Propagation_characteristics_of_an_elastic_bar_coup_acoustic_metamaterials_数学计算

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/638a1c5a9f0e6d0d655fab5c/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/638a1c5a9f0e6d0d655fab5c/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Nonlinear Dyn</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls1 ws0">https://doi.org/10.1007/s11071-022-08028-8</div><div class="t m0 x2 h3 y3 ff1 fs1 fc1 sc0 ls2 ws0">ORIGINAL PAPER</div><div class="t m0 x1 h4 y4 ff2 fs2 fc0 sc0 ls3 ws0">Propagation characteristics of an elastic bar coupled with a</div><div class="t m0 x1 h4 y5 ff2 fs2 fc0 sc0 ls3 ws0">discrete snap-through element</div><div class="t m0 x1 h5 y6 ff2 fs3 fc0 sc0 ls4 ws0">Aneesh Bhattacharyya · Sondipon Adhikari</div><div class="t m0 x1 h2 y7 ff1 fs0 fc0 sc0 ls5 ws0">Received: 18 April 2022 / Accepted: 14 October 2022</div><div class="t m0 x1 h2 y8 ff1 fs0 fc0 sc0 ls6 ws0">© The Author(s) 2022</div><div class="t m0 x1 h6 y9 ff2 fs3 fc0 sc0 ls7 ws0">Abstract Wave propagation characteristics of an</div><div class="t m0 x1 h6 ya ff1 fs3 fc0 sc0 ls7 ws0">elastic bar coupled at one end with a single degree of</div><div class="t m0 x1 h6 yb ff1 fs3 fc0 sc0 ls8 ws0">freedom, bi-stable, essentially nonlinear snap-through</div><div class="t m0 x1 h6 yc ff1 fs3 fc0 sc0 ls7 ws0">element are considered. The free end of the bar is</div><div class="t m0 x1 h6 yd ff1 fs3 fc0 sc0 ls7 ws0">subjected to sinusoidal excitations. A novel approach</div><div class="t m0 x1 h6 ye ff1 fs3 fc0 sc0 ls7 ws0">based on multiple time scales and harmonic balance</div><div class="t m0 x1 h6 yf ff1 fs3 fc0 sc0 ls8 ws0">method has been proposed to analytically investigate</div><div class="t m0 x1 h6 y10 ff1 fs3 fc0 sc0 ls7 ws0">the reﬂected wave from the nonlinear interface and the</div><div class="t m0 x1 h6 y11 ff1 fs3 fc0 sc0 ls7 ws0">dynamic response of the snap-through element. A uni-</div><div class="t m0 x1 h6 y12 ff1 fs3 fc0 sc0 ls7 ws0">ﬁed approach to the non-dimensional representation of</div><div class="t m0 x1 h6 y13 ff1 fs3 fc0 sc0 ls7 ws0">the governing equations of motion, boundary condi-</div><div class="t m0 x1 h6 y14 ff1 fs3 fc0 sc0 ls8 ws0">tions and system parameters, which is consistent across</div><div class="t m0 x1 h6 y15 ff1 fs3 fc0 sc0 ls7 ws0">all the externally applied excitation frequencies and</div><div class="t m0 x1 h6 y16 ff1 fs3 fc0 sc0 ls7 ws0">excitation amplitudes, has been developed. Through</div><div class="t m0 x1 h6 y17 ff1 fs3 fc0 sc0 ls7 ws0">Taylor series expansion of the non-autonomous forc-</div><div class="t m0 x1 h6 y18 ff1 fs3 fc0 sc0 ls8 ws0">ing functions arising in the governing differential equa-</div><div class="t m0 x1 h6 y19 ff1 fs3 fc0 sc0 ls7 ws0">tions and natural boundary condition about an initial</div><div class="t m0 x1 h6 y1a ff1 fs3 fc0 sc0 ls7 ws0">stable conﬁguration of the system and the proposed</div><div class="t m0 x1 h6 y1b ff1 fs3 fc0 sc0 ls8 ws0">asymptotic method, approximate closed-form analyti-</div><div class="t m0 x1 h6 y1c ff1 fs3 fc0 sc0 ls7 ws0">cal solutions have been derived for sufﬁciently small</div><div class="t m0 x1 h2 y1d ff2 fs0 fc0 sc0 ls0 ws0">Supplementary Information The online version contains</div><div class="t m0 x1 h2 y1e ff1 fs0 fc0 sc0 ls9 ws0">supplementary material available at https://doi.org/10.1007/</div><div class="t m0 x1 h2 y1f ff1 fs0 fc2 sc0 lsa ws0">s11071-022-08028-8.