Spectral energy scattering and targeted energy transfer in phononic lattices with local vibroimpact nonlinearities.

We propose a method for manipulating wave propagation in phononic lattices by employing local vibroimpact (VI) nonlinearities to scatter energy across the underlying linear band structure of the lattice, and transfer energy from lower to higher optical bands. First, a one-dimensional, two-band phononic lattice with embedded VI unit cells is computationally studied to demonstrate that energy is scattered in the wave number domain, and this nonlinear scattering mechanism depends on the energy of the propagating wave. Next, a four-band lattice is studied with a similar technique to demonstrate the concept of nonresonant interband targeted energy transfer (IBTET) and to establish analogous scaling relations with respect to energy. Both phononic lattices are shown to exhibit a maximum energy transfer at moderate input energies, followed by a power-law decay of relative energy transfer either to the wave number domain or between bands on input energy. Last, the nonlinear normal modes (NNMs) of a reduced order model (ROM) of a VI unit cell are computed with the method of numerical continuation to provide a physical interpretation of the IBTET scaling with respect to energy. We show that the slope of the ROM's frequency-energy evolution for 1:1 resonance matches well with IBTET scaling in the full lattice. Moreover, the phase-space trajectories of the NNM solutions elucidate how the power-law scaling is related to the nonlinear dynamics of the VI unit cell.

Buckling-induced transmission switching in phononic waveguidesa).

On-chip phononic circuits tailor the transmission of elastic waves and couple to electronics and photonics to enable new signal manipulation capabilities. Phononic circuits rely on waveguides that transmit elastic waves within desired frequency passbands, which are typically designed based on the Bloch modes of the constitutive unit cell of the waveguide, assuming periodicity. Acoustic microelectromechanical system waveguides composed of coupled drumhead resonators offer megahertz operation frequencies for applications in acoustic switching. Here, we construct a reduced-order model (ROM) to demonstrate the mechanism of transmission switching in coupled drumhead-resonator waveguides. The ROM considers the mechanics of buckling under the effect of temperature variation. Each unit cell has two degrees of freedom: translation to capture the symmetric bending modes and angular motion to capture the asymmetric bending modes of the membranes. We show that thermoelastic buckling induces a phase transition triggered by temperature variation, causing the localization of the first-passband modes, similar to Anderson localization caused by disorders. The proposed ROM is essential to understanding these phenomena since Bloch mode analysis fails for weakly disordered (<5%) finite waveguides due to the disorder amplification caused by the thermoelastic buckling. The illustrated transmission control can be extended to two-dimensional circuits in the future.

Predictive Helmet Optimization Framework Based on Reduced-Order Modeling of the Brain Dynamics.

Sports-related traumatic brain injuries (TBIs) are among the leading causes of head injuries in the world. Use of helmets is the main protective measure against this epidemic. The design criteria for the majority of the helmets often only consider the kinematics of the head. This approach neglects the importance of regional deformations of the brain especially near the deep white matter structures such as the corpus callosum (CC) which have been implicated in mTBI studies. In this work, we develop a dynamical reduced-order model of the skull-brain-helmet system to analyze the effect of various helmet parameters on the dynamics of the head and CC. Here, we show that the optimal head-helmet coupling values that minimize the CC dynamics are different from the ones that minimize the skull and brain dynamics (at some kinematics, up to two times stiffer for the head motion mitigation). By comparing our results with experimental impact tests performed on seven different helmets for five different sports, we found that the football helmets with an absorption of about 65-75% of the impact energy had the best performance in mitigating the head motion. Here, we found that none of the helmets are effective in protecting the CC from harmful impact energies. Our computational results reveal that the origin of the difference between the properties of a helmet mitigating the CC motion vs. the head motion is nonlinear vs. linear dynamics. Unlike the globally linear behavior of the head dynamics, we demonstrate that the CC exhibits nonlinear mechanical response similar to an energy sink. This means that there are scenarios where, at the instant of impact, the CC does not undergo extreme motions, but these may occur with a time delay as it absorbs shock energy from other parts of the brain. These findings hint at the importance of considering tissue level dynamics in designing new helmets.

Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry.

