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Heat propagation in quasi-one-dimensional materials (Q1DMs) often appears puzzling. For example, while an isolated Q1DM, such as a nanowire, a carbon nanotube, or a polymer, can exhibit a high thermal conductivity κ, forests of the same materials can show a reduction in κ. Until now, the complex structures of these assemblies have hindered the emergence of a clear molecular picture for this intriguing phenomenon. We combine coarse-grained simulations with concepts known from polymer physics and thermal transport to unveil a generic microscopic picture of κ reduction in molecular forests. We show that a delicate balance among the persistence length of the Q1DM, the segment orientations, and the flexural vibrations governs the reduction in κ.

The application of a polymer brush in sensing, actuation, self-folding, among others acutely depends on the tuneable bending of a brush-grafted substrate caused by the stress in the brush. However, the stress in a stimuli-responsive brush has not been investigated. In this work, we study the stress in the stimuli-responsive planar polymer brushes of neutral water-soluble polymers with low to very high graft densities using strong stretching theory (SST). First, SST with the Langevin force-extension relation for a polymer chain is extended to the study of stimuli-responsive brushes. Stress profile and other properties of a Poly(N-isopropylacrylamide) (PNIPAm) brush are then obtained using the extended SST and an empirical Flory-Huggins parameter. The model predicts that the stress in a PNIPAm brush is inhomogeneous and compressive at all temperatures and graft densities. The resultant stress is predicted to increase in magnitude with increasing graft density. Moreover, it decreases in magnitude with an increase in temperature before plateauing in low graft density brushes. In contrast, its magnitude increases weakly with increasing temperature in high density brushes. This contrasting behavior is traced to the minimum in interaction free energy density \emph{vs} polymer volume fraction curve for PNIPAm solution at a large volume fraction, and stiffening of chains due to finite extensibility. Furthermore, our results indicate that the ability to tune the resultant stress by changing temperature diminishes with increasing graft density.

Dissipation alone can produce counterintuitive topological wave transport that is otherwise absent in a non-dissipative system. This work demonstrates the influence of dissipation on degeneracies that arise in the context of elastic wave transport. The conditions on the parameters necessary to observe non-Hermitian degeneracies in the Bloch spectrum are precisely derived. It will be shown—contrary to the expectation from singularity theory of a linear eigenvalue problem—that a proportionally damped system with commutative damping does not exhibit non-Hermitian degeneracies. The necessity of a non-commutative and non-proportional dissipation model to observe non-Hermitian degeneracies (or exceptional points) is emphasized. Non-proportional dissipation is used to induce a non-Hermitian degeneracy in a local resonance sub-Bragg bandgap of a linear chain, without using negative damping. While Bloch waves are chosen to illustrate the influence of dissipation, the results readily extend to waves in non-periodic media as well as other wave and vibration transport problems.

The manufacturing of low-density tissue paper involves a key process called creping, which consists of pressing and bonding a wet web onto a drying cylinder rotating at a high speed, and scraping it off subsequently. Creping is understood as a periodic debonding and buckling process which creates a series of micro-folds in the paper. Despite the fact that fibers in the web undergo significant plastic deformation during creping, previous models treat the web as a single thin elastic layer. In this paper we apply a particle dynamics model to investigate the effects of plasticity on creping. A bilinear elastoplastic material model with associated kinematic hardening rule is used to describe the constitutive behavior of the web and a discrete cohesive zone model is implemented for the interfacial delamination. Inclusion of the plasticity of the web leads to significant decrease in creping force and wavelength. A virtual tensile test is performed to predict the stretch and stiffness of the simulated tissue paper. It is found that the stretch increases and the stiffness decreases as the ratio between the creping amplitude and the wavelength increases, thus leading to a higher softness. The simulated tensile stress-strain curve shows significant nonlinearity and qualitatively agrees with experiments. Finally, we explore the “explosive bulk” regime by modeling the web as three individual layers connected by inter-layer bonds. A phase diagram for the creping regimes is constructed. Our simulations indicate that the “explosive bulk” is more likely to occur when the interfacial fracture energy is high and the cohesion of the web is weak.