</div><div class="t m0 x1 h2 y20 ff1 fs0 fc0 sc0 ls0 ws0">A. Bhattacharyya</div><div class="t m0 x1 h2 y21 ff1 fs0 fc0 sc0 ls5 ws0">Viterbi School of Engineering, University of Southern</div><div class="t m0 x1 h2 y22 ff1 fs0 fc0 sc0 ls3 ws0">California (USC), Los Angeles, California, USA</div><div class="t m0 x1 h2 y23 ff1 fs0 fc0 sc0 ls0 ws0">e-mail: aneeshbh@usc.edu</div><div class="t m0 x1 h2 y24 ff1 fs0 fc0 sc0 ls5 ws0">S. Adhikari (</div><div class="t m0 x3 h7 y25 ff4 fs4 fc0 sc0 ls3 ws0">B</div><div class="t m0 x4 h2 y26 ff1 fs0 fc0 sc0 ls3 ws0">)</div><div class="t m0 x1 h2 y27 ff1 fs0 fc0 sc0 ls0 ws0">James Watt School of Engineering, University of Glasgow</div><div class="t m0 x1 h2 y28 ff1 fs0 fc0 sc0 ls1 ws0">(UoG), Glasgow, UK</div><div class="t m0 x1 h2 y29 ff1 fs0 fc0 sc0 ls0 ws0">e-mail: sondipon.adhikari@glasgow.ac.uk</div><div class="t m0 x5 h6 y9 ff1 fs3 fc0 sc0 ls8 ws0">amplitudes of the excitation pulse. Numerical results</div><div class="t m0 x5 h6 ya ff1 fs3 fc0 sc0 ls7 ws0">obtained through a ﬁnite difference algorithm validate</div><div class="t m0 x5 h6 yb ff1 fs3 fc0 sc0 ls8 ws0">the asymptotic model for the same small amplitudes</div><div class="t m0 x5 h6 yc ff1 fs3 fc0 sc0 ls8 ws0">of the excitation pulse. A stability analysis has been</div><div class="t m0 x5 h6 yd ff1 fs3 fc0 sc0 ls8 ws0">subsequently performed for the discrete snap-through</div><div class="t m0 x5 h6 ye ff1 fs3 fc0 sc0 ls7 ws0">element by using the extended Floquet theory for suf-</div><div class="t m0 x5 h6 yf ff1 fs3 fc0 sc0 ls8 ws0">ﬁciently large amplitudes of the excitation pulse by</div><div class="t m0 x5 h6 y10 ff1 fs3 fc0 sc0 ls7 ws0">approximating the displacement at the nonlinear inter-</div><div class="t m0 x5 h6 y11 ff1 fs3 fc0 sc0 ls7 ws0">face as a sinusoidal function of time, and the Math-</div><div class="t m0 x5 h6 y12 ff1 fs3 fc0 sc0 ls7 ws0">ieu plot of the excitation frequency vs the excitation</div><div class="t m0 x5 h6 y13 ff1 fs3 fc0 sc0 ls7 ws0">amplitude showing the stable and unstable regions for</div><div class="t m0 x5 h6 y14 ff1 fs3 fc0 sc0 ls4 ws0">the motion of the snap-through element has been gen-</div><div class="t m0 x5 h6 y15 ff1 fs3 fc0 sc0 ls7 ws0">erated. The expressions derived here give the most</div><div class="t m0 x5 h6 y16 ff1 fs3 fc0 sc0 ls7 ws0">comprehensive and consistent description of the wave</div><div class="t m0 x5 h6 y17 ff1 fs3 fc0 sc0 ls7 ws0">propagation characteristics and the motion of the snap-</div><div class="t m0 x5 h6 y18 ff1 fs3 fc0 sc0 ls7 ws0">through element, which can be directly used in ﬁnite</div><div class="t m0 x5 h6 y19 ff1 fs3 fc0 sc0 ls7 ws0">difference analysis over a wide range of parameter val-</div><div class="t m0 x5 h6 y1a ff1 fs3 fc0 sc0 ls7 ws0">ues of the excitation pulse.