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Perturbations of completely integrable Hamiltonian systems with three or more degrees of freedom are studied. In particular, the unperturbed systems are assumed to be separable into a product of simple oscillator-type systems and a system containing homo- or heteroclinic connections consisting of stable and unstable manifolds of saddle points. Under a perturbation, the manifolds persist but separate and may no longer intersect. In this paper we show how, with proper choices for initial conditions, one may solve the variational equations to obtain analytical expressions for orbits on the perturbed manifolds in the form of expansions in the small parameter characterizing the perturbation. The derivation also shows how the distance between the manifolds can be uniquely defined, and thus provides an alternative to the traditional higher dimensional Melnikov method. It is finally argued that the approximate knowledge of the shape and position of the perturbed manifolds could be utilized for the study of large-scale phase-space motions, such as those associated with Arnold diffusion. The theory is further illuminated in two example problems.

A new time-dependent normal-form procedure for dynamical equilibria (undergoing parametric excitation) of one-dimensional (1D) Hamiltonian systems is developed with the method of Lie transforms. The expansion is based on the Lewis invariant for the linearized motion. The time-dependent Hamiltonian normal form reduces smoothly to the usual representation in the autonomous limit. Illustrative examples of the formalism are focused on time-periodic systems and the dynamics of Hamiltonian switching.

This paper derives an alternative approach to the Melnikov method, which greatly reduces the amount of algebra involved in higher-order calculations. To illustrate this, a particular system is studied for which such a higher-order analysis is necessary, due to an identically vanishing first-order Melnikov function. The results of a second-order calculation imply the existence of transverse homoclinic orbits and, consequently, the existence of a horseshoe.

We study a perturbed Newtonian two-body problem, in which the perturbation is due to a force field of constant magnitude but rotating direction. By considering this system as a perturbation of the non-rotating case a Melnikov-type analysis allows us to show the existence of horseshoes in the level sets of the Hamiltonian and the subsequent sensitive dependence on initial conditions and non-integrability. We discuss the consequences of these results for a particular planar restricted three-body problem.

Computational modeling is a growing area of mechanical engineering that focuses on the use of numerical simulation to examine complex phenomena. Computational modeling includes work in finite element analysis, computational fluid dynamics, and multi-body dynamics modeling. Intrinsic to most of these modeling efforts are common elements including: 1. Assumptions that are made to reduce a problem to a solvable mathematical problem, 2. Formulation of a mathematical representation based on scientific principles, 3. Reduction of the mathematical representation through the removal of terms of small effect (neglected terms) 4. Collection and use of input data, 5. Algorithm development using numerical methods and discretization of the mathematical problem, 6. Implementation of the algorithm in computer code, 7. Creating representations of the model results, 8. External validation of the model, and 9. Dissemination of the model and model computer code.

Recent research in ethics education shows a potentially problematic variation in content, curricular materials, and instruction. While ethics instruction is now widespread, studies have identified significant variation in both the goals and methods of ethics education, leaving researchers to conclude that many approaches may be inappropriately paired with goals that are unachievable. This paper speaks to these concerns by demonstrating the importance of aligning classroom-based assessments to clear ethical learning objectives in order to help students and instructors track their progress toward meeting those objectives. Two studies at two different universities demonstrate the usefulness of classroom-based, formative assessments for improving the quality of students’ case responses in computational modeling and research ethics.

In this paper, we consider model examples of dynamical systems with only a few degrees of freedom and with desirable symmetry properties and explore compensating control strategies for retaining robust symmetric system response even under symmetry-breaking defects. The analysis demonstrates the distinct differences between linear versions of these models, in which open-loop fault-compensating strategies are always found, and weakly nonlinear counterparts with varying degrees of asymmetry, for which a multitude of locally optimal solutions may co-exist. We further formulate a candidate open-loop optimization protocol for fault compensation applied to self-healing systems, which respond to symmetry-breaking defects by a continuous process of fault correction. The analysis shows that such a protocol exhibits discontinuous changes in the control strategy as the self-healing system successively regains its original symmetry properties. In addition, it is argued that upon return to a symmetric configuration, such a protocol may result in a different control strategy from that applied prior to the occurrence of a fault.

