Mass-spring Chain Research Articles

Dynamic fracture of a dissimilar chain.

In this paper, we study the dynamic fracture of a dissimilar chain composed of two different mass-spring chains and connected with other springs. The propagation of the fault (crack) is realized under externally applied moving forces. In comparison with a homogeneous double chain, the considered structure displays some new essential features of steady-state crack propagation. Specifically, the externally applied forces are of a different strength, unlike a static case, and should be appropriately chosen to satisfy the equilibrium of the structure. Moreover, there exists a gap in the range of crack speeds where the steady-state fracture cannot occur. We analyse the admissibility of solutions for different model parameters and crack speeds. We complement analytical findings with numerical simulations to validate our results.This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

Non-reciprocal wave propagation in mechanically-modulated continuous elastic metamaterials.

Acoustic and elastic metamaterials with time- and space-dependent effective material properties have recently received significant attention as a means to induce non-reciprocal wave propagation. Recent analytical models of spring-mass chains have shown that external application of a nonlinear mechanical deformation, when applied on time scales that are slow compared to the characteristic times of propagating linear elastic waves, may induce non-reciprocity via changes in the apparent elastic modulus for perturbations around that deformation. Unfortunately, it is rarely possible to derive analogous analytical models for continuous elastic metamaterials due to complex unit cell geometry. The present work derives and implements a finite element approach to simulate elastic wave propagation in a mechanically-modulated metamaterial. This approach is implemented on a metamaterial supercell to account for the modulation wavelength. The small-on-large approximation is utilized to separate the nonlinear mechanical deformation (the "large" wave) from superimposed linear elastic waves (the "small" waves), which are then analyzed via Bloch wave analysis with a Fourier expansion in the harmonics of the modulation frequency. Results on non-reciprocal wave propagation in a negative stiffness chain, a structure exhibiting large stiffness modulations due to the presence of mechanical instabilities, are then shown as a case example.

Dramatic bandwidth enhancement in nonlinear metastructures via bistable attachments

We report amplitude-dependent substantial enhancement of the frequency bandwidth in locally resonant metamaterial-based finite structures (metastructures) via bistable attachments. The bistable magnetoelastic beam attachments of the unit cells exhibit linear intrawell, nonlinear intrawell, and nonlinear interwell oscillations for low, moderate, and sufficiently high intensity excitations, respectively. As a result, the overall metastructure leverages linear locally resonant bandgaps under low amplitudes and nonlinear attenuation due to wideband chaotic vibrations of the bistable attachments under large amplitudes. The concept was first demonstrated through a linear mass-spring chain with bistable attachments in a numerical case study. Experimental results and validations are then presented for a base-excited cantilever beam hosting seven bistable unit cells. Transition from linear locally resonant bandgaps to nonlinear attenuation is observed, and the amplitude-dependent bandwidth enhancement is shown. The bandwidth offered by nonlinear interwell oscillations is substantially wider than the linear locally resonant bandgap that is limited by the added mass.

Lower band gaps of longitudinal wave in a one-dimensional periodic rod by exploiting geometrical nonlinearity

Abstract In this paper, a local resonant (LR) rod with high-static-low-dynamic-stiffness (HSLDS) resonators is proposed to create a very low-frequency band gap for longitudinal wave propagation along the rod. The HSLDS resonator is devised by employing geometrical nonlinearity, and attached onto a periodic rod composed of rigid frames and rubbers to construct a LR rod. To reveal the band gaps, the LR rod is modeled as a lumped mass-spring chain. The effects of damping and nonlinearity of the HSLDS resonator on the dispersion relation is studied analytically by the Harmonic Balance method. The analytical results indicate that the damping mainly affects the width and depth of the band gap, while the nonlinearity can influence the central frequency and width of the band gap. In addition, both multi-body dynamic analyses and numerical simulations are conducted to predict longitudinal wave propagation along the LR rod, and thus to validate the very low-frequency band gap. The results show that the periodic rod with HSLDS resonators can create a very low-frequency band gap for longitudinal waves propagating along the rod.

