Fig 23 - uploaded by Michael Leamy
Content may be subject to copyright.
Source publication
Wave propagation in one-dimensional nonlinear periodic structures is investigated through a novel perturbation analysis and accompanying numerical simulations. Several chain unit cells are considered featuring a sequence of masses connected by linear and cubic springs. Approximate closed-form, ?rst-order dispersion relations capture the effect of n...
Similar publications
We study a one-dimensional nonlinear periodic structure which contains two different spring stiffness and an identical mass in each period. The linear dispersion relationship we obtain indicates that our periodic structure has obvious advantages compared to other kinds of periodic structures (i.e. those with the same spring stiffness but two differ...
A new amplitude equation is derived for high-frequency acoustic waves propagating through an incompressible vortical flow using multi-time-scale asymptotic analysis. The reduced model is derived without an explicit spatial-scale separation ansatz between the wave and vortical fields. As a consequence, the model is seen to capture very well the feat...
The steady solution of a solitary wave propagating in the presence of a linear shear background
current is investigated by the Green-Naghdi (GN) equations. The steady solution is
obtained by use of the Newton-Raphson method. Three aspects are investigated; they are
the wave speed, wave profile and velocity field. The converged GN results are compar...
This paper presents a compressive sensing based experimental method for detecting spinning modes of sound waves propagating inside a cylindrical duct system. This method requires fewer dynamic pressure sensors than the number required by the Shannon-Nyquist sampling theorem so long as the incident waves are sparse in spinning modes. In this work, t...
The effect of current on the formation of freak waves was investigated by conducting
laboratory experiments in a wave flume. The generation of freak-type waves in a wave flume is
based on the method of focusing of waves propagating in a wave group. The investigation
show that the formation of a freak waves is strongly affected by the direction of a...
Citations
... The Lindstedt-Poincaré method is another perturbation technique that has been used, as with the method of multiple scales, to predict amplitude-dependent shifts to the dispersion curves of nonlinear lattices. Its basic framework follows closely to that of the method of multiple scales and has been applied successfully to 1-D [30] and 2-D [23] lattices. ...
... This multi-harmonic solution can then be used in conjunction with the 0th-order solution to update higher-order equations of the asymptotic analysis. Interestingly, the authors in [30] determined this solution but neither extended the procedure beyond the first order nor commented on the significance of the particular solution at O(ε 1 ). However, as will be detailed in Sect. ...
... The method proceeds by substituting Eq. (31) into Eq. (30) and determining the coefficient vectors c l and s l via a Galerkin projection, reducing the set of nonlinear differential equations to nonlinear algebraic equations. Since the number of unknowns (harmonic coefficients, amplitude, frequency, and wavenumber) exceeds the number of equations, a general solution is unavailable. ...
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 followon
studies. Where appropriate, we include details on experimental explorations and envisioned applications,
both of which have received relatively sparse attention
to date.
... With regard to (iii), a number of approximate analytical methods have been devoted to the prediction of nonlinear dispersion curves. Among others, we recall those based on the harmonic balance [55,56], the perturbation method [45,46,57], the homotopy harmonic balance [58], the homotopy Padé technique [59], the multiple scales [60], and the factorization of the spatial and temporal parts of the solution [61]. Analytical methods are advantageous because they provide closed-form solutions that can be easily compared with numerical predictions in the validity range of linearization. ...
... Subsequently, following the Lindstedt-Poincaré method [57], we consider the following asymptotic expansions: More precisely, we consider (0) (0) * = | (0) | 2 in (A.7). The forcing terms on the right-hand side of the first equation in (A.6) are secular; therefore, we determine that 1 is equal to 0. We then apply the F-B boundary conditions on the left and right-hand bounds via controlled displacements to the ( − 1) ℎ and ( + 1) ℎ masses, respectively. ...
... The total energy can be calculated via Eq. (29) by substitution of asymptotic expansion for the velocity of zeroth particle up to order of β F 2 0 /c 3 , which can be obtained by means of the perturbation analysis (see, e.g., [59,60]). However, this approach involves technical difficulties and, therefore, we propose the following strategy below. ...
