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We investigate the spectral properties of one-dimensional spatially modulated nonlinear phononic lattices, and their evolution as a function of amplitude. In the linear regime, the stiffness modulations define a family of periodic and quasiperiodic lattices whose bandgaps host topological edge states localized at the boundaries of finite domains. With cubic nonlinearities, we show that edge states whose eigenvalue branch remains within the gap as amplitude increases remain localized, and therefore appear to be robust with respect to amplitude. In contrast, edge states whose corresponding branch approaches the bulk bands experience de-localization transitions. These transitions are predicted through continuation studies on the linear eigenmodes as a function of amplitude, and are confirmed by direct time domain simulations on finite lattices. Through our predictions, we also observe a series of amplitude-induced localization transitions as the bulk modes detach from the nonlinear bulk bands and become discrete breathers that are localized in one or more regions of the domain. Remarkably, the predicted transitions are independent of the size of the finite lattice, and exist for both periodic and quasiperiodic lattices. These results highlight the co-existence of topological edge states and discrete breathers in nonlinear modulated lattices. Their interplay may be exploited for amplitude-induced eigenstate transitions, for the assessment of the robustness of localized states, and as a strategy to induce discrete breathers through amplitude tuning.

A substantial challenge in guiding elastic waves is the presence of reflection and scattering at sharp edges, defects, and disorder. Recently, mechanical topological insulators have sought to overcome this challenge by supporting back-scattering resistant wave transmission. In this paper, we propose and experimentally demonstrate a reconfigurable electroacoustic topological insulator exhibiting an analog to the quantum valley Hall effect (QVHE). Using programmable switches, this phononic structure allows for rapid reconfiguration of domain walls and thus the ability to control back-scattering resistant wave propagation along dynamic interfaces for phonons lying in static and finite-frequency regimes. Accordingly, a graphene-like polyactic acid (PLA) layer serves as the host medium, equipped with periodically arranged and bonded piezoelectric (PZT) patches, resulting in two Dirac cones at the K points. The PZT patches are then connected to negative capacitance external circuits to break inversion symmetry and create nontrivial topologically protected bandgaps. As such, topologically protected interface waves are demonstrated numerically and validated experimentally for different predefined trajectories over a broad frequency range.

Directing and controlling flexural waves in thin plates along a curved trajectory over a broad frequency range is a significant challenge that has various applications in imaging, cloaking, wave focusing, and wireless power transfer circumventing obstacles. To date, all studies appeared controlling elastic waves in structures using periodic arrays of inclusions where these structures are narrowband either because scattering is efficient over a small frequency range, or the arrangements exploit Bragg scattering bandgaps, which themselves are narrowband. Here, we design and experimentally test a wave-bending structure in a thin plate by smoothly varying the plate’s rigidity (and thus its phase velocity). The proposed structures are (i) broadband, since the approach is frequency-independent and does not require bandgaps, and (ii) capable of bending elastic waves along convex trajectories with an arbitrary curvature.

In this paper we review recent progress on the analysis, experimental exploration, and application of elastic wave propagation in weakly nonlinear media and metamaterials. We provide a detailed technical discussion overviewing two broad areas of active research: (1) discrete nonlinear periodic systems and metamaterials, and (2) continuous nonlinear systems with a focus on nonlinear guided waves. The specific intent is to introduce the reader to asymptotic analysis methods currently being employed in the field of study, to highlight their results to date, and to motivate follow-on studies. Where appropriate, we include details on experimental explorations and envisioned applications, both of which have received relatively sparse attention to date.

We present analysis of dispersive wave propagation through a spatial interface between a linear and nonlinear monatomic chain using a proposed multiple scales perturbation approach. As such, we solve interface problems at each perturbation order (up to and including the second order) and assemble multi-harmonic solutions for transmitted and back-scattered waves. The perturbation approach predicts the existence of multiple nonlinear dispersion curves in the nonlinear subdomain. Using these curves, we further predict spatially-varying, higher-harmonic generation in the transmitted field. For propagating higher-harmonic waves, their amplitude is predicted to experience oscillatory spatial modulation due to the presence of multiple wavenumbers at each frequency, whereas for evanescent waves, their amplitude is predicted to undergo a saturating modulation. A transmission analysis quantifies the increase of the extra-harmonic frequency transmission, and the decrease of the fundamental frequency transmission, as the level of nonlinearity increases. Using direct numerical integration, we show that the perturbation predictions agree closely with numerical simulations for weakly nonlinear wave propagation. Lastly, informed by the perturbation results, we suggest a wave device which tailors the transmission of higher harmonics through the choice of the nonlinear subdomain’s length and/or the signal amplitude.