Nonreciprocal wave phenomena in spring-mass chains with effective stiffness modulation induced by geometric nonlinearity.

Acoustic nonreciprocity has been shown to enable a plethora of effects analogous to phenomena seen in quantum physics and electromagnetics, such as immunity from backscattering and unidirectional band gaps, which could lead to the design of direction-dependent acoustic devices. One way to break reciprocity is by spatiotemporally modulating material properties, which breaks parity and time-reversal symmetries. In this paper, we present a model for a medium in which a slow nonlinear deformation modulates the effective material properties for small overlaid disturbances (often referred to as "small-on-large" propagation). The medium is modeled as a discrete spring-mass chain that undergoes large deformation via prescribed displacements of certain points in the unit cell. A multiple-scale perturbation analysis shows that, for sufficiently slow modulations, the small-scale waves can be described by a linear monatomic chain with time- and space-dependent on-site stiffness. The modulation depth can be tuned by changing the geometric and stiffness parameters of the unit cell. The accuracy of the small-on-large approximation is demonstrated using direct numerical simulations.

Strictly supersonic solitary waves in lattices with second-neighbor interactions

We consider a monatomic nonlinear mass–spring chain with first and second-neighbor interactions and show that there is a parameter range where solitary waves in this system are strictly supersonic. In these regimes standard quasicontinuum theories, targeting long-wave limits of lattice models, are not adequate since even weak strictly supersonic solitary waves are of envelope type and crucially involve a microscopic scale in addition to the mesoscopic scale of the envelope. To capture this effect in a continuum setting it is necessary to employ unconventional, higher-order quasicontinuum approximations carrying more than one length scale.

The effect of long-range coulomb interaction on the phonon transport of an ionized mass-spring chain

The effect of long-range coulomb interaction on the phonon transport of an ionized mass-spring chain

Tuning frequency band gaps of tensegrity mass-spring chains with local and global prestress

This work studies the acoustic band structure of tensegrity mass-spring chains, and the possibility to tune the dispersion relation of such systems by suitably varying local and global prestress variables. Building on established results of the Bloch–Floquet theory, the paper first investigates the linearized response of chains composed of tensegrity units and lumped masses, which undergo small oscillations around an initial equilibrium state. The stiffness of the units in such a state varies with an internal self-stress induced by prestretching the cables forming the tensegrity units, and the global prestress induced by the application of compression forces to the terminal bases. The given results show that frequency band gaps of monoatomic and biatomic chains can be effectively altered by the fine tuning of local and global prestress parameters, while keeping material properties unchanged. Numerical results on the wave dynamics of chains under moderately large displacements confirm the presence of frequency band gaps of the examined systems in the elastically hardening regime. Novel engineering uses of the examined systems are discussed.

A study of deformation localization in nonlinear elastic square lattices under compression.

The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These studies are conducted on other lattice structures, with the objective of identifying and investigating minimal models that exhibit localization, hysteresis and path-dependent behaviour. To this end, we first consider a two-dimensional square lattice consisting of point masses connected by in-plane axial springs and vertical ground springs, which may be considered as a discrete description of an elastic membrane supported by an elastic substrate. Results illustrate that, depending on the relative values of the spring constants, the lattice exhibits in-plane or out-of-plane instabilities leading to localized deformations. This model is further simplified by considering the one-dimensional case of a spring-mass chain sitting on an elastic foundation. A bifurcation analysis of this lattice identifies the stable and unstable branches and sheds light on the mechanism of transition from affine deformation to global or diffuse deformation to localized deformation. Finally, the lattice is further reduced to a minimal four-mass model, which exhibits a deformation qualitatively similar to that in the central part of a longer chain. In contrast to the widespread assumption that localization is induced by defects or imperfections in a structure, this work illustrates that such phenomena can arise in perfect lattices as a consequence of the mode shapes at the bifurcation points.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Band gap synthesis in elastic monatomic lattices via input shaping