We study nonreciprocal wave transmission across the interface of two dissimilar granular media separated by an elastic solid medium. Specifically, a left, larger-scale and a right smaller-scale granular media composed of two-dimensional, initially uncompressed hexagonally packed granules are interfacing with an intermediate linearly elastic solid, modeled either as a thin elastic plate or a linear Euler-Bernoulli beam. The granular media are modeled by discrete elements and the elastic solid by finite elements assuming a plane stress approximation for the thin plate. Accounting for the combined effects of Hertzian, frictional and rotational interactions in the granular media, as well as the highly discontinuous interfacial effects between the (discrete) granular media and the (continuous) intermediate elastic solid, the nonlinear acoustics of the integrated system is computationally studied subject to a half-sine shock excitation applied to a boundary granule of either the left or right granular medium. The highly discontinuous and nonlinear interaction forces coupling the granular media to the elastic solid are accurately computed through an algorithm with interrelated iteration and interpolation at successive adaptive time steps. Numerical convergence is ensured by monitoring the (linearized) eigenvalues of a nonlinear map of interface forces at each (variable) time step. Due to the strong nonlinearity and hierarchical asymmetry of the left and right granular media, time scale disparity occurs in the response of the interface which breaks acoustic reciprocity. Specifically, depending on the location and intensity of the applied shock, propagating wavefronts are excited in the granular media, which, in turn, excite either (slow) low-frequency vibrations or (fast) high-frequency acoustics in the intermediate elastic medium. This scale disparity is due to the size disparity of the left and right granular media, which yields drastically different wave speeds in the resulting propagating wavefronts. As a result, the continuum part of the interface responds with either low-frequency vibrations-when the shock is applied to the larger-scale granular medium, or high-frequency waves-when the shock is applied to the smaller-scale granular medium. This provides the fundamental mechanism for breaking reciprocity in the interface. The nonreciprocal interfacial acoustics studied here apply to a broad class of asymmetric hybrid (discrete-continuum) nonlinear systems and can inform predictive designs of highly effective granular shock protectors or granular acoustic diodes.

Buckling-Mediated Phase Transitions in Nano-Electromechanical Phononic Waveguides.

Waveguides for mechanical signal transmission in the megahertz to gigahertz regimes enable on-chip phononic circuitry, which brings new capabilities complementing photonics and electronics. Lattices of coupled nano-electromechanical drumhead resonators are suitable for these waveguides due to their high Q-factor and precisely engineered band structure. Here, we show that thermally induced elastic buckling of such resonators causes a phase transition in the waveguide leading to reversible control of signal transmission. Specifically, when cooled, the lowest-frequency transmission band associated with the primary acoustic mode vanishes. Experiments show the merging of the lower and upper band gaps, such that signals remain localized at the excitation boundary. Numerical simulations show that the temperature-induced destruction of the pass band is a result of inhomogeneous elastic buckling, which disturbs the waveguide's periodicity and suppresses the wave propagation. Mechanical phase transitions in waveguides open opportunities for drastic phononic band reconfiguration in on-chip circuitry and computing.

A Nonlinear Reduced-Order Model of the Corpus Callosum Under Planar Coronal Excitation.

Traumatic brain injury (TBI) is often associated with microstructural tissue damage in the brain, which results from its complex biomechanical behavior. Recent studies have shown that the deep white matter (WM) region of the human brain is susceptible to being damaged due to strain localization in that region. Motivated by these studies, in this paper, we propose a geometrically nonlinear dynamical reduced order model (ROM) to model and study the dynamics of the deep WM region of the human brain under coronal excitation. In this model, the brain hemispheres were modeled as lumped masses connected via viscoelastic links, resembling the geometry of the corpus callosum (CC). Employing system identification techniques, we determined the unknown parameters of the ROM, and ensured the accuracy of the ROM by comparing its response against the response of an advanced finite element (FE) model. Next, utilizing modal analysis techniques, we determined the energy distribution among the governing modes of vibration of the ROM and concluded that the demonstrated nonlinear behavior of the FE model might be predominantly due to the special geometry of the brain deep WM region. Furthermore, we observed that, for sufficiently high input energies, high frequency harmonics at approximately 45 Hz, were generated in the response of the CC, which, in turn, are associated with high-frequency oscillations of the CC. Such harmonics might potentially lead to strain localization in the CC. This work is a step toward understanding the brain dynamics during traumatic injury.

