Nicholas Boechler

Bifurcation-based acoustic switching and rectification

Switches and rectification devices are fundamental components used for controlling the flow of energy in numerous applications. Thermal and acoustic rectifiers have been proposed for use in biomedical ultrasound applications, thermal computers, energy- saving and -harvesting materials, and direction-dependent insulating materials. In all these systems the transition between transmission states is smooth with increasing signal amplitudes. This limits their effectiveness as switching and logic devices, and reduces their sensitivity to external conditions as sensors. Here we overcome these limitations by demonstrating a new mechanism for tunable rectification that uses bifurcations and chaos. This mechanism has a sharp transition between states, which can lead to phononic switching and sensing. We present an experimental demonstration of this mechanism, applied in a mechanical energy rectifier operating at variable sonic frequencies. The rectifier is a granular crystal, composed of a statically compressed one-dimensional array of particles in contact, containing a light mass defect near a boundary. As a result of the defect, vibrations at selected frequencies cause bifurcations and a subsequent jump to quasiperiodic and chaotic states with broadband frequency content. We use this combination of frequency filtering and asymmetrically excited bifurcations to obtain rectification ratios greater than 10(4). We envisage this mechanism to enable the design of advanced photonic, thermal and acoustic materials and devices.

Discrete Breathers in One-Dimensional Diatomic Granular Crystals

We report the experimental observation of modulational instability and discrete breathers in a one-dimensional diatomic granular crystal composed of compressed elastic beads that interact via Hertzian contact. We first characterize their effective linear spectrum both theoretically and experimentally. We then illustrate theoretically and numerically the modulational instability of the lower edge of the optical band. This leads to the dynamical formation of long-lived breather structures, whose families of solutions we compute throughout the linear spectral gap. Finally, we experimentally observe the manifestation of the modulational instability and the resulting generation of localized breathing modes with quantitative characteristics that agree with our numerical results.

Tunable Vibrational Band Gaps in One-Dimensional Diatomic Granular Crystals with Three-Particle Unit Cells

We investigate the tunable vibration filtering properties of statically compressed one-dimensional diatomic granular crystals composed of arrays of stainless steel spheres and cylinders interacting via Hertzian contact. The arrays consist of periodically repeated three-particle unit cells (sphere-cylinder-sphere) in which the length of the cylinder is varied systematically. We investigate the response of these granular crystals, given small amplitude dynamic displacements relative to those due to the static compression, and characterize their linear frequency spectrum. We find good agreement between theoretical dispersion relation analysis (for infinite systems), state-space analysis (for finite systems), and experiments. We report the observation of three distinct pass bands separated by two finite band gaps, and show their tunability for variations in cylinder length and static compression.

Interaction of a contact resonance of microspheres with surface acoustic waves.

We study the interaction of surface acoustic waves (SAWs) with a contact-based vibrational resonance of 1 μm silica microspheres forming a two-dimensional granular crystal adhered to a substrate. The laser-induced transient grating technique is used to excite SAWs and measure their dispersion. The measured dispersion curves exhibit avoided crossing behavior due to the hybridization of the SAWs with the microsphere resonance. We compare the measured dispersion curves with those predicted by our analytical model and find excellent agreement. The approach presented can be used to study the contact mechanics and adhesion of micro- and nanoparticles as well as the dynamics of microscale granular crystals.

Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals

We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy profile. We investigate their existence, structure, and stability throughout the band gap of the linear spectrum and classify them into four regimes: a regime near the lower optical band edge of the linear spectrum, a moderately discrete regime, a strongly discrete regime that lies deep within the band gap of the linearized version of the system, and a regime near the upper acoustic band edge. We contrast discrete breathers in anharmonic Fermi-Pasta-Ulam (FPU)-type diatomic chains with those in diatomic granular crystals, which have a tensionless interaction potential between adjacent particles, and note that the asymmetric nature of the tensionless interaction potential can lead to hybrid bulk-surface localized solutions.

Nonlinear waves in disordered diatomic granular chains.

We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a spin, so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: in low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatiotemporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for a transition between the two regimes. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves.