This work describes control-theoretic methods for inducing a band gap-like behavior in elastic monatomic lattices. The dynamics of the system under consideration are derived in detail. Open-loop pre-filtering techniques, in the form of Posicast and user-selected time delay filters, are then utilized to eliminate specific frequency contents from the response of the multi-degree of freedom spring-mass chain. The input shaping approach is used to synthesize one or more band gaps in a homogenous lattice that can be designed to resemble Bragg-scattering in phononic crystals, as well as local resonance effects in acoustic metamaterials. The presence of the synthesized band gaps in the lattice’s dispersion behavior is validated using a spatiotemporal Fourier transform of the system’s impulse response as well as the frequency response of the end-to-end transfer function. Finally, an analysis of the control effort required to cancel the targeted system poles is carried out.

Experimental demonstration of excitation and propagation of intrinsic localized modes in a mass–spring chain

Excitation and propagation of nonlinear localized oscillations are demonstrated experimentally by using a mass–spring chain to emulate those in the celebrated Fermi–Pasta–Ulam chain of β type. Two types of experiments are performed, one corresponding to a boundary-value problem and the other to an initial-value one. In the former experiment, one end of the chain is driven sinusoidally in the direction of the chain above a cut-off frequency. In the latter experiment, only a number of weights centered around the weight in the middle of the chain are held displaced initially and released suddenly. The experimental results show quantitatively good agreements with the numerical ones. Two videos linked also show visually mobile and stationary localized oscillations.

Asymptotic analysis of high-frequency modulation in periodic systems. Analytical study of discrete and continuous structures

This paper addresses the modeling of large scale modulations of high-frequency mechanical waves propagating in discrete and continuum periodic reticulated structures. By means of an asymptotic approach implemented using a multi-cells approach, we derive (i) the equations governing the large scale modulations, (ii) the specific features of the modulation propagation, and (iii) the domain of validity of the description. This approach provides specific information on the physics of large scale modulations that is not directly accessible from the Bloch wave decomposition nor from a multi-scattering approach. The theoretical formulation enables (i) the identification of the frequency range for which evolutions of the rapidly oscillating wave motions arise over long distances, and (ii) simple calculation of the high-frequency wave field based on a two-step procedure that separate the (multi-) cell and the large modulation scales.The asymptotic method is introduced by studying the academic case of a 1D spring-mass chain. This allows us to establish the correspondence between the modulation phenomena and the classical approaches of homogenization and Floquet–Bloch waves.High-frequency modulations in reticulated beams are then investigated. The dispersion equation for longitudinal and transversal vibrations are first established. Then, from the asymptotic analysis, the equations governing the structure behavior in the standard homogenization regime and in the modulation regimes are derived. These results are illustrated and supported by numerical simulations. General comments on modulation and potential application are developed in the conclusions.

Almost-dispersionless pulse transport in long quasiuniform spring-mass chains: A different kind of Newton's cradle.

Almost-dispersionless pulse transfer between the extremal masses of a uniform harmonic spring-mass chain of arbitrary length can be induced by suitably modifying two masses and their spring's elastic constant at both extrema of the chain. It is shown that a deviation (or a pulse) imposed to the first mass gives rise to a wave packet that, after a time of the order of the chain length, almost perfectly reproduces the same deviation (pulse) at the opposite end, with an amplitude loss that is as small as 1.3% in the infinite-length limit; such a dynamics can continue back and forth again for several times before dispersion cleared the effect. The underlying coherence mechanism is that the initial condition excites a bunch of normal modes with almost equal frequency spacing. This constitutes a possible mechanism for efficient energy transfer, e.g., in nanofabricated structures.