Breather arrest, localization, and acoustic non-reciprocity in dissipative nonlinear lattices.

The effect of on-site damping on breather arrest, localization, and non-reciprocity in strongly nonlinear lattices is analytically and numerically studied. Breathers are localized oscillatory wavepackets formed by nonlinearity and dispersion. Breather arrest refers to breather disintegration over a finite "penetration depth" in a dissipative lattice. First, a simplified system of two nonlinearly coupled oscillators under impulsive excitation is considered. The exact relation between the number of beats (energy exchanges between oscillators), the excitation magnitude, and the on-site damping is derived. Then, these analytical results are correlated to those of the semi-infinite extension of the simplified system, where breather penetration depth is governed by a similar law to that of the finite beats in the simplified system. Finally, motivated by the experimental results of Bunyan, Moore, Mojahed, Fronk, Leamy, Tawfick, and Vakakis [Phys. Rev. E 97, 052211 (2018)], breather arrest, localization, and acoustic non-reciprocity in a non-symmetric, dissipative, strongly nonlinear lattice are studied. The lattice consists of repetitive cells of linearly grounded large-scale particles nonlinearly coupled to small-scale ones, and linear intra-cell coupling. Non-reciprocity in this lattice yields either energy localization or breather arrest depending on the position of excitation. The nonlinear acoustics governing non-reciprocity, and the surprising effects of existence of linear components in the coupling nonlinear stiffnesses, in the acoustics, are investigated.

Realization by impedance discontinuity of a unidirectional wave in a duct with harmonically perturbed uniform mean flow.

This paper presents a study of intentionally induced acoustic mode complexity in rigid-walled ducts of separable geometry and with uniform mean flow. An intermediately located perforated plate conceptualized as an impedance discontinuity is employed to maximize the acoustic mode complexity, in turn producing a unidirectional traveling wave from the source to the impedance discontinuity. The impedance of the perforated plate for realization of a unidirectional traveling wave is derived analytically and is found to be a function of the modal wavenumbers, the Mach number of the mean flow, the position of the perforated plate, and the termination impedance. The conditions derived analytically are verified computationally by finite element analysis. A measure of acoustic mode complexity is defined and also evaluated from the finite element analysis. It is found that the realization of a unidirectional traveling wave is robust at low Mach number mean flows, except at the occurrence of resonances. The method presented in this work provides a strategy to control the transmission of acoustic energy in rigid-walled ducts of separable geometry in the presence of uniform mean flow.

Passive nonlinear targeted energy transfer.

Nonlinearity in dynamics and acoustics may be viewed as scattering of energy across frequencies/wavenumbers. This is in contrast with linear systems when no such scattering exists. Motivated by irreversible large-to-small-scale energy transfers in turbulent flows, passive targeted energy transfers (TET) in mechanical and structural systems incorporating intentional strong nonlinearities are considered. Transient or permanent resonance captures are basic mechanisms for inducing TET in such systems, as well as nonlinear energy scattering across scales caused by strongly nonlinear resonance interactions. Certain theoretical concepts are reviewed, and some TET applications are discussed. Specifically, it is shown that the addition of strongly nonlinear local attachments in an otherwise linear dynamical system may induce energy scattering across scales and 'redistribution' of input energy from large to small scales in the linear modal space, in similarity to energy cascades that occur in turbulent flows. Such effects may be intentionally induced in the design stage and may lead to improved performance, e.g. it terms of vibration and shock isolation or energy harvesting. In addition, a simple mechanical analogue in the form of a nonlinear planar chain of particles composed of linear stiffness elements but exhibiting strong nonlinearity due to kinematic and geometric effects is discussed, exhibiting similar energy scattering across scales in its acoustics. These results demonstrate the efficacy of intentional utilization of strong nonlinearity in design to induce predictable and controlled intense multi-scale energy transfers in the dynamics and acoustics of a broad class of systems and structures, thus achieving performance objectives that would be not possible in classical linear settings.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Acoustic nonreciprocity in a lattice incorporating nonlinearity, asymmetry, and internal scale hierarchy: Experimental study.