Localization of normal modes in weakly coupled resonators is used in the recent MEMS sensors to achieve orders of magnitude improvement in sensitivity. It is known that the sensitivity ( $S$ ) of the modal amplitude ratio varies with the strength of the normalized coupling ( $\kappa$ ) as $S \propto \kappa^{-\textbf {1}}$ in the case of two identical coupled oscillators, and as $S \propto \kappa^{-\textbf {2}}$ in the case of three coupled oscillators with the middle oscillator of a much higher natural frequency. We have recently shown that an asymmetry measure $(\alpha)$ will also affect the sensitivity according to $S \propto \alpha^{-\textbf {1}}$ . Here, we prove, using an energy-based analysis, that in the general case of $n-2$ uniformly coupled oscillators chained between two terminal oscillators, $S \propto \alpha^{-\textbf {1}}{\kappa }^{\textbf {1}-{n}}$ , if the blocked resonance spectra of the embedded oscillators and the end oscillators are well-separated. We further show that coupled asymmetric oscillators will also enhance the operating linear range, besides the increased sensitivity when compared with their symmetric counterparts, in a three coupled oscillators system. Our sensitivity and linear range expressions, not based on a (small) perturbation approach, have a wider range of accuracy. These analytical results are validated through numerical computation in the illustrative cases of three and four cell resonator chains.

Aneurysms are localized bulges of arteries; and they can rupture with fatal consequences. Complex mechanobiological factors preclude in vivo testing to assess the rupture risk of an aneurysm, and size based criteria are often used in clinical practice to guide surgical interventions. It is often found that tortuous and buckled aneurysms can exceed the size recommended for surgical intervention, and yet do not rupture. This study addresses why buckled aneurysms exhibit this intriguing behavior by combining in vitro inflation experiments on hyperelastic cylindrical tubes with finite element calculations. Using a biologically relevant material model for an arterial wall, we show that buckled aneurysms can grow in size without rupture under favorable arterial pre-tensions. Stretch reversal phenomenon exhibited by arteries governs whether buckling or bulging occurs first. Exponential stiffening favors the axial propagation of an aneurysm instead of radial growth or size. The choice of failure criteria based on Ogden's strain energy function, Gent's first stretch invariant, and Cauchy stress are discussed. Failure maps incorporating post-bifurcation (bulging and buckling) response are constructed to delineate the regimes of growth, buckling and rupture of an aneurysm.

The maximal aortic diameter has been used as a key indication for whether to repair abdominal aortic aneurysms (AAAs). Aortic tortuosity has been proposed as another factor to consider. In the present study, we compared the degree of aortic tortuosity in ruptured AAAs with that of unruptured AAAs using computed tomography.We performed a retrospective review of a prospectively maintained database of patients who had undergone AAA repair from December 2014 to December 2019. Patients with a ruptured aneurysm (rAAA) were matched with patients with a nonruptured AAA (nrAAA) with the same maximal aneurysm diameter and age. The degree of aortic tortuosity, defined as the maximum lateral deviation from the aortic centerline, was measured on preoperative coronal computed tomography scans.During a 5-year period, 572 AAA cases were identified. The aortic tortuosity of the 25 rAAA cases was compared with that of a matched control group of 31 nrAAAs, selected by the same mean maximum diameter of 8.4 cm and similar patient age. In the rAAA group, the mean age was 74.8 years (84% men). In the nrAAA group, the mean age was 76.3 years (88% men). The mean aortic tortuosity for the rAAA and nrAAA groups was 9.3 ± 7.9 mm and 18.0 ± 11.2 mm, respectively (P .01).Greater aortic tortuosity was seen in the nrAAA cases compared with the rAAA cases at the same matched aneurysm size. Thus, aortic tortuosity might confer a reduced rupture risk. Further studies with larger cohorts are needed to verify this observation.