This paper discusses the development of a method for on-the-fly, self-calibration of a mass-flow sensing system on a harvest combine in order to enable accurate mass-flow sensing and, consequently, yield mapping, under varying input conditions, for example due to changes in the moisture content of harvested grains or the aging of elevator paddles. The proposed method relies on a known or estimated physics-based relationship between the grain dynamics through the grain elevator and a measured force imparted on an impact plate through collisions with the grains. The paper provides a summary description of the modeled relationship between the rate of mass flow and impact-plate sensor readout and conceptualizes the self-calibration scheme through the introduction of an additional, controllable, degree of freedom of the plate. The validity of the self-calibration technique is illustrated through off-line application to bench-top and full-scale experimental test data.Copyright © 2012 by ASME

This paper proposes a conceptual design for a microscale experimental device that could be used to investigate oblique, intermittent contact while allowing for tangential slip and adhesion/stiction at the contact interface. The discussion includes a formulation of a reduced-order model of the nonlinear device dynamics. Emphasis is given to a possible contact model that incorporates normal and tangential compliance, friction-like limits on the ratio of tangential to normal contact forces, adhesive interactions, and irreversible energy loss in the normal contact interactions. A preliminary theoretical and numerical analysis is performed on this contact model in order to assess the influence of adhesion and energy loss on the contact phase. The results consider both the case of a normal collision in the absence of coupling between normal and tangential degrees of freedom as well as the general case of an oblique impact with such coupling. The paper argues that there is great value in an experimental validation of such a model for microscale contact, as a growing number of devices either rely on contact directly or must accommodate and design for its possible occurrence. In addition, mention is made of a number of inconsistencies and paradoxes observed in macroscopic models of rigid-body contact with friction that warrant validation or resolution through appropriate experimental designs.Copyright © 2012 by ASME

A computational framework is proposed to perform parameter continuation of periodic solutions of nonlinear, distributed-parameter systems represented by partial differential equations with time-dependent coefficients and excitations. The path-following procedure, encoded in the general-purpose Matlab-based computational continuation core (referred to below as coco), employs only the evaluation of the vector field of an appropriate spatial discretization; for example as formulated through an explicit finite-element discretization or through reliance on a black-box discretization. An original contribution of this paper is a systematic treatment of the coupling of coco with Comsolmultiphysics, demonstrating the great flexibility afforded by this computational framework. Comsolmultiphysics provides embedded discretization algorithms capable of accommodating a great variety of mechanical/physical assumptions and multiphysics interactions. Within this framework, it is shown that a concurrent bifurcation analysis may be carried out together with parameter continuation of the corresponding monodromy matrices. As a case study, we consider a nonlinear beam, subject to a harmonic, transverse direct excitation for two different sets of boundary conditions and demonstrate how the proposed approach may be able to generate results for a variety of structural models with great ease. The numerical results include primary-resonance, frequency-response curves together with their stability and two-parameter analysis of multistability regions bounded by the loci of fold bifurcations that occur along the resonance curves. In addition, the results of comsol are validated for the Mettler model of slender beams against an in-house constructed finite-element discretization scheme, the convergence of which is assessed for increasing number of finite elements.

The motion has been studied of a particle in a gravitational field perturbed by radiation pressure. By combining the formulation in the physical space variables with the KS variables we obtained explicit evidence for the existence of a surface of stable circular orbits with centers on an axis through the primary body. Furthermore, the effects of a sharp shadow on the two-dimensional unstable parabolic orbits were investigated. It was found that they do not survive the introduction of a shadow.

Most previous works on responsible conduct of research have focused on good practices in laboratory experiments. Because computation now rivals experimentation as a mode of scientific research, we sought to identify the responsibilities of researchers who develop or use computational modeling and simulation. We interviewed nineteen experts to collect examples of ethical issues from their experiences in conducting research with computational models. We gathered their recommendations for guidelines for computational research. Informed by these interviews, we describe the respective professional responsibilities of developers and users of computational models in research. In particular, we examine whether developers should disclose the full computational codes, and we explain how developers and users should minimize harms from improper uses of models.