In linear time-invariant systems acoustic reciprocity holds by the Onsager-Casimir principle of microscopic reversibility, and it can be broken only by odd external biases, nonlinearities, or time-dependent properties. Recently it was shown that one-dimensional lattices composed of a finite number of identical nonlinear cells with internal scale hierarchy and asymmetry exhibit nonreciprocity both locally and globally. Considering a single cell composed of a large scale nonlinearly coupled to a small scale, local dynamic nonreciprocity corresponds to vibration energy transfer from the large to the small scale, but absence of energy transfer (and localization) from the small to the large scale. This has been recently proven both theoretically and experimentally. Then, considering the entire lattice, global acoustic nonreciprocity has been recently proven theoretically, corresponding to preferential energy transfer within the lattice under transient excitation applied at one of its boundaries, and absence of similar energy transfer (and localization) when the excitation is applied at its other boundary. This work provides experimental validation of the global acoustic nonreciprocity with a one-dimensional asymmetric lattice composed of three cells, with each cell incorporating nonlinearly coupled large and small scales. Due to the intentional asymmetry of the lattice, low impulsive excitations applied to one of its boundaries result in wave transmission through the lattice, whereas when the same excitations are applied to the other end, they lead in energy localization at the boundary and absence of wave transmission. This global nonreciprocity depends critically on energy (i.e., the intensity of the applied impulses), and reduced-order models recover the nonreciprocal acoustics and clarify the nonlinear mechanism generating nonreciprocity in this system.

Ultimate Decoupling between Surface Topography and Material Functionality in Atomic Force Microscopy Using an Inner-Paddled Cantilever.

Atomic force microscopy (AFM) has been widely utilized to gain insight into various material and structural functionalities on the nanometer scale, leading to numerous discoveries and technologies. Despite the phenomenal success in applying AFM to the simultaneous characterization of topological and functional properties of materials, it has continuously suffered from the crosstalk between the observables, causing undesirable artifacts and complicated interpretations. Here, we introduce a two-field AFM probe, namely an inner-paddled cantilever integrating two discrete pathways such that they respond independently to the variations in surface topography and material functionality. Hence, the proposed design allows reliable and potentially quantitative determination of functional properties. In this paper, the efficacy of the proposed design has been demonstrated via piezoresponse force microscopy of periodically poled lithium niobate and collagen, although it can also be applied to other AFM methods such as AFM-based infrared spectroscopy and electrochemical strain microscopy.

Inducing a nonreflective airborne discontinuity in a circular duct by using a nonresonant side branch to create mode complexity.

A nonreflective airborne discontinuity is created in a one-dimensional rigid-walled duct when the mode complexity introduced by a nonresonant side branch reaches a maximum, so that a sound wave can be spatially separated into physical regions of traveling and standing waves. The nonresonance of the side branch is demonstrated, the mode complexity is quantified, and a computational method to optimize side-branch parameters to maximize mode complexity in the duct in the presence of three-dimensional effects is presented. The optimal side-branch parameters that maximize the mode complexity and thus minimize reflection are found using finite element analysis and a derivative-free optimization routine. Sensitivity of mode complexity near the optimum with respect to side-branch parameters is then examined. The results show reflection from the impedance discontinuity in the duct can be reduced nearly to zero, providing a practical means of achieving a nonreflective discontinuity for a plane wave propagating in a duct of finite length.

Nonreciprocity in the dynamics of coupled oscillators with nonlinearity, asymmetry, and scale hierarchy.

In linear time-invariant dynamical and acoustical systems, reciprocity holds by the Onsager-Casimir principle of microscopic reversibility, and this can be broken only by odd external biases, nonlinearities, or time-dependent properties. A concept is proposed in this work for breaking dynamic reciprocity based on irreversible nonlinear energy transfers from large to small scales in a system with nonlinear hierarchical internal structure, asymmetry, and intentional strong stiffness nonlinearity. The resulting nonreciprocal large-to-small scale energy transfers mimic analogous nonlinear energy transfer cascades that occur in nature (e.g., in turbulent flows), and are caused by the strong frequency-energy dependence of the essentially nonlinear small-scale components of the system considered. The theoretical part of this work is mainly based on action-angle transformations, followed by direct numerical simulations of the resulting system of nonlinear coupled oscillators. The experimental part considers a system with two scales-a linear large-scale oscillator coupled to a small scale by a nonlinear spring-and validates the theoretical findings demonstrating nonreciprocal large-to-small scale energy transfer. The proposed study promotes a paradigm for designing nonreciprocal acoustic materials harnessing strong nonlinearity, which in a future application will be implemented in designing lattices incorporating nonlinear hierarchical internal structures, asymmetry, and scale mixing.