Acoustic phonons play a special role in lattice heat transport, and confining these low-energy modes in low-dimensional materials may enable nontrivial transport phenomena. By applying lowest-order anharmonic perturbation theory to an atomistic model of a carbon nanotube, we investigate numerically and analytically the spectrum of three-phonon scattering channels in which at least one phonon is of low energy. Our calculations show that acoustic longitudinal (LA), flexural (FA), and twisting (TW) modes in nanotubes exhibit a distinct dissipative behavior in the long-wavelength limit, $|k|\ensuremath{\rightarrow}0$, which manifests itself in scattering rates that scale as ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{LA}}\ensuremath{\sim}{|k|}^{\ensuremath{-}1/2}, {\mathrm{\ensuremath{\Gamma}}}_{\mathrm{FA}}\ensuremath{\sim}{k}^{0}$, and ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{TW}}\ensuremath{\sim}{|k|}^{1/2}$. These scaling relations are a consequence of the harmonic lattice approximation and critically depend on the condition that tubes are free of mechanical strain. In this regard, we show that small amounts of tensile lattice strain $\ensuremath{\epsilon}$ reduce the strength of anharmonic scattering, resulting in strain-modulated rates that, in the long-wavelength limit, obey $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}{\ensuremath{\epsilon}}^{r}{|k|}^{s}$ with $r\ensuremath{\le}0$ and $s\ensuremath{\ge}1$, irrespectively of acoustic mode polarization. Under the single-mode relaxation time approximation of the linearized Peierls-Boltzmann equation (PBE), the long-tube limit of lattice thermal conductivity in stress-free and stretched tube configurations can be unambiguously characterized. Going beyond relaxation time approximations, analytical results obtained in the present study may help to benchmark numerical routines which aim at deriving the thermal conductivity of nanotubes from an exact solution of the PBE.

Mechanical instability in a pre-tensioned finite hyperelastic tube subjected to a slowly increasing internal pressure produces a spatially localized bulge at a critical pressure. This instability is studied in controlled experiments on inflated latex rubber tubes, from the perspective of buckling observed in aneurysms and their rupture risk. The fate of the bulge under continued inflation is governed by the end-conditions and the initial tension in the tube. In a tube with one end fixed and a weight attached to the other freely moving end, the bulge propagates axially at low initial tension, growing in length, and the tube relaxes by extension without buckling. Rupture occurs when the tension is high. By contrast, the bulge formed in an initially stretched tube held fixed at both its ends can buckle or rupture, depending on the amount of initial tension. Experiments are reported for different initial tensions and boundary conditions (BCs). Failure maps in the stretch parameter space and in stretch–tension space are constructed by extending existing theories for bulge formation and buckling analyses to the experimentally relevant BCs. Failure maps deduced from the theory are compared against experiments, and the underlying assumptions are critically assessed. Experiments reveal that buckling provides an alternative route to relieve the stress built up during inflation. Hence, buckling, when it occurs, can be a protective fail-safe mechanism against the rupture of a bulge in an inflated elastic tube.

Deformation properties of venous stents based on braided design, chevron design, Z design, and diamond design are compared using in vitro experiments coupled with analytical and finite element modelling. Their suitability for deployment in different clinical contexts is assessed based on their deformation characteristics. Self-expanding stainless steel stents possess superior collapse resistance compared to Nitinol stents. Consequently, they may be more reliable to treat diseases like May-Thurner syndrome in which resistance against a concentrated (pinching) force applied on the stent is needed to prevent collapse. Braided design applies a larger radial pressure particularly for vessels of diameter smaller than 75% of its nominal diameter, making it suitable for a long lesion with high recoil. Z design has the least foreshortening, which aids in accurate deployment. Nitinol stents are more compliant than their stainless steel counterparts, which indicates their suitability in veins. The semi-analytical method presented can aid in rapid assessment of topology governed deformation characteristics of stents and their design optimization, Comment: 30 pages, 9 figures

A viscoelastic solid sheet fed from a certain height towards a rigid horizontal plane folds on itself provided that there is no slip. This phenomenon commonly occurs in the manufacturing process of textile and paper products. In this paper we apply a particle dynamics model to investigate this phenomenon. At a low feeding velocity and low viscosity, the inertial effect and the viscous dissipation within the sheet are negligible, and our model successfully reproduces the existing quasistatic results in the gravitational regime. As the feeding velocity and the viscosity of the sheet increase, the folding process changes significantly. The length of the folds decrease and the "rolling back" motion of the sheet vanishes. In the inertial regime, a scaling law between the fold length and the feeding velocity is derived by balancing the kinetic energy and the elastic bending energy involved in folding, which is verified by the simulation. It is found that above a critical feeding velocity, the folding morphology transforms from line contact into point contact with the sheet exhibiting a lemniscate-like pattern. Finally, a phase diagram for the folding morphology is constructed. The results presented in this work may offer some insights into the high-speed manufacturing of paper and fabric sheets.