Nonlinear Periodic Phononic Structures and Granular Crystals

This chapter describes the dynamic behavior of nonlinear periodic phononic structures, along with how such structures can be utilized to affect the propagation of mechanical waves. Granular crystals are one type of nonlinear periodic phononic structure and are the focus of this chapter. The chapter begins with a brief history of nonlinear lattices and an introduction to granular crystals. This is followed by a summary of past and recent work on one-dimensional (1D) and two-dimensional (2D) granular crystals, which is categorized according to the crystals’ periodicity and dynamical regime. The chapter is concluded with a commentary by the authors, which discusses several possible future directions relating to granular crystals and other nonlinear periodic phononic structures. Throughout this chapter, a richness of nonlinear dynamic effects that occur in granular crystals is revealed, including a plethora of phenomena with no linear analog such as solitary waves, discrete breathers, tunable frequency band gaps, bifurcations, and chaos. Furthermore, in addition to the description of fundamental nonlinear phenomena, the authors describe how such phenomena can enable novel engineering devices and be applied to other nonlinear periodic systems.

Hysteresis loops and multi-stability: From periodic orbits to chaotic dynamics (and back) in diatomic granular crystals

We consider a statically compressed diatomic granular crystal, consisting of alternating aluminum and steel spheres. The combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. Through both numerical simulations and experiments, we find that the interplay of nonlinear surface modes with modes caused by the driver create the possibility, as the driving amplitude is increased, of limit cycle saddle-node bifurcations beyond which the dynamics of the system becomes chaotic. In this chaotic state, part of the applied energy can propagate through the chain. We also find that the chaotic branch depends weakly on the driving frequency, and speculate a connection between the chaotic dynamics with the gap openings between the spheres. Finally, we observe hysteretic dynamics and an interval of multi-stability involving stable periodic solutions and chaotic ones.

Complex Contact-Based Dynamics of Microsphere Monolayers Revealed by Resonant Attenuation of Surface Acoustic Waves.

Contact-based vibrations play an essential role in the dynamics of granular materials. Significant insights into vibrational granular dynamics have previously been obtained with reduced-dimensional systems containing macroscale particles. We study contact-based vibrations of a two-dimensional monolayer of micron-sized spheres on a solid substrate that forms a microscale granular crystal. Measurements of the resonant attenuation of laser-generated surface acoustic waves reveal three collective vibrational modes that involve displacements and rotations of the microspheres, as well as interparticle and particle-substrate interactions. To identify the modes, we tune the interparticle stiffness, which shifts the frequency of the horizontal-rotational resonances while leaving the vertical resonance unaffected. From the measured contact resonance frequencies we determine both particle-substrate and interparticle contact stiffnesses and find that the former is an order of magnitude larger than the latter. This study paves the way for investigating complex contact-based dynamics of microscale granular crystals and yields a new approach to studying micro- to nanoscale contact mechanics in multiparticle networks.

Analytical and Experimental Analysis of Bandgaps in Nonlinear one Dimensional Periodic Structures

Wave propagation characteristics of nonlinear one-dimensional periodic structures are investigated analytically, numerically and experimentally. A novel perturbation analysis is first applied to predict the band gap location and extent in terms of linear and nonlinear system parameters. Approximate closed-form expressions capture the effect of nonlinearities on dispersion and depict amplitude dependent cut-off frequencies. The predictions from the perturbation analysis are verified through numerical simulations of harmonic wave motion. Results indicate the possibility of input amplitude as a tuning parameter through which cut-off frequencies can be adjusted to achieve filtering properties over selected frequency ranges. A periodic diatomic chain of stainless steel spheres alternating with aluminium spheres is experimentally investigated. The dynamic behavior of the chain is governed by Hertzian interaction of spheres and by a compressive pre-load which can be adjusted to obtain linear, weakly nonlinear and highly nonlinear behavior. For a weakly nonlinear case, preliminary results in experiments show the tendency for a shift in the band gap edges by varying input amplitude. The paper is a work in progress, for which the experimental results for a weakly nonlinear system are interpreted by the perturbation analysis developed for a specific case of linear and nonlinear power law interaction of exponent 3/2