear structural dynamics: an introduction.

Nonlinear behaviour is ever-present in vibrations and other dynamical motions of engineering structures. Manifestations of nonlinearity include amplitude-dependent natural frequencies, buzz, squeak and rattle, self-excited oscillation and non-repeatability. This article primarily serves as an extended introduction to a theme issue in which such nonlinear phenomena are highlighted through diverse case studies. More ambitiously though, there is another goal. Both the engineering context and the mathematical techniques that can be used to identify, analyse, control or exploit these phenomena in practice are placed in the context of a mind-map, which has been created through expert elicitation. This map, which is available in software through the electronic supplementary material, attempts to provide a practitioner's guide to what hitherto might seem like a vast and complex research landscape.

Improving the signal-to-noise ratio of high-speed contact mode atomic force microscopy.

During high-speed contact mode atomic force microscopy, higher eigenmode flexural oscillations of the cantilever have been identified as the main source of noise in the resultant topography images. We show that by selectively filtering out the frequencies corresponding to these oscillations in the time domain prior to transforming the data into the spatial domain, significant improvements in image quality can be achieved.

Modelling oscillatory flexure modes of an atomic force microscope cantilever in contact mode whilst imaging at high speed.

Understanding the modal response of an atomic force microscope is important for the identification of image artefacts captured using contact-mode atomic force microscopy (AFM). As the scan rate of high speed AFM increases, these modes present themselves as ever clearer noise patterns as the frequency of cantilever vibration falls under the frequency of pixel collection. An Euler-Bernoulli beam equation is used to simulate the flexural modes of the cantilever of an atomic force microscope as it images a hard surface in contact mode. Theoretical results are compared with experimental recordings taken in the high speed regime, as well as previous analytical results. It is shown that the model can capture the mode shapes and resonance properties of the first four eigenmodes.

High-speed atomic force microscopy in slow motion--understanding cantilever behaviour at high scan velocities.

Using scanning laser Doppler vibrometer we have identified sources of noise in contact mode high-speed atomic force microscope images and the cantilever dynamics that cause them. By analysing reconstructed animations of the entire cantilever passing over various surfaces, we identified higher eigenmode oscillations along the cantilever as the cause of the image artefacts. We demonstrate that these can be removed by monitoring the displacement rather than deflection of the tip of the cantilever. We compare deflection and displacement detection methods whilst imaging a calibration grid at high speed and show the significant advantage of imaging using displacement.

Nonlinear models of development, amplification and compression in the mammalian cochlea.

This paper reviews current understanding and presents new results on some of the nonlinear processes that underlie the function of the mammalian cochlea. These processes occur within mechano-sensory hair cells that form part of the organ of Corti. After a general overview of cochlear physiology, mathematical modelling results are presented in three parts. First, the dynamic interplay between ion channels within the sensory inner hair cells is used to explain some new electrophysiological recordings from early development. Next, the state of the art is reviewed in modelling the electro-motility present within the outer hair cells (OHCs), including the current debate concerning the role of cell body motility versus active hair bundle dynamics. A simplified model is introduced that combines both effects in order to explain observed amplification and compression in experiments. Finally, new modelling evidence is presented that structural longitudinal coupling between OHCs may be necessary in order to capture all features of the observed mechanical responses.

Experimental observation of contact mode cantilever dynamics with nanosecond resolution.

We report the use of a laser Doppler vibrometer to measure the motion of an atomic force microscope contact mode cantilever during continuous line scans of a mica surface. With a sufficiently high density of measurement points the dynamics of the entire cantilever beam, from the apex to the base, can be reconstructed. We demonstrate nanosecond resolution of both rectangular and triangular cantilevers. This technique permits visualization and quantitative measurements of both the normal and lateral tip sample interactions for the first and higher order eigenmodes. The ability to derive quantitative lateral force measurements is of interest to the field of microtribology/nanotribology while the comprehensive understanding of the cantilever's dynamics also aids new cantilever designs and simulations.

Catastrophic sliding bifurcations and onset of oscillations in a superconducting resonator.