A computational framework is proposed to path follow the periodic solutions of nonlinear spatially continuous systems and more general coupled multiphysics problems represented by systems of partial differential equations with time-dependent excitations. The set of PDEs is cast in first order differential form (in time) ?u = f(u; s; t; c) where u(s; t) is the vector collecting all state variables including the velocities/time rates, s is a space co-ordinate (here, one-dimensional systems are considered without lack of generality for the space dependence) and t denotes time. The vector field f depends, in general, not only on the classical state variables (such as positions and velocities) but also on the space gradients of the leading unknowns. The space gradients are introduced as part of the state variables. This is justified by the mathematical and computational requirements on the continuity in space up to the proper differential order of the space gradients associated with the unknown position vector field. The path following procedure employs, for the computation of the periodic solutions, only the evaluation of the vector field f. This part of the path following procedure within the proposed combined scheme was formerly implemented by Dankowicz and co-workers in a MATLAB software package called COCO. The here proposed procedure seeks to discretize the space dependence of the variables using finite elements based on Lagrangian polynomials which leads to a discrete form of the vector field f: A concurrent bifurcation analysis is carried out by calculating the eigenvalues of the monodromy matrix. A hinged-hinged nonlinear beam subject to a primary-resonance harmonic transverse load or to a parametric-resonance horizontal end displacement is considered as a case study. Some primary-resonance frequency-response curves are calculated along with their stability to assess the convergence of the discretization scheme. The frequency-response curves are shown to be in close agreement with those calculated by direct integration of the PDEs through the FE software called COMSOL Multiphysics. Besides primary-resonance direct forcing conditions, also parametric forcing causing the principal parametric resonance of the lowest two bending modes is considered through construction of the associated transition curves. The proposed approach integrates algorithms from the finite element and bifurcation domains thus enabling an accurate and effective unfolding of the bifurcation and post-bifurcation scenarios of nonautonomous PDEs with the underlying structures. Copyright © 2011 by ASME.

Limit-switch sensors are input-output devices that switch operating state in reaction to the crossing of a threshold value of their input. These are used to monitor and control critical values of temperature, voltage, pressure, etc., in both consumer and industrial settings. This paper argues for exploiting nonsmooth fold bifurcations in the design of ultrafast and robust, resettable, electromechanical limit switches. Specifically, the discussion emphasizes the dramatic changes in system response associated with the onset of near-grazing, low-velocity contact in vibro-impacting systems. These include rapid transient dynamics away from a pre-grazing, periodic, steady-state trajectory following the onset of impacts and post-grazing steady-state trajectories with a distinctly different amplitude and frequency content. The results reported here include a review of an experimental and computational verification of the ultrafast transient growth rates that show a significant potential for dramatic improvement in sensor performance. Moreover, two novel candidate sensor designs are discussed that rely on the post-grazing response characteristics for device function. In the first instance, transduction of a change in the response periodicity following grazing in a mechanical device is detected in a coupled electromagnetic circuit. In the second instance, a snap-through post-grazing response forms the operating principle of a capacitively driven circuit protection device.

This paper compares the frequency-amplitude relationship for nonlinear oscillations of a geometrically nonlinear model of a slender beam in the absence of damping with the corresponding predictions from the Mettler model for the transverse motion. In particular, the analysis shows that the Mettler model fails to account for a constant, amplitude-independent shift in the nonlinear frequency relative to the linear frequency caused by rotary inertia terms.Copyright © 2010 by ASME

This paper presents a lumped-parameter, planar, physical model for capturing steady-state interactions between a deformable tire and a deformable terrain. The model includes effects due to tangential and normal compliance of the terrain surface as well as radial and circumferential compliance of the tire. In addition, shear failure and dry friction limits are introduced through a distinction between regions along the circumferential direction of the tire that are in local stick or slip, respectively. The time history of interactions between individual infinitesimal patches of the tire and the terrain is then described by a hybrid dynamical system, in which changes between individual phases of motion are triggered by characteristic events. The paper further illustrates the application of a non-linear regression technique for identification of the seven model parameters and, in selected cases additional unknown kinematic variables. Specifically, the model is fit to experimental load-deflection, gross-thrust, and net-pull data demonstrating good quantitative agreement.Copyright © 2010 by ASME

This paper considers discontinuity-induced bifurcations due to the onset and termination of hysteretic, capillary tip-sample interaction forces in a single-mechanical-mode model of intermittent-contact atomic-force microscopy. The theoretical analysis generalizes earlier results for a piecewise-linear hybrid dynamical system to establish the singular termination of branches of steady-state oscillations of the AFM cantilever at critical equilibrium separations corresponding to the grazing contact of the cantilever tip with a fluid layer deposited on the sample. It is shown that this termination is preceded by rapid changes in linearized stability characteristics with one characteristic multiplier going to plus or minus infinity in the deterministic model. The paper describes the application of a discontinuity-mapping technique that allows for unfolding the system response in the vicinity of the grazing condition and the critical equilibrium separation. Numerical simulations and results of parameter continuation are shown to closely agree with the predictions of the theoretical analysis.Copyright © 2010 by ASME