Utilizing intentional internal resonance to achieve multi-harmonic atomic force microscopy.

During dynamic atomic force microscopy (AFM), the deflection of a scanning cantilever generates multiple frequency terms due to the nonlinear nature of AFM tip-sample interactions. Even though each frequency term is reasonably expected to encode information about the sample, only the fundamental frequency term is typically decoded to provide topographic mapping of the measured surface. One of main reasons for discarding higher harmonic signals is their low signal-to-noise ratio. Here, we introduce a new design concept for multi-harmonic AFM, exploiting intentional nonlinear internal resonance for the enhancement of higher harmonics. The nonlinear internal resonance, triggered by the non-smooth tip-sample dynamic interactions, results in nonlinear energy transfers from the directly excited fundamental bending mode to the higher-frequency mode and, hence, enhancement of the higher harmonic of the measured response. It is verified through detailed theoretical and experimental study that this AFM design can robustly incorporate the required internal resonance and enable high-frequency AFM measurements. Measurements on an inhomogeneous polymer specimen demonstrate the efficacy of the proposed design, namely that the higher harmonic of the measured response is capable of enhanced simultaneous topography imaging and compositional mapping, exhibiting less crosstalk with an abrupt height change.

Mechanisms for impulsive energy dissipation and small-scale effects in microgranular media.

We study impulse response in one-dimensional homogeneous microgranular chains on a linear elastic substrate. Microgranular interactions are analytically described by the Schwarz contact model which includes nonlinear compressive as well as snap-to and from-contact adhesive effects forming a hysteretic loop in the force deformation relationship. We observe complex transient dynamics, including disintegration of solitary pulses, local clustering, and low-to-high-frequency energy transfers resulting in enhanced energy dissipation. We study in detail the underlying dynamics of cluster formation in the impulsively loaded medium and relate enhanced energy dissipation to the rate of cluster formation. These unusual and interesting dynamical phenomena are shown to be robust over a range of physically feasible conditions and are solely scale effects since they are attributed to surface forces, which have no effect at the macroscale. We establish a universal relation between the reclustering rate and the effective damping in these systems. Our findings demonstrate that scale effects generating new nonlinear features can drastically affect the dynamics and acoustics of microgranular materials.

Complex nonlinear dynamics in the limit of weak coupling of a system of microcantilevers connected by a geometrically nonlinear tunable nanomembrane.

Intentional utilization of geometric nonlinearity in micro/nanomechanical resonators provides a breakthrough to overcome the narrow bandwidth limitation of linear dynamic systems. In past works, implementation of intentional geometric nonlinearity to an otherwise linear nano/micromechanical resonator has been successfully achieved by local modification of the system through nonlinear attachments of nanoscale size, such as nanotubes and nanowires. However, the conventional fabrication method involving manual integration of nanoscale components produced a low yield rate in these systems. In the present work, we employed a transfer-printing assembly technique to reliably integrate a silicon nanomembrane as a nonlinear coupling component onto a linear dynamic system with two discrete microcantilevers. The dynamics of the developed system was modeled analytically and investigated experimentally as the coupling strength was finely tuned via FIB post-processing. The transition from the linear to the nonlinear dynamic regime with gradual change in the coupling strength was experimentally studied. In addition, we observed for the weakly coupled system that oscillation was asynchronous in the vicinity of the resonance, thus exhibiting a nonlinear complex mode. We conjectured that the emergence of this nonlinear complex mode could be attributed to the nonlinear damping arising from the attached nanomembrane.

Silicon nano-mechanical resonators fabricated by using tip-based nanofabrication.