We deal with dynamics of the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-Fermi–Pasta–Ulam–Tsingou chain with one free end, subjected to the sudden sinusoidal force. We examine the evolution of the total energy supplied into the chain at large times. In the harmonic case (β=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =0$$\end{document}), the energy grows in time linearly at the driving frequencies, corresponding to nonzero group velocities. The rate of energy supply is shown to decrease with increasing excitation frequency. Loading with the cut-off frequency, corresponding to zero group velocity, results in the energy growth proportionally to t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{t}$$\end{document}. Explanation of behavior in time of the energy is proposed by analysis of the obtained approximate closed-form expressions for the particle velocity. In the weak anharmonic case, large-time asymptotic approximation for the total energy is obtained by using the renormalized dispersion relation. The approximation allows one to estimate rate of the energy supply at the driving frequencies, which belong both in the pass-band and in the stop-band of the harmonic chain. Consistency of the asymptotic estimates with the results of numerical simulations is discussed.
... This approach aligns with a well-established tradition dating back to the early 1980s [7]. In the past decades, different perturbation methods have been employed to study the nonlinear dispersion properties of harmonic waves in periodic systems, including the ordered Harmonic Balance [8,9], Lindstedt-Poincaré [10][11][12][13] and the Method of Multiple Scales [14][15][16][17][18]. Most of the studies in the literature are focused on determining the amplitude-dependent softening or hardening behaviour of the wavefrequency backbone curves. ...
This paper deals with the free propagation problem of resonant and close-to-resonance waves in one-dimensional lattice metamaterials endowed with nonlinearly viscoelastic resonators. The resonators' constitutive and geometric nonlinearities imply a cubic coupling with the lattice. The analytical treatment of the nonlinear wave propagation equations is carried out via a perturbation approach. In particular, after a suitable reformulation of the problem in the Hamiltonian setting, the approach relies on the well-known resonant normal form techniques from Hamiltonian perturbation theory. It is shown how the constructive features of the Lie Series formalism can be exploited in the explicit computation of the approximations of the invariant manifolds. A discussion of the metamaterial dynamic stability, either in the general or in the weak dissipation case, is presented.
... A perturbation method [38] and the original IHB method [39] were used to verify the improved IHB , ...
... (11) and (12) with use of the improved IHB method are given in Appendix D and Appendix E, respectively. Since there is no mechanism used in the structure in the literature [38], the stiffness of the torsional spring in Eqs. (11) and (12) is set to be zero (i.e., φ = 0), and the system in Fig. 4 degenerates to a conventional cubic-spring model. ...
... A = 0.002. In the case, the system has a weak nonlinearity [38], and A represents the amplitude, which can also represent the nonlinear intensity. The larger the amplitude, the stronger the nonlinear. ...
Reciprocity is a fundamental property of wave propagations, and many researchers devoted their efforts to breaking the reciprocity and implementing unidirectional wave propagations. At present, the main method to realize non-reciprocal waves uses aperiodic structure as the wave propagation medium. The non-reciprocal bandgap achieved by this method is narrow and difficult to adjust actively. To improve the controllability of non-reciprocal bandwidth, a one-dimensional (1D) periodic lattice structure based on linkage element is proposed in the work. The linkage element enables the lattice structure to have nonlinear stiffness with respect to the asymmetry of the equilibrium position. This stiffness asymmetry leads to the non-reciprocity of wave propagation, which provides a new idea for the design of non-reciprocal structures. To deal with the strong nonlinearity and high dimensional characteristics of the structure, the improved incremental harmonic balance (IHB) method is used to analyze the dispersion and bandgap characteristics of the structure. The results show that the structure has two bidirectional bandgaps (high and low frequency) and four unidirectional bandgaps, and the position, width and direction of the bandgap can be adjusted by the equilibrium position and mechanical parameters of the structure. The obtained structural properties are verified by numerical experiments.
... The analysis strategy used in this study could be generalized to investigate other wave dynamics problems concerning nonlinear discrete lattices in which only on-site nonlinearity arises, for instance, metamaterials with local nonlinear resonators [42] and local nonlinear chirality-induced autoparametric metastructures [43]. Whereas for lattices with inter-site nonlinearity, such as phononic crystals containing nonlinear internal forces between adjacent cells [44], errors generate if one directly invokes the Floquet-Bloch theorem on the original nonlinear wave problem. Moreover, previous analytical study on the nonlinear systems with quadratic nonlinearity has indicated that, the multiple-time scales perturbation technique with at least second-order accuracy ought to be used to capture the evolution behaviors of amplitude and phase with multiple harmonic contents [45], unless one to two internal resonance is triggered [46]. ...