The focus of this paper is on the possibility of formulating a consistent and unambiguous forward-simulation model of planar rigid-body mechanical systems with isolated points of intermittent or sustained contact with rigid constraining surfaces in the presence of dry friction. In particular, the analysis considers paradoxical ambiguities associated with the coexistence of sustained contact and one or several alternative forward trajectories that include phases of free-flight motion. Special attention is paid to the so-called Painlev´e paradoxes where sustained contact is possible even if the contact-independent contribution to the normal acceleration would cause contact to cease. Here, through taking the infinite-sti ness limit of a compliant contact model, the ambiguity in the case of a condition of sustained stick is resolved in favour of sustained contact, whereas the ambiguity in the case of a condition of sustained slip is resolved by eliminating the possibility of reaching such a condition from an open set of initial conditions. A more significant challenge to the goal of an unambiguous forward-simulation model is a orded by the discovery of open sets of initial conditions and parameter values associated with the possibility of a left accumulation point of impacts or reverse chatter a transition to free flight through an infinite sequence of impacts with impact times accumulating from the right on a limit point and with impact velocities diverging exponentially away from the limit point, even where the contact-independent normal acceleration supports sustained contact. In this case, the infinite-sti ness limit of the compliant formulation establishes that, under a specific set of open conditions, the possibility of reverse chatter in the rigid-contact model is an irresolvable ambiguity in the forward dynamics based at the terminal point of a phase of sustained slip. Indeed, as the existence of a left-accumulation point of impacts is associated with a one-parameter family of possible forward trajectories, the ambiguity is of infinite multiplicity. The conclusions of the theoretical analysis are illustrated and validated through numerical analysis of an example single-rigid-body mechanical model.

This paper presents the application of the discontinuity-mapping technique to the analysis of discontinuity-induced bifurcations of periodic trajectories in an example piecewise smooth system in which changes in the vector field associated with the crossing of a discontinuity-surface depend on the direction of crossing. The analysis is motivatived by a hysteretic model of the capillary force interactions between an atomic-force-microscope cantilever probe tip and a nanoscale sample surface in the presence of a thin liquid film on the tip and the surface and operating in intermittent-contact mode. The analysis predicts the sudden termination of branches of periodic system responses at parameter values corresponding to grazing contact with the onset of the hysteretic force interactions. It further establishes the increase beyond all bounds of the magnitude of one of the eigenvalues of the linearization of a suitably defined Poincare mapping indicating the destabilizing influence of near-grazing contact.Copyright © 2008 by ASME

This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in Rn. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.

The onset of low-velocity collisions in vibro-impacting systems induces instabilities in the system dynamics that, when not checked, may result in sudden, and unanticipated discontinuous transitions between distinct steady-state responses. This paper illustrates this phenomenology in an example system that includes dry friction. Here, the instability is associated with the zero-velocity contact of an oscillatory unilateral constraint and a stationary mass suspended through a preloaded spring. The analysis summarizes observations on the passive response of the mass under variations in the oscillation amplitude of the constraint. A control strategy is subsequently shown to successfully suppress the instability. The paper concludes with suggestions for applications of this phenomenology as well as a description of similar observation in mechanical systems with or without friction and with rigid as well as compliant contact.

Near-grazing, low-velocity contact in vibro-impacting systems has been shown to result in dramatic changes in steady-state system response following rapid transient growth of deviations away from the pre-grazing steady-state response. In low-dimensional example systems such transitions are often associated with large jumps in response amplitude. Coupled with the rapidity of the transient dynamics, this phenomenology supports the design of limit-switch sensors that trigger at the onset of grazing contact. A particularly exciting area of application of such sensors, and one in which their implementation might offer particular advantages, is in the context of microelectromechanical structures. Here, desirable scaling effects, such as increased system frequencies, low damping, batch fabrication, and decreased packaging size, can be leveraged. Fabricating simple beam structures at the microscale is relatively easier than fabricating proof-mass-based lumped-parameter systems with elaborate suspension structures. Consequently, it often becomes necessary to account for the flexibility of participating mechanical members, for example doubly-clamped, silicon-based beam elements. Physical contact further poses modeling challenges, as the flexibility of the beam elements and that of the contact region necessitate a compliant, but very stiff model description. The present work investigates a sequence of reduced-order models for such a doubly-clamped beam, subject to capacitive electrostatic actuation and a low-compliance physical constraint localized at a point along the span of the beam. The objective is to determine whether grazing-induced transitions, characteristic of lumped-mass models, are retained in the flexible structure. Specifically, numerical simulations are employed to quantify the variations in the response amplitude following the onset of contact and to contrast these to a spreading of system energy across mechanical modes.Copyright © 2009 by ASME