</div><div class="t m0 x5 h6 y2a ff2 fs3 fc0 sc0 lsb ws0">Keywords Essentially nonlinear systems · Bi-stable</div><div class="t m0 x5 h6 y2b ff1 fs3 fc0 sc0 ls7 ws0">snap-through element · Multiple time scales analysis ·</div><div class="t m0 x5 h6 y2c ff1 fs3 fc0 sc0 ls7 ws0">Harmonic balance method · Extended Floquet theory ·</div><div class="t m0 x5 h6 y2d ff1 fs3 fc0 sc0 ls7 ws0">Finite difference method</div><div class="t m0 x5 h5 y2e ff2 fs3 fc0 sc0 ls7 ws0">1 Introduction</div><div class="t m0 x5 h6 y2f ff1 fs3 fc0 sc0 ls8 ws0">Wave propagation in an elastic continuum is a sub-</div><div class="t m0 x5 h6 y30 ff1 fs3 fc0 sc0 ls7 ws0">ject of interest in many ﬁelds of engineering [1] and</div><div class="t m0 x5 h6 y31 ff1 fs3 fc0 sc0 ls7 ws0">has been explored for several decades from a theoret-</div><div class="t m0 x5 h6 y32 ff1 fs3 fc0 sc0 ls7 ws0">ical, computational and experimental perspective. In</div><div class="t m0 x6 h8 y33 ff6 fs5 fc0 sc0 ls3 ws0">123</div><div class="t m0 x7 h9 y34 ff7 fs6 fc3 sc0 ls3 ws0">Content courtesy of Springer Nature, terms of use apply. Rights reserved.</div></div><div class="pi" data-data='{"ctm":[1.754739,0.000000,0.000000,1.754739,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/638a1c5a9f0e6d0d655fab5c/bg2.jpg"><div class="t m0 x8 h2 y35 ff1 fs0 fc0 sc0 ls0 ws0">A. Bhattacharyya, S. Adhikari</div><div class="t m0 x1 h6 y36 ff1 fs3 fc0 sc0 ls8 ws0">this context, a one/two-dimensional elastic continuum</div><div class="t m0 x1 h6 y37 ff1 fs3 fc0 sc0 ls8 ws0">is more colloquially called a waveguide since they min-</div><div class="t m0 x1 h6 y38 ff1 fs3 fc0 sc0 ls8 ws0">imize energy loss by restricting the wave propagation in</div><div class="t m0 x1 h6 y39 ff1 fs3 fc0 sc0 ls7 ws0">a speciﬁc direction/plane. These one-dimensional elas-</div><div class="t m0 x1 h6 y3a ff1 fs3 fc0 sc0 ls7 ws0">tic waveguides can be dispersion free, such as an elas-</div><div class="t m0 x1 h6 y3b ff1 fs3 fc0 sc0 ls7 ws0">tic bar propagating longitudinal waves, or can exhibit</div><div class="t m0 x1 h6 y3c ff1 fs3 fc0 sc0 ls7 ws0">dispersion, such as ﬂexural waves in Euler–Bernoulli</div><div class="t m0 x1 h6 y3d ff1 fs3 fc0 sc0 ls7 ws0">or Timoshenko beams. However, in practical engi-</div><div class="t m0 x1 h6 y3e ff1 fs3 fc0 sc0 ls8 ws0">neering structures, such waveguides invariably inter-</div><div class="t m0 x1 h6 y3f ff1 fs3 fc0 sc0 ls7 ws0">act with structural components whose s patial scales</div><div class="t m0 x1 h6 y40 ff1 fs3 fc0 sc0 ls7 ws0">are much smaller than the elastic continuum and the</div><div class="t m0 x1 h6 y41 ff1 fs3 fc0 sc0 ls4 ws0">wavelengthof the wavephenomena that theyencounter.