This paper explores a clearance-type nonlinear energy sink (NES) for increasing electrical energy harvested from non-stationary mechanical waves, such as those encountered during impact and intermittent events. The key idea is to trap energy in the NES such that it can be harvested over a time period longer than that afforded by the passing disturbance itself. Analytical, computational, and experimental techniques are employed to optimize the energy sink, explore qualitative behavior (to include bifurcations), and verify enhanced performance. Unlike traditionally studied single-DOF NESs, both subdomains of the NES (i.e., on either side of the clearance) contain displaceable degrees of freedom, increasing the complexity of the analytical solution approach. However, closed-form solutions are found which quantify the relationship between the impact amplitude and the energy produced, parameterized by system properties such as the harvester effective resistance, the clearance gap, and the domain mass and stiffness. Bifurcation diagrams and trends therein provide insight into the number and state of impact events at the NES as excitation amplitude increases. Moreover, a closed-form Poincaré map is derived which maps one NES impact location to the next, greatly simplifying the analysis while providing an important tool for follow-on bifurcation studies. Finally, a series of representative experiments are carried out to realize the benefits of using clearance-type nonlinearities to trap wave energy and increase the net harvested energy.

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass–spring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave–wave interaction problems.

This paper presents a feedforward technique for arresting motion in nonlinear systems based on two-scale command shaping (TSCS). The advantages of the proposed technique arise from its feedforward nature and ease of implementation in linear and nonlinear systems. Using the TSCS strategy, the control input required to arrest motion is decomposed into two scales—the first arrests dynamics associated with the linear subproblem, while the second eliminates response from the nonlinearities. Using direct numerical integration, the method is assessed using a traditional Duffing system and multi-degree-of-freedom nonlinear systems. Experiments are conducted on a compound pendulum attached to a servomotor, documenting effective arrest of the system in close agreement with theoretical predictions.

In this paper, we develop two wave-based approaches for predicting the nonlinear periodic response of jointed elastic bars. First, we present a nonlinear wave-based vibration approach (WBVA) for studying jointed systems informed by re-usable, perturbation-derived scattering functions. This analytical approach can be used to predict the steady-state, forced response of jointed elastic bar structures incorporating any number and variety of nonlinear joints. As a second method, we present a nonlinear Plane-Wave Expansion (PWE) approach for analyzing periodic response in the same jointed bar structures. Both wave-based approaches have advantages and disadvantages when compared side-by-side. The WBVA results in a minimal set of equations and is re-usable following determination of the reflection and transmission functions, while the PWE formulation can be easily applied to new joint models and maintains solution accuracy to higher levels of nonlinearity. For example cases of two and three bars connected by linearly damped joints with linear and cubic stiffness, the two wave-based approaches accurately predict the expected Duffing-like behavior in which multiple periodic responses occur in the near-resonant regime, in close agreement with reference finite element simulations. Lastly, we discuss extensions of the work to jointed structures composed of beam-like members, and propose follow-on studies addressing opportunities identified in the application of the methods presented.

In this paper, we propose and numerically study a nonlinear, asymmetric, passive metamaterial that achieves giant non-reciprocity with (i) broadband frequency operation and (ii) robust signal integrity. Previous studies have shown that nonlinearity and geometric asymmetry are necessary to break reciprocity passively. Herein, we employ strongly nonlinear coupling, triangle-shaped asymmetric cell topology, and spatial periodicity to break reciprocity with minimal frequency distortion. To investigate the nonlinear band structure of this system, we propose a new representation, namely a wavenumber–frequency–amplitude band structure, where amplitude-dependent dispersion is quantitatively computed and analyzed. Additionally, we observe and document the new nonlinear phenomenon of time-delayed wave transmission, whereby wave propagation in one direction is initially impeded and resumes only after a duration delay. Based on numerical evidence, we construct a nonlinear reduced-order model (ROM) to further study this phenomenon and show that it is caused by energy accumulation, instability, and a transition between distinct branches of certain nonlinear normal modes of the ROM. The implications and possible practical applications of our findings are discussed.