This paper presents a general analysis and a concrete example of the catastrophic case of a discontinuity-induced bifurcation in so-called Filippov nonsmooth dynamical systems. Such systems are characterized by discontinuous jumps in the right-hand sides of differential equations across a phase space boundary and are often used as physical models of stick-slip motion and relay control. Sliding bifurcations of periodic orbits have recently been shown to underlie the onset of complex dynamics including chaos. In contrast to previously analyzed cases, in this work a periodic orbit is assumed to graze the boundary of a repelling sliding region, resulting in its abrupt destruction without any precursive change in its stability or period. Necessary conditions for the occurrence of such catastrophic grazing-sliding bifurcations are derived. The analysis is illustrated in a piecewise-smooth model of a stripline resonator, where it can account for the abrupt onset of self-modulating current fluctuations. The resonator device is based around a ring of NbN containing a microbridge bottleneck, whose switching between normal and super conducting states can be modeled as discontinuous, and whose fast temperature versus slow current fluctuations are modeled by a slow-fast timescale separation in the dynamics. By approximating the slow component as Filippov sliding, explicit conditions are derived for catastrophic grazing-sliding bifurcations, which can be traced out as parameters vary. The results are shown to agree well with simulations of the slow-fast model and to offer a simple explanation of one of the key features of this experimental device.

Mathematical modelling of the active hearing process in mosquitoes.

Insects have evolved diverse and delicate morphological structures in order to capture the inherently low energy of a propagating sound wave. In mosquitoes, the capture of acoustic energy and its transduction into neuronal signals are assisted by the active mechanical participation of the scolopidia. We propose a simple microscopic mechanistic model of the active amplification in the mosquito species Toxorhynchites brevipalpis. The model is based on the description of the antenna as a forced-damped oscillator coupled to a set of active threads (ensembles of scolopidia) that provide an impulsive force when they twitch. This twitching is in turn controlled by channels that are opened and closed if the antennal oscillation reaches a critical amplitude. The model matches both qualitatively and quantitatively with recent experiments: spontaneous oscillations, nonlinear amplification, hysteresis, 2 : 1 resonances, frequency response and gain loss owing to hypoxia. The numerical simulations presented here also generate new hypotheses. In particular, the model seems to indicate that scolopidia located towards the tip of Johnston's organ are responsible for the entrainment of the other scolopidia and that they give the largest contribution to the mechanical amplification.

Investigation of a multi-ball, automatic dynamic balancing mechanism for eccentric rotors.

This paper concerns an analytical and experimental investigation into the dynamics of an automatic dynamic balancer (ADB) designed to quench vibration in eccentric rotors. This fundamentally nonlinear device incorporates several balancing masses that are free to rotate in a circumferentially mounted ball race. An earlier study into the steady state and transient response of the device with two balls is extended to the case of an arbitrary number of balls. Using bifurcation analysis allied to numerical simulation of a fully nonlinear model, the question is addressed of whether increasing the number of balls is advantageous. It is found that it is never possible to perfectly balance the device at rotation speeds comparable with or below the first natural, bending frequency of the rotor. When considering practical implementation of the device, a modification is suggested where individual balls are contained in separate arcs of the ball race, with rigid partitions separating each arc. Simulation results for a partitioned ADB are compared with those from an experimental rig. Close qualitative and quantitative match is found between the theory and the experiment, confirming that for sub-resonant rotation speeds, the ADB at best makes no difference to the imbalance, and can make things substantially worse. Further related configurations worthy of experimental and numerical investigation are proposed.

Radiationless traveling waves in saturable nonlinear Schrödinger lattices.

The long-standing problem of moving discrete solitary waves in nonlinear Schrödinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities.

Accumulation of embedded solitons in systems with quadratic nonlinearity.

Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multiwave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model. An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter epsilon, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of Wentzel-Kramers-Brillouin type yields an asymptotic formula for the distribution of discrete values of epsilon at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, -65, and a fairly good approximation for the numerical factor in front of the scaling term.

Walking cavity solitons.

A family of walking solitons is obtained for the degenerate optical parametric oscillator below threshold. The loss-driven mechanism of velocity selection for these structures is described analytically and numerically. Our approach is based on understanding the role played by the field momentum and generic symmetry properties and, therefore, it can be easily generalized to other dissipative multicomponent models with walk off.

Grazing and border-collision in piecewise-smooth systems: a unified analytical framework.

A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes. This links grazings with border-collisions in nonsmooth maps. Contrary to previous literature, piecewise linear maps correspond only to nonsmooth discontinuity boundaries. All other maps have either square-root or (3/2)-type singularities.