</div><div class="t m0 x1 h6 y42 ff1 fs3 fc0 sc0 ls7 ws0">Such structural components can be modelled as dis-</div><div class="t m0 x1 h6 y43 ff1 fs3 fc0 sc0 ls8 ws0">crete elements by considering point masses, stiffness</div><div class="t m0 x1 h6 y44 ff1 fs3 fc0 sc0 ls7 ws0">and damping. The wave propagation characteristics</div><div class="t m0 x1 h6 y45 ff1 fs3 fc0 sc0 ls8 ws0">when elastic waveguides interact with linear discrete</div><div class="t m0 x1 h6 y46 ff1 fs3 fc0 sc0 ls8 ws0">elements are well known [2]. The present work primar-</div><div class="t m0 x1 h6 y47 ff1 fs3 fc0 sc0 ls8 ws0">ily dwells on the effect of weakly/essentially nonlinear</div><div class="t m0 x1 h6 y48 ff1 fs3 fc0 sc0 ls7 ws0">discrete elements on the wave propagation character-</div><div class="t m0 x1 h6 y49 ff1 fs3 fc0 sc0 ls3 ws0">istics in non-dispersive elastic waveguides. Since, in</div><div class="t m0 x1 h6 y4a ff1 fs3 fc0 sc0 ls8 ws0">general, closed-form analytical solutions are seldom</div><div class="t m0 x1 h6 y4b ff1 fs3 fc0 sc0 ls7 ws0">available for wave propagation problems in nonlinear</div><div class="t m0 x1 h6 y4c ff1 fs3 fc0 sc0 ls8 ws0">dynamical systems, in this paper, we propose a sys-</div><div class="t m0 x1 h6 y4d ff1 fs3 fc0 sc0 ls7 ws0">tematic approach based on classical perturbation tech-</div><div class="t m0 x1 h6 y4e ff1 fs3 fc0 sc0 ls7 ws0">niques like the method of multiple scales (MMS) to ﬁnd</div><div class="t m0 x1 h6 y4f ff1 fs3 fc0 sc0 ls7 ws0">the solutions. The analytical solutions are validated by</div><div class="t m0 x1 h6 y50 ff1 fs3 fc0 sc0 ls8 ws0">numerical results obtained computationally.</div><div class="t m0 x9 h6 y51 ff1 fs3 fc0 sc0 ls7 ws0">This paper focuses on the propagation characteris-</div><div class="t m0 x1 h6 y52 ff1 fs3 fc0 sc0 ls8 ws0">tics in an elastic bar coupled with a discrete nonlin-</div><div class="t m0 x1 h6 y53 ff1 fs3 fc0 sc0 ls7 ws0">ear end attachment. In general discrete nonlinear end</div><div class="t m0 x1 h6 y54 ff1 fs3 fc0 sc0 ls7 ws0">attachments can be mathematically modelled as the</div><div class="t m0 x1 h6 y55 ff1 fs3 fc0 sc0 ls8 ws0">Dufﬁng oscillator with a stiffness nonlinearity or a Van</div><div class="t m0 x1 h6 y56 ff1 fs3 fc0 sc0 ls7 ws0">der Pol oscillator with nonlinear damping [3] or a geo-</div><div class="t m0 x1 h6 y57 ff1 fs3 fc0 sc0 ls8 ws0">metrically nonlinear snap-through truss. The general</div><div class="t m0 x1 h6 y58 ff1 fs3 fc0 sc0 ls7 ws0">governing differential equation of motion of such non-</div><div class="t m0 x1 h6 y59 ff1 fs3 fc0 sc0 ls4 ws0">linear end attachments can be given by:</div><div class="t m0 xa ha y5a ff8 fs3 fc0 sc0 ls3 ws0">d</div><div class="t m0 xb hb y5b ff1 fs7 fc0 sc0 ls3 ws0">2</div><div class="t m0 x9 ha y5c ff8 fs3 fc0 sc0 ls3 ws0">x(t)</div><div class="t m0 xc ha y5d ff8 fs3 fc0 sc0 lsc ws0">dt</div><div class="t m0 xd hb y5e ff1 fs7 fc0 sc0 ls3 ws0">2</div><div class="t m0 xe hc y5f ff5 fs3 fc0 sc0 ls3 ws0">+ f</div><div class="t m0 xf hd y60 ff9 fs3 fc0 sc0 ls3 ws0">(</div><div class="t m0 x4 hc y5f ff8 fs3 fc0 sc0 ls3 ws0">x(t), ˙x(t), t</div><div class="t m0 x10 hd y60 ff9 fs3 fc0 sc0 ls3 ws0">)</div><div class="t m0 x11 h6 