We report fabrication of silicon nano-mechanical resonators where the key nanolithography step is performed by using tip-based nanofabrication (TBN). Specifically, a heated atomic force microscope tip deposited polystyrene nanowires that were used together with a lithographically patterned aluminum to serve as an etch mask for silicon resonators their anchors. Using this nanofabrication technique, we demonstrate the fabrication of different types of silicon nano-mechanical resonator devices, including those that are either singly or doubly clamped and having either straight or curvilinear features. Typical dimensions for the width and thickness of these devices is in the range of several hundred nanometers. We characterized the mechanical resonance properties of these devices by using laser Doppler vibrometry and compared the measured response with finite element simulations. Typical resonance frequency values ranged from 1 to 3 MHz and typical quality factor values ranged from 100 to 150. The combination of TBN along with conventional microfabrication processes could help to realize new types of nano-devices.

Improved atomic force microscope infrared spectroscopy for rapid nanometer-scale chemical identification.

Atomic force microscope infrared spectroscopy (AFM-IR) can perform IR spectroscopic chemical identification with sub-100 nm spatial resolution, but is relatively slow due to its low signal-to-noise ratio (SNR). In AFM-IR, tunable IR laser light is incident upon a sample, which results in a rise in temperature and thermomechanical expansion of the sample. An AFM tip in contact with the sample senses this nanometer-scale photothermal expansion. The tip motion induces cantilever vibrations, which are measured either in terms of the peak-to-peak amplitude of time-domain data or the integrated magnitude of frequency-domain data. Using a continuous Morlet wavelet transform to the cantilever dynamic response, we show that the cantilever dynamics during AFM-IR vary as a function of both time and frequency. Based on the observed cantilever response, we tailor a time-frequency-domain filter to identify the region of highest vibrational energy. This approach can increase the SNR of the AFM cantilever signal, such that the throughput is increased 32-fold compared to state-of-the art procedures. We further demonstrate significant increases in AFM-IR imaging speed and chemical identification of nanometer-scale domains in polymer films.

Modeling and measurement of geometrically nonlinear damping in a microcantilever-nanotube system.

Nonlinear mechanical systems promise broadband resonance and instantaneous hysteretic switching that can be used for high sensitivity sensing. However, to introduce nonlinear resonances in widely used microcantilever systems, such as AFM probes, requires driving the cantilever to an amplitude that is too large for any practical applications. We introduce a novel design for a microcantilever with a strong nonlinearity at small cantilever oscillation amplitude arising from the geometrical integration of a single BN nanotube. The dynamics of the system was modeled theoretically and confirmed experimentally. The system, besides providing a practical design of a nonlinear microcantilever-based probe, demonstrates also an effective method of studying the nonlinear damping properties of the attached nanotube. Beyond the typical linear mechanical damping, the nonlinear damping contribution from the attached nanotube was found to be essential for understanding the dynamical behavior of the designed system. Experimental results obtained through laser microvibrometry validated the developed model incorporating the nonlinear damping contribution.

Frequency bands of strongly nonlinear homogeneous granular systems.

Recent numerical studies on an infinite number of identical spherical beads in Hertzian contact showed the presence of frequency bands [Jayaprakash, Starosvetsky, Vakakis, Peeters, and Kerschen, Nonlinear Dyn. 63, 359 (2011)]. These bands, denoted here as propagation and attenuation bands (PBs and ABs), are typically present in linear or weakly nonlinear periodic media; however, their counterparts are not intuitive in essentially nonlinear periodic media where there is a complete lack of classical linear acoustics, i.e., in "sonic vacua." Here, we study the effects of PBs and ABs on the forced dynamics of ordered, uncompressed granular systems. Through numerical and experimental techniques, we find that the dynamics of these systems depends critically on the frequency and amplitude of the applied harmonic excitation. For fixed forcing amplitude, at lower frequencies, the oscillations are large in amplitude and governed by strongly nonlinear and nonsmooth dynamics, indicating PB behavior. At higher frequencies the dynamics is weakly nonlinear and smooth, in the form of compressed low-amplitude oscillations, indicating AB behavior. At the boundary between the PB and the AB large-amplitude oscillations due to resonance occur, giving rise to collisions between beads and chaotic dynamics; this renders the forced dynamics sensitive to initial and forcing conditions, and hence unpredictable. Finally, we study asymptotically the near field standing wave dynamics occurring for high frequencies, well inside the AB.