Amplitude-dependent topological edge states in nonlinear phononic lattices

This work investigates the effect of nonlinearities on topologically protected edge states in one- and two-dimensional phononic lattices. We first show that localized modes arise at the interface between two spring-mass chains that are inverted copies of each other. Explicit expressions derived for the frequencies of the localized modes guide the study of the effect of cubic nonlinearities on the resonant characteristics of the interface, which are shown to be described by a Duffing-like equation. Nonlinearities produce amplitude-dependent frequency shifts, which in the case of a softening nonlinearity cause the localized mode to migrate to the bulk spectrum. The case of a hexagonal lattice implementing a phononic analog of a crystal exhibiting the quantum spin Hall effect is also investigated in the presence of weakly nonlinear cubic springs. An asymptotic analysis provides estimates of the amplitude dependence of the localized modes, while numerical simulations illustrate how the lattice response transitions from bulk-to-edge mode-dominated by varying the excitation amplitude. In contrast with the interface mode of the first example studies, this occurs both for hardening and softening springs. The results of this study provide a theoretical framework for the investigation of nonlinear effects that induce and control topologically protected wave modes through nonlinear interactions and amplitude tuning.

Higher-order FRFs and their applications to the identifications of continuous structural systems with discrete localized nonlinearities

Abstract Many modern structural systems are found to contain localized areas, often around structural joints and boundaries, where the actual dynamic behavior is far from linear. Such nonlinearities need to be properly identified so that they can be incorporated into the improved mathematical model used for design and operation. One approach to this task is to use higher-order frequency response functions (FRFs). Identification of structural nonlinearities using higher-order FRFs is currently an active and growing emerging research area and to date, much research has been conducted. However, most existing research seems to be confined to very simple nonlinear system models such as SDOF mass-spring-damper models or to the most, MDOF mass-spring chain models. For more general continuous nonlinear structures with discrete localized nonlinearities however, analytical derivation of higher-order FRFs remains unknown and as a result, identification of nonlinearities of such systems using measured higher-order FRFs becomes impossible to achieve. This missing link between higher-order FRFs and physical parameters of nonlinearities of continuous structures has to be established before any real progress can be possibly made. In this paper, a new novel method is developed which can be used to derive analytical higher-order FRFs of continuous structural systems with discrete localized nonlinearities. The method is generally applicable and theoretically exact, and it serves exactly as that missing link. Having established analytical higher-order FRFs which serve as the theoretical foundation for any subsequent identification, method of parameter identification of nonlinearity is then further developed. Important characteristics of higher-order FRFs are discussed, some of which are revealed the first time since there has never been higher-order FRFs derived from nonlinear continuous structures. Various numerical aspects on how to improve accuracy of identified nonlinear system parameters are discussed.

Frequency graded 1D metamaterials: A study on the attenuation bands

Depending on the frequency, waves can either propagate (transmission band) or be attenuated (attenuation band) while travelling through a one-dimensional spring-mass chain with internal resonators. The literature on wave propagation through a 1D mass-in-mass chain is vast and continues to proliferate because of its versatile applicability in condensed matter physics, optics, chemistry, acoustics, and mechanics. However, in all these areas, a uniformly periodic arrangement of identical linear resonating units is normally used which limits the attenuation band to a narrow frequency range. To counter this limitation of linear uniformly periodic metamaterials, the attenuation bandwidth in a one-dimensional finite chain with frequency graded linear internal resonators are investigated in this paper. The result shows that a properly tuned frequency graded arrangement of resonating units can extend the upper part of the attenuation band of 1D metamaterial theoretically up to infinity and also increases the lower part of the attenuation bandwidth by around 40% of an equivalent uniformly periodic metamaterial without increasing the mass. Therefore, the frequency graded metamaterials can be a potential solution towards low frequency and wideband acoustic or vibration insulation. In addition, this paper provides analytical expressions for the attenuation and transmission frequency limits for a periodic mass-in-mass metamaterial and demonstrates the attenuation band is generated by the high absolute value of the effective mass not only due to the negative effective mass.