In this paper, we address lattice heat transport in single-walled carbon nanotubes (CNTs) by a quantum mechanical calculation of three-phonon scattering rates in the framework of the Boltzmann transport equation (BTE) and classical molecular dynamics (MD) simulation. Under a consistent choice of an empirical, realistic atomic interaction potential, we compare the tube-length dependence of the lattice thermal conductivity (TC) at room temperature determined from an iterative solution of the BTE and from a nonequilibrium MD (NEMD) approach. Qualitatively similar trends are found in the limit of short tubes, where an extensive regime of ballistic heat transport prevailing in CNTs of lengths $L\ensuremath{\lesssim}1\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$ is independently confirmed. In the limit of long tubes, the BTE approach suggests a saturation of TC with tube length, whereas direct NEMD simulations of tubes extending up to $L=10\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$ are demonstrated to be insufficient to settle the question of whether a fully diffusive heat transport regime and an intrinsic value of TC exist for CNTs. Noting that acoustic phonon lifetimes lie at the heart of a saturation of TC with tube length as per the BTE framework, we complement the quantum mechanical prediction of acoustic phonon lifetimes with an analysis of phonon modes in the framework of equilibrium MD. A normal mode analysis with an emphasis on long-wavelength acoustic modes corroborates the BTE prediction that heat transport in CNTs in the long tube limit is governed by the low attenuation rates of longitudinal and twisting phonons.

A brush-like structure emerges from stretching of long polymer chains, densely grafted on to the surface of an impermeable substrate. The structure arises due to the competition between conformational entropic elasticity of polymer chains and excluded volume interactions leading to intra and interchain monomer repulsions. Recently, soft materials based on stimuli responsive polymer brushes have been developed to produce controllable and reversible large bending deformation of the host substrates. To understand such systems and improve their functional properties, we study the stress distribution in a brush, and develop surface stress-curvature relation for an elastic beam of a soft material grafted with a neutral polymer brush. In the strongly stretched brush regime, we combine mean field theory from polymer physics with a continuum mechanics model and show that the residual stress variation in a brush is a quartic function of distance from the grafting surface, with a maximum occurring at the grafting surface. By idealizing a brush as a continuum elastic surface layer with residual stress, we derive a closed form expression for surface stress and surface elasticity of the layer as a function of brush parameters, such as graft density and molecular weight. A generalized continuum beam model accounts for the Young–Laplace and Steigmann–Ogden curvature elasticity correction terms, and yields a surface stress-curvature relation, that contains existing relations in the literature as special cases. Further, we report experiments on a thermoresponsive random copolymer brush, Poly(N- isopropylacrylamide)-co-Poly(N,N-Dimethylacrylamide) (PNIPAm-co-PDMA) brush, grafted on one side of a plasticized poly(vinyl chloride) (pPVC) thin film. Estimated surface stress from measured curvature is on the order of − 10 N/m , and its magnitude decreases gradually, and reversibly, on increasing ambient temperature from 15 °C to 55 °C.

The manufacturing of low-density paper such as tissue and towel typically involves a key operation called creping. In this process, the wet web is continuously pressed onto the hot surface of a rotating cylinder sprayed with adhesive chemicals, dried in place, and then scraped off by a doctor blade. The scraping process produces periodic microfolds in the web, which enhance the bulk, softness, and absorbency of the final tissue products. Various parameters affect the creping process and finding the optimal combination is currently limited to costly full-scale experiments. In this paper, we apply a one-dimensional (1D) particle dynamics model to systematically study creping. The web is modeled as a series of discrete particles connected by viscoelastic elements. A mixed-mode discrete cohesive zone model (CZM) is embedded to describe the failure of the adhesive layer. Self-contact of the web is incorporated in the model using a penalty method. Our simulation results delineate three typical stages during the formation of a microfold: interfacial delamination, web buckling, and post-buckling deformation. The effects of key control parameters on creping are then studied. The creping angle and the web thickness are found to have the highest impact on creping. An analytical solution for the maximum creping force applied by the blade is derived and is found to be consistent with the simulation. The proposed model is shown to be able to capture the mechanism of crepe formation in the creping process and may provide useful insights into the manufacturing of tissue paper.