Limit-switch sensors are input-output devices that switch operating state in reaction to the crossing of a threshold value of their input. These are used to monitor and control critical values of temperature, voltage, pressure, etc. in both consumer and industrial settings. This paper argues for exploiting nonsmooth fold bifurcations in the design of ultrafast and robust, resettable, electromechanical limit switches. Specifically, the discussion emphasizes the dramatic changes in system response associated with the onset of near-grazing, low-velocity contact in vibro-impacting systems. These include rapid transient dynamics away from a pre-grazing, periodic, steady-state trajectory following the onset of impacts and post-grazing steady-state trajectories with distinctly different amplitude and frequency content. The results reported here include an experimental and computational verification of the ultrafast transient growth rates that show a significant potential for dramatic improvement in sensor performance. Moreover, two novel candidate sensor designs are discussed that rely on the post-grazing response characteristics for device function. In the first instance, transduction of a change in the response periodicity following grazing in a mechanical device is detected in a coupled electromagnetic circuit. In the second instance, a snap-through post-grazing response forms the operating principle of a capacitively-driven circuit protection device.Copyright © 2009 by ASME

Vehicle manufacturers are constantly pushed to reduce the aerodynamic drag of vehicles, for example by constructing lower vehicles with less road clearance. This, however, reduces the available margin for oscillations within the suspension. If the oscillation amplitude exceeds a critical value, the suspension will impact a bumpstop. Under periodic excitation, the onset of low-velocity impacts is associated with a strong instability in favor of high-velocity impacts. Such impacts reduce comfort and could be damaging to the vehicle. Efforts should therefore be made to limit impact velocities with the bumpstop, for example by suppressing the instability associated with low-velocity impacts. This paper proposes a low-cost feedback-control strategy, based on making small adjustments to the position of the bumpstop, that serve to suppress the transition to high-velocity impacts with the bumpstop in the case of periodic excitation. The control law is derived from the theory of discontinuity maps. The results demonstrate that the feedback strategy works even when wheel-hop is present.

We study the power dissipated by the tip of an oscillating micron-scale cantilever as it interacts with a sample using a nonlinear model of the tip-surface force interactions that includes attractive, adhesive, repulsive, and capillary contributions. The force interactions of the model are entirely conservative and the dissipated power is due to the hysteretic nature of the interaction with the capillary fluid layer. Using numerical techniques tailored for nonlinear and discontinuous dynamical systems we compute the exact dissipated power over a range of experimentally relevant conditions. This is accomplished by computing precisely the fraction of oscillations that break the fluid meniscus. We find that the dissipated power as a function of the equilibrium cantilever-surface separation has a characteristic shape that we directly relate to the cantilever dynamics. Even for regions where the cantilever dynamics are highly irregular the fraction of oscillations breaking the fluid meniscus exhibits a simple trend. Using our results we also explore the accuracy of the often used harmonic approximation in determining dissipated power. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.2980057]

We study the nonlinear dynamics of a tapping mode atomic force microscope with tip-surface interactions that include attractive, repulsive, and capillary force contributions using numerical techniques tailored for hybrid or discontinuous dynamical systems that include forward-time simulation with event handling and numerical pseudo-arclength continuation. We find four branches of periodic solutions that are separated by windows of complex and irregular dynamics. The branches of periodic solutions end where the cantilever comes into grazing contact with event surfaces in state space, corresponding to the onset of capillary interactions and the onset of repulsive forces associated with contact. These windows of irregular dynamics are found to coexist with the periodic branches of solutions as well as exist beyond the termination of the periodic solution. Finally, we show that these details can be overlooked unless one is careful to sample the dynamics appropriately. (C) 2008 American Institute of Physics.

