Synchronization properties of self-sustained mechanical oscillators

We study, both analytically and numerically, the dynamics of mechanical oscillators kept in motion by a feedback force, which is generated electronically from a signal produced by the oscillators themselves. This kind of self-sustained systems may become standard in the design of frequency-control devices at microscopic scales. Our analysis is thus focused on their synchronization properties under the action of external forces, and on the joint dynamics of two to many coupled oscillators. Existence and stability of synchronized motion are assessed in terms of the mechanical properties of individual oscillators --namely, their natural frequencies and damping coefficients-- and synchronization frequencies are determined. Similarities and differences with synchronization phenomena in other coupled oscillating systems are emphasized.Comment: To appear in Phys. Rev.

Duffing revisited: Phase-shift control and internal resonance in self-sustained oscillators

We address two aspects of the dynamics of the forced Duffing oscillator which are relevant to the technology of micromechanical devices and, at the same time, have intrinsic significance to the field of nonlinear oscillating systems. First, we study the stability of periodic motion when the phase shift between the external force and the oscillation is controlled -contrary to the standard case, where the control parameter is the frequency of the force. Phase-shift control is the operational configuration under which self-sustained oscillators -and, in particular, micromechanical oscillators- provide a frequency reference useful for time keeping. We show that, contrary to the standard forced Duffing oscillator, under phase-shift control oscillations are stable over the whole resonance curve. Second, we analyze a model for the internal resonance between the main Duffing oscillation mode and a higher-harmonic mode of a vibrating solid bar clamped at its two ends. We focus on the stabilization of the oscillation frequency when the resonance takes place, and present preliminary experimental results that illustrate the phenomenon. This synchronization process has been proposed to counteract the undesirable frequency-amplitude interdependence in nonlinear time-keeping micromechanical devices

Large N reduction on a twisted torus

We consider SU(N) lattice gauge theory at infinite N defined on a torus with a CP invariant twist. Massless fermions are incorporated in an elegant way, while keeping them quenched. We present some numerical results which suggest that twisting can make numerical simulations of planar QCD more efficient.Comment: 14 pages, 2 figures, 1 tabl

Combinatorics of lattice paths with and without spikes

We derive a series of results on random walks on a d-dimensional hypercubic lattice (lattice paths). We introduce the notions of terse and simple paths corresponding to the path having no backtracking parts (spikes). These paths label equivalence classes which allow a rearrangement of the sum over paths. The basic combinatorial quantities of this construction are given. These formulas are useful when performing strong coupling (hopping parameter) expansions of lattice models. Some applications are described.Comment: Latex. 25 page

Problem Solvers, INC.

Our unit centers on building our students\u27 social-emotional skills. The skills we focus on are listening, following directions, self-control, and problem solving. Using various read-alouds, kid-friendly videos, and visuals, we hope to provide students with tools necessary to become problem-solvers in their daily lives.Students will be introduced to different age-appropriate solutions and practice them in an interactive and fun way. Ideally, this unit should be taught during the first weeks of school in effort to build a positive learning community

Large N reduction with overlap fermions

We revisit quenched reduction with fermions and explain how some old problems can be avoided using the overlap Dirac operator.Comment: Lattice2002(chiral) 3 pages, no figure

Synchronization of a forced self-sustained Duffing oscillator

We study the dynamics of a mechanical oscillator with linear and cubic forces -the Duffing oscillator- subject to a feedback mechanism that allows the system to sustain autonomous periodic motion with well-defined amplitude and frequency. First, we characterize the autonomous motion for both hardening and softening nonlinearities. Then, we analyze the oscillator's synchronizability by an external periodic force. We find a regime where, unexpectedly, the frequency range where synchronized motion is possible becomes wider as the amplitude of oscillations grows. This effect of nonlinearities may find application in technological uses of mechanical Duffing oscillators -for instance, in the design of time-keeping devices at the microscale- which we briefly review.Comment: To appear in Eur. Phys. J. Special Topic

Two-loop critical mass for Wilson fermions

We have redone a recent two-loop computation of the critical mass for Wilson fermions in lattice QCD by evaluating Feynman integrals with the coordinate-space method. We present the results for different types of infrared regularization. We confirm both the previous numerical estimates and the power of the coordinate-space method whenever high accuracy is needed.Comment: 13 LaTeX2e pages, 2 ps figures include

Diverse corrugation pattern in radially shrinking carbon nanotubes

Stable cross-sections of multi-walled carbon nanotubes subjected to electron-beam irradiation are investigated in the realm of the continuum mechanics approximation. The self-healing nature of sp$^2$ graphitic sheets implies that selective irradiation of the outermost walls causes their radial shrinkage with the remaining inner walls undamaged. The shrinking walls exert high pressure on the interior part of nanotubes, yielding a wide variety of radial corrugation patterns ({\it i.e.,} circumferentially wrinkling structures) in the cross section. All corrugation patterns can be classified into two deformation phases for which the corrugation amplitudes of the innermost wall differ significantly.Comment: 8 pages, 4 figure