We study nonlinear transmission of wave energy in a finite dissipative periodic structure, which is harmonically driven at one end at a forcing frequency lying within its stop band. We show that there is a threshold for the driving amplitude above which there is a sudden increase in the energy transmitted across the finite structure. This generic phenomenon for discrete nonlinear periodic systems is due to a loss of stability of the periodic solutions that are initially localized to the driven end of the structure. The transmission threshold is therefore predicted analytically based on the corresponding saddle-node bifurcation. The influence of damping, strength of coupling and the type of nonlinearity (hardening or softening) are assessed. In particular, we show that damping may eliminate the transmission phenomenon within a frequency range in the stop band. Increasing the strength of coupling between the units is found to increase the minimum force required for the onset of transmission, and the type of nonlinearity determines on which side of the pass band the enhanced transmission may occur.

We study the stress distribution in a polymer brush material over a range of graft densities using molecular dynamics (MD) simulations and theory. Flexible polymer chains are treated as beads connected by nonlinear springs governed by a modified finitely extensible nonlinear elastic (FENE) potential in MD simulations. Simulations confirmed the quartic variation of the normal stress parallel to the substrate, within the bulk of the brush, as predicted in our previous work, for low graft densities. However, in the high graft density regime, the Gaussian chain elasticity assumption is violated by finite extensibility effects (force-extension divergence) and the restriction to binary interaction among monomers is insufficient. This motivated us to extend a semi-analytical strong stretching mean field theory (SST) for polymer brushes, based on Langevin chains and a modified Carnahan-Starling equation of state to model monomer interactions. Our extended theory elucidates the stress and monomer density profiles obtained from MD simulations, as well as reproduces Gaussian chain results for small graft densities. A good agreement is observed between predictions of MD and Langevin chain SST for monomer density profile, end density profile and stress profile in high graft density regime, without fitting parameters (virial coefficients). Quantitative comparisons of MD results with various available theories suggest that excluded volume correlations may be important.

Sandwich panels are extensively used in constructional, naval, and aerospace structures due to the high stiffness and strength-to-weight ratios. In contrast, the sound transmission properties are adversely influenced by the low effective mass. Phase velocity matching of structural waves propagating within the panel and the incident pressure waves from the fluid medium leads to coincidence effects resulting in reduced impedance and high sound transmission. Truss-like lattice cores with porous microarchitecture and reduced inter panel connectivity offer the potential to satisfy the conflicting structural and vibroacoustic response requirements. This study combines Bloch-wave analysis and the finite element method to understand wave propagation and hence sound transmission in sandwich panels with a truss lattice core. Three dimensional coupled fluid-structure finite element simulations are conducted to compare the performance of a representative set of lattice core topologies. Potential advantages of sandwich structures with a lattice core are identified. The significance of partial band gaps is evident in the sound transmission loss characteristics of the panels studied. This work demonstrates that, even without optimization, significant enhancements in sound transmission loss performance can be achieved in truss lattice core sandwich panels compared to a traditional sandwich panel employing a honeycomb core under constant mass constraint.