This paper presents the application of the discontinuity-mapping technique to the analysis of discontinuity-induced bifurcations of periodic trajectories in an example piecewise smooth system in which changes in the vector field associated with the crossing of a discontinuity-surface depend on the direction of crossing. The analysis is motivatived by a hysteretic model of the capillary force interactions between an atomic-force-microscope cantilever probe tip and a nanoscale sample surface in the presence of a thin liquid film on the tip and the surface and operating in intermittent-contact mode. The analysis predicts the sudden termination of branches of periodic system responses at parameter values corresponding to grazing contact with the onset of the hysteretic force interactions. It further establishes the increase beyond all bounds of the magnitude of one of the eigenvalues of the linearization of a suitably defined Poincare´ mapping indicating the destabilizing influence of near-grazing contact.

In this paper, dynamical systems analysis and optimization tools are used to investigate the local dynamic stability of periodic task-related motions of simple models of the lower-body musculoskeletal apparatus and to seek parameter values guaranteeing their stability. In particular, the dynamics of a two-link model of a leg undergoing periodic excitation through one or several contractile muscle elements corresponding to a simple knee-bending motion is studied. Several muscle models incorporating various active and passive elements are included and the notion of self-stabilization of the rigid-body dynamics through the imposition of muscle-like actuation is investigated. It is found that self-stabilization depends both on muscle architecture and configuration as well as the properties of the reference motion. Additionally, antagonistic muscles (flexor-extensor muscle couples) are shown to enable stable motions over larger ranges in parameter space and that even the simplest neuronal feedback mechanism can stabilize the repetitive motions. The work provides a review of the necessary concepts of stability and a commentary on existing incorrect results that have appeared in the literature on muscle self-stabilization.Copyright © 2007 by ASME

We numerically explore the nonlinear dynamics of the oscillating cantilever tip in tapping mode atomic force microscopy. The cantilever dynamics are determined by complex force interactions between the sample surface and the oscillating cantilever tip which are dominated by attractive, adhesive, and repulsive contributions depending on the instantaneous position of the cantilever. We use a model proposed by Zitzler et al that includes a capillary force interaction due to the thin film of water that covers all surfaces as a result of ambient humidity. As the cantilever approaches the surface a meniscus is formed and as the cantilever retracts this water layer forms a neck and eventually breaks. This introduces hysteresis since the formation of the meniscus and the breaking of the water neck occur at different spatial locations during an oscillation of the cantilever. Using forward-time simulation with event handling techniques tailored for situations with rapid changes in force interactions we find three classes of steady-state dynamics: (i) a branch of solutions with periodic dynamics and large amplitude of oscillation; (ii) a branch of solutions with periodic dynamics and small amplitude of oscillation; (iii) windows of irregular aperiodic dynamics. We quantify the global basins of attraction for these solutions by performing a large set of numerical simulations over a wide range of initial conditions. Our findings provide a useful framework for further studies interested in controlling these dynamics.

This paper presents the application of a newly developed computational toolbox, TC-HAT (TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of an example hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, perioddoubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.Copyright © 2007 by ASME

Tapping-mode atomic force microscopy provides a means for successful and non-intrusive characterization of soft physical and biological structures at the nanoscale. Its full potential can only be realized, provided that the response of the oscillating probe tip to the strongly nonlinear, near-field force interactions with the structure and the intermittency of contact can be accurately modelled, analysed, controlled and interpreted. To this end, this paper reviews some experimental observations of fundamentally nonlinear behaviour of the tip dynamics. It discusses the nonlinear phenomenology that explains their presence in the tapping-mode operation of the atomic force microscope. Particular emphasis is placed on the coexistence of different steady-state responses and their origin in transitions across regions of rapidly varying force characteristics. The heuristics of a recently developed method for treating such transitions are presented and insights into its implications are drawn from related micro- and nanoscale applications.

This paper considers the dependence of the performance of a Braille printer impact hammer on system parameters, specifically the lag time between subsequent current pulses and the location of a back stop constraining the oscillations of the hammer core. Here, the goal is to improve the printer speed while maintaining readability of the Braille type and preventing tears in the paper. Based on observations of the limit sets for the hammer-core dynamics, a low-cost feedback control algorithm is proposed for affecting the stability of otherwise unstable periodic oscillations with more desirable operating characteristics.