Segmentation in cohesive systems constrained by elastic environments.

The complexity of fracture-induced segmentation in elastically constrained cohesive (fragile) systems originates from the presence of competing interactions. The role of discreteness in such phenomena is of interest in a variety of fields, from hierarchical self-assembly to developmental morphogenesis. In this paper, we study the analytically solvable example of segmentation in a breakable mass-spring chain elastically linked to a deformable lattice structure. We explicitly construct the complete set of local minima of the energy in this prototypical problem and identify among them the states corresponding to the global energy minima. We show that, even in the continuum limit, the dependence of the segmentation topology on the stretching/pre-stress parameter in this problem takes the form of a devil's type staircase. The peculiar nature of this staircase, characterized by locking in rational microstructures, is of particular importance for biological applications, where its structure may serve as an explanation of the robustness of stress-driven segmentation.This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications.'

Normal modes of a defected linear system of beaded springs

A model of a one-dimensional mass-spring chain with mass or spring defects is investigated. With a mass defect, all oscillators except the central one have the same mass, and with a spring defect, all the springs except those connected to the central oscillator have the same stiffness constant. The motion is assumed to be one-dimensional and frictionless, and both ends of the chain are assumed to be fixed. The system vibrational modes are obtained analytically, and it is shown that if the defective mass is lighter than the others, then a high frequency mode appears in which the amplitudes decrease exponentially with the distance from the defect. In this sense, the mode is localized in space. If the defect mass is greater than the others, then there will be no localized mode and all modes are extended throughout the system. Analogously, for some values of the defective spring constant, there may be one or two localized modes. If the two defected spring constants are less than that of the others, there is no localized mode.

Modal interaction and higher harmonic generation in a weakly nonlinear, periodic mass–spring chain

Wave propagation in a nonlinear periodic material is investigated, by considering an infinite chain of two-mass unit cells with cubic stiffness nonlinearity. The chain is analysed using the method of multiple scales, predicting the dispersion shift in the band structure due to nonlinear self-interaction. The solution further reveals modest higher harmonic generation within the limits of the solution approach, proportional to the strength of nonlinearity and energy level in the chain. The possibility for controlling the higher harmonic generation by changing the distribution of the cubic nonlinearity is investigated. The predictions based on the analytical model are verified by numerical simulations, which also explores the limits of the infinite, analytical model.

One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology

There are two classes of phononic structures that can support elastic waves with non-conventional topology, namely intrinsic and extrinsic systems. The non-conventional topology of elastic wave results from breaking time reversal symmetry (T-symmetry) of wave propagation. In extrinsic systems, energy is injected into the phononic structure to break T-symmetry. In intrinsic systems symmetry is broken through the medium microstructure that may lead to internal resonances. Mass-spring composite structures are introduced as metaphors for more complex phononic crystals with non-conventional topology. The elastic wave equation of motion of an intrinsic phononic structure composed of two coupled one-dimensional (1D) harmonic chains can be factored into a Dirac-like equation, leading to antisymmetric modes that have spinor character and therefore non-conventional topology in wave number space. The topology of the elastic waves can be further modified by subjecting phononic structures to externally-induced spatio-temporal modulation of their elastic properties. Such modulations can be actuated through photo-elastic effects, magneto-elastic effects, piezo-electric effects or external mechanical effects. We also uncover an analogy between a combined intrinsic-extrinsic systems composed of a simple one-dimensional harmonic chain coupled to a rigid substrate subjected to a spatio-temporal modulation of the side spring stiffness and the Dirac equation in the presence of an electromagnetic field. The modulation is shown to be able to tune the spinor part of the elastic wave function and therefore its topology. This analogy between classical mechanics and quantum phenomena offers new modalities for developing more complex functions of phononic crystals and acoustic metamaterials.