Shape–memory materials (SMMs) and actuators possess the ability to respond to external stimuli such as temperature, electricity, magnetic field, and light, and change their shapes. During the process, energy is converted into mechanical deformation which makes them attractive for various applications in biomedical devices, deployable structures, artificial muscles, microdevices, sensors, etc. Traditional SMMs and actuators rely on the properties of bulk material irrespective of their nature (e.g. polymers, metallic alloys, composite materials). Very recently, polymer-brush-based nanoscale bending actuators were reported. As a consequence of the strong interchain repulsion, the overcrowded polymer chains within the polymer brush can exert forces onto the underlying substrate and deform the substrate. However, there is no empirical evidence that bending observed on the nanoscale can be adapted for actuator applications on the macroscale. Furthermore, bending alone may not be sufficient to provide the desired macroscopic actuation; axial stretching may also be required. Herein we demonstrate the bending and stretching of a soft polymeric substrate, plasticized poly(vinyl chloride) (pPVC, thickness 400 mm, Young!s Modulus 6.89 MPa), on the macroscale by grafting a model hydrophilic polymer, poly(N,N-dimethylacrylamide) (PDMA), at high graft density on the pPVC surface. As shown in Figure 1A, when PDMA

An experimental investigation of uncertainty in natural frequencies of linear structures estimated from measured frequency response function (FRF) under dynamic loading is presented. Experiments were conducted on one hundred nominally identical realizations of a fixed-fixed beam; each realisation was obtained by placin g twelve identical masses at random spatial locations (generated by a computer) along the length of the beam. The total random mass is about 2% of the total mass of the beam. These experiments contain parametric uncertainty in the mass matrix, and hence may be useful for the validation of many random eigenvalue analysis and prediction methods currently available to structural dynamicist. Predictions from Monte Carlo simulation of deterministic finite elements model are compared with experiments. It is co ncluded that the method of estimation of natural frequencies from FRFs and the spatial location of the measurements has significant influence upon the first two moments (mean and standard deviation) of the natural frequency ensemble. Furthermore, whilst the Monte Carlo simulation estimates of the mean and standard deviation are in reasonable agreement with experiments at higher frequencies, the probability density function differ appreciably, within the limits of the sample size investigated in this study.

Friction control at the wheel–rail interface, using on-board solid stick friction modifier systems can lead to enhanced track life, reduced wear, and increased fuel economy in railroads. Frictional contact between the solid stick and the railway wheel itself can potentially cause vibrations within the modifier systems, influencing their stability and performance. A frequency domain linearized stability analysis of the state of steady sliding at the frictional contact between the solid stick and the wheel is performed. The proposed approach relies on individual frequency response functions (FRFs) of the wheel and the applicator–bracket subsystems of the on-board friction modifier. Stability characteristics of three representative bracket designs are qualitatively compared, using the FRFs generated by their respective finite element (FE) models. The FE models are validated by comparing the predicted natural frequencies with corresponding experimentally measured values on a full wheel test rig (FWTR) facility. The validated FE models are then used to compute stability maps which delineate stable and unstable regions of operation in the design parameter space, defined by train speed, angle of applicator, friction coefficient, and bracket design. Strong dependence of stability upon the bracket designs is observed. The methodology developed here can be used by design engineers to assess the effectiveness of design changes on the stability of the applicator–bracket assembly in a virtual environment—thus avoiding costly retrofitting and prototyping. Directions for further model refinement and testing are provided.

The need for a simple and effective characterization technique for thin-film materials which are widely used in MEMS (micro-electro-mechanical systems), using commonly available equipment, has prompted consideration of cantilever beam-based methods. The advantages of this class of techniques which employ a scanning surface profiler to deform micro-cantilevers are simplicity, speed, cost and wide applicability. A technique for extracting Young's modulus from static deflection data is developed in this paper and validated in experiments on thin-film specimens of silicon nitride deposited on a silicon substrate under different conditions. Finite element analysis is used to assess the influence of factors affecting the bending of thin films, and thus guide the analysis of micro-cantilever deflection data for reliable characterization of the material.

General theorems concerning the simultaneous diagonalization of two real symmetric matrices using a real orthogonal transformation matrix are presented to place the Caughey and O?Kelly?s theorem on classical normal modes in damped linear dynamic systems in context. It was also brought to attention that equivalent theorems similar to Caughey and O?Kelly?s existed in linear algebra since the works of Weirestrass. The examples given by Liang et al. are shown to be inadequate to modify Caughey and O?Kelly?s classification of damped systems. The classification of damped systems into proportional/non-proportional systems based on Caughey and O?Kelly?s criterion still remains valid if proportional damping is meant to convey that all the modes of the damped system are real, not just one or a few and that all the three system matrices are simultaneously diagonalised by the same transformation matrix.