Inspired by the spatiotemporally modulated metamaterials, we propose a one-dimensional autoparametric periodic structure induced by nonlinear inertia that is realized simply by constructing an array of pendula coupled to local oscillators. The mechanism of autoparametric modulation is revealed using a unit cell. The evolution equations for a superlattice describing the global dynamics of the wave-vibration coupling are obtained analytically by combining the Floquet-Bloch theorem and the multiple-time scales perturbation technique. The band structure exists corresponding to the fixed point in the phase space, wherein large amplitude local vibrations with distributed traveling phases modulate the inertial properties of the host pendula propagating with a small amplitude signal wave by playing the role of a pump wave. Non-reciprocal dispersion relations are demonstrated both by numerical results and perturbation results with two-order accuracy. Selective wave amplification with two-mode directional emission is observed in a two-port system which is composed of a sandwich composite medium. The present research based on the proposed autoparametric structure paves a way towards building tunable wave manipulators, vibration energy migrators and mechanical amplifiers.
... To analyze nonlinear wave propagation in periodic structures, many analytical tools have been developed, including those based on the method of harmonic balance [31,32], transfer matrix approaches [29,33,34], and perturbation methods [35][36][37][38][39][40]. In the weakly nonlinear regime, perturbation methods derive closed-form nonlinear wave approximations without a priori knowl-edge of the solution sought [41]. ...
... The simplest perturbation method, the straightforward expansion, applies only in limited circumstances where the expansion of nonlinear terms does not generate secular terms at the leading order [28], or is only required to hold for short times and/or spatial extents [42]. The Lindstedt-Poincaré method [35,40] and the method of multiple scales (MMS) [36,37,39,43] introduce additional expansions for the frequency and time, respectively. They yield solutions to a wider class of problems and, as such, can be used to study amplitude-dependent dispersion corrections [37,39], invariant multi-harmonic waves and their stability [36], and internally resonant waves [39,43,44]. ...
... The majority of existing perturbation studies consider weakly nonlinear wave propagation in an infinite medium, as excited (for example) by initial conditions spanning the entire spatial domain [36,44]. Infinite medium studies identify a nonlinear frequency and wavenumber pair that shifts with amplitude away from the linear band structure [35][36][37]44,45], without distinguishing which of the two quantities (frequency or wavenumber) is fixed and which is shifted. However, for semi-infinite nonlinear media with boundaryexcited nonlinear waves, the frequency is fixed by the source and thus the wavenumber is considered to shift [46]. ...
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.
... In the literature, study of nonlinear metamaterials is often focused on obtaining the band structure analytically or numerically [26,27]. The former uses perturbation techniques (such as Lindstedt-Poincare [28], multiple scales [29], and homotopy analysis [30]), while the latter applies frequency and spectro-spatial analysis [31][32][33][34]. ...
Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasiperiodic metamaterial with periodic local resonator attachments. We model a 1-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasiperiodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasiperiodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps.
... In this context, the investigation of periodic lattices featuring bi-stable elements has received substantial attention recently. Notably, emphasis is placed on amplitude-dependent behavior [6][7][8], dispersion analysis and propagation of solitary waves [9][10][11][12] and tunability of wave propagation characteristics [13]. The comprehensive investigation of chaotic behavior in such systems remains uncommon. ...
The interest for the study of metamaterials and metastructures has been rapidly increasing in the scientific community. A number of different approaches for the configuration of such systems have already shown to produce unique and potentially useful dynamic behavior, such as broadband vibration attenuation and negative mechanical properties. Locally resonant metamaterials are of special interest in this domain because of its versatility in creating wide bandgaps for wave propagation while being relatively simple in terms of analysis and experimental reproduction. Two specific subcategories of this type of systems have shown to enable specially promising bandgap formation behavior: nonlinear resonators and functionally graded, or rainbow, metamaterials. This work presents the yet unexplored combination of these two characteristics, showing that graded bistable nonlinear resonators can further widen the broadband vibration attenuation for a simple 1-D periodic lattice system due to chaotic response. This work also presents an contribution on the phenomenology of the vibration attenuation mechanisms with chaotic behavior.
... In the literature, study of nonlinear metamaterials is often focused on obtaining the band structure analytically or numerically [24,25]. The former uses perturbation techniques (such as Lindstedt-Poincare [26], multiple scales [27] and homotopy analysis [28]), while the latter applies frequency and spectrospatial analysis [29][30][31][32]. ...
Metamaterials have been shown to benefit from the addition of local resonators, nonlinear elements, or topological properties, gaining features such as additional bandgaps and localized vibration modes. However, there is currently no work in the literature that examines a metamaterial system including all three elements. In this work, we model a 1-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasiperiodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The infinite chain response of the proposed system is solved through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasiperiodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators.