Recent studies have demonstrated mode localized resonant micro-electro-mechanical systems (MEMS) sensing devices with orders of magnitude improvement in sensitivity. Avoided crossings or eigenvalue veering is the physical mechanism exploited to achieve the enhancement in sensitivity of devices operating either in vacuum or in air. The mode localized MEMS devices are typically designed to be symmetric and use gap-varying electrostatic springs to couple motions of two or more resonators. The role of asymmetry in the design of devices and its influence on sensitivity is not fully understood. Furthermore, gap-varying electrostatic springs suffer from nonlinearities when gap variation between coupling plates becomes large due to mode localization, imposing limitations on the device performance. To address these shortcomings, this contribution has two principal objectives. The first objective is to critically assess the role of asymmetry in the device design and operation. We show, based on energy analysis, that carefully designed asymmetry in devices can lead to even higher sensitivities than reported in the literature. Our second objective is to design and implement linear, tunable, electrostatic springs, using shaped combs, which allow large vibration amplitudes of resonators thereby increasing the signal to noise ratio. We experimentally demonstrate linear electrostatic coupling in a two oscillator device. Our study suggests that a future avenue for progress in the mode localized resonant sensing technology is to combine asymmetric devices with tunable linear coupling designs.

We study nonlinear transmission of energy in a finite dissipative periodic structure, which is externally driven at one end with a harmonic force. We show that when the driving frequency is within the stop band of the periodic structure, there is a threshold for the driving amplitude, above which there is a sudden increase in the energy transmitted through the finite structure. This is a generic phenomenon for discrete nonlinear periodic systems. The onset of transmission is due to loss of stability of the harmonic solutions localized at the driven end of the structure. The transmission threshold is predicted analytically based on the underlying bifurcation. The influence of both hardening and softening types of nonlinearity on the transmission phenomenon is discussed.

Periodic composites such as acoustic metamaterials use local resonance phenomenon in designing low frequency sub-Bragg bandgaps. These bandgaps emerge from a resonant scattering interaction between a propagating wave and periodically arranged resonators. This paper develops a receptance coupling technique to combine the dynamics of the resonator with the unit cell dynamics of the background medium to analyze flexural wave transmission in a periodic structure, involving a single degree of freedom coupling between the medium and the resonator. Receptance techniques allow for a straightforward extension to higher dimensional systems with multiple degrees of freedom coupling and for easier experimental measurements. Closed-form expressions for the location and width of sub-Bragg bandgaps are obtained. Rigid body modes of the unit cell of the background medium are shown to set the bounding frequencies for local resonance bandgaps. Results from the receptance analysis compare well with Bloch wave analysis and experiments performed on a finite structural beam with periodic masses and resonators. Stronger coupling and inertia of the resonator increase the local resonance bandgap width. Two-fold periodicity widens the Bragg bandgap, narrowed by local resonators, thus expanding the design space and highlighting the advantages of hierarchical periodicity.

Bloch waves in viscously damped periodic material and structural systems are analyzed using a perturbation method originally developed by Rayleigh for vibration analysis of finite structures. The extended method, called the Bloch–Rayleigh perturbation method here, utilizes the Bloch waves of an undamped unit cell as basis functions to provide approximate closed-form expressions for the complex eigenvalues and eigenvectors of the damped unit cell. In doing so, we circumvent the solution of a quadratic Bloch eigenvalue problem and subsequent computationally intensive transformation to first order/state-space form. Dispersion curves of a one-dimensional damped spring-mass chain and a two-dimensional phononic crystal with square inclusions are calculated using the state-space method and the proposed method. They are compared and found to be in excellent quantitative agreement for both proportional and nonproportional viscous damping models. The perturbation method is able to capture anomalous dispersion phenomena—branch overtaking, branch cut-on/cut-off, and frequency contour transformation—in parametric ranges where state-space formulations encounter numerical issues. Generalization to other linear nonviscous damping models is permissible.