y5f ff5 fs3 fc0 sc0 ls3 ws0">= 0(1)</div><div class="t m0 x1 h6 y61 ff1 fs3 fc0 sc0 ls7 ws0">There are quite a few known methods to solve non-</div><div class="t m0 x1 h6 y62 ff1 fs3 fc0 sc0 ls7 ws0">linear equations of the form (1). Some of them are</div><div class="t m0 x1 h6 y63 ff1 fs3 fc0 sc0 ls7 ws0">eigen/modal analysis, perturbation techniques, har-</div><div class="t m0 x1 h6 y64 ff1 fs3 fc0 sc0 ls4 ws0">monic balance method and method of averaging [4].</div><div class="t m0 x1 h6 y65 ff1 fs3 fc0 sc0 ls7 ws0">Analysis of dynamical systems comprising of non-</div><div class="t m0 x1 h6 y66 ff1 fs3 fc0 sc0 ls8 ws0">linear oscillators can be found in the existing litera-</div><div class="t m0 x1 h6 y67 ff1 fs3 fc0 sc0 ls7 ws0">ture dating back as early as the 80s. In 1982, Nayfeh</div><div class="t m0 x1 h6 y68 ff1 fs3 fc0 sc0 ls3 ws0">[5] investigated the response of single degree of free-</div><div class="t m0 x1 h6 y69 ff1 fs3 fc0 sc0 ls8 ws0">dom systems with quadratic and cubic nonlineari-</div><div class="t m0 x1 h6 y6a ff1 fs3 fc0 sc0 ls7 ws0">ties to a sub-harmonic excitation. In this paper, one</div><div class="t m0 x5 h6 y6b ff1 fs3 fc0 sc0 ls4 ws0">can get a general idea of how to apply the multiple</div><div class="t m0 x5 h6 y6c ff1 fs3 fc0 sc0 ls7 ws0">time scales analysis and harmonic balance method to</div><div class="t m0 x5 h6 y6d ff1 fs3 fc0 sc0 ls7 ws0">solve a nonlinear equation. In the context of more</div><div class="t m0 x5 h6 y6e ff1 fs3 fc0 sc0 ls7 ws0">recent works, Young Sup Lee et al. in [6] studied the</div><div class="t m0 x5 h6 y6f ff1 fs3 fc0 sc0 ls8 ws0">dynamics of a two degree of freedom system compris-</div><div class="t m0 x5 h6 y70 ff1 fs3 fc0 sc0 ls7 ws0">ing of a grounded linear oscillator coupled to a light</div><div class="t m0 x5 h6 y71 ff1 fs3 fc0 sc0 ls8 ws0">mass through an essentially nonlinear stiffness. This</div><div class="t m0 x5 h6 y72 ff1 fs3 fc0 sc0 ls8 ws0">seemingly simple system showed a very complicated</div><div class="t m0 x5 h6 y73 ff1 fs3 fc0 sc0 ls7 ws0">dynamical behaviour due to the essential nonlinear-</div><div class="t m0 x5 h6 y74 ff1 fs3 fc0 sc0 ls7 ws0">ity of the coupling stiffness even though the nonlinear</div><div class="t m0 x5 h6 y75 ff1 fs3 fc0 sc0 ls7 ws0">attachment was chosen to be very light, compared to the</div><div class="t m0 x5 h6 y76 ff1 fs3 fc0 sc0 ls8 ws0">main linear oscillator. In the context of energy harvest-</div><div class="t m0 x5 h6 y77 ff1 fs3 fc0 sc0 ls8 ws0">ing using nonlinear oscillators, A. F. Vakakis [7] studied</div><div class="t m0 x5 h6 y78 ff1 fs3 fc0 sc0 ls7 ws0">the effect of inducing passive nonlinear energy sinks</div><div class="t m0 x5 h6 y79 ff1 fs3 fc0 sc0 ls8 ws0">in linear vibrating systems. He considered a system</div><div class="t m0 x5 h6 y7a ff1 fs3 fc0 sc0 ls7 ws0">composed of strongly coupled, grounded damped lin-</div><div class="t m0 x5 h6 y7b ff1 fs3 fc0 sc0 ls7 ws0">ear oscillators with a strongly nonlinear end attachment</div><div class="t m0 x5 h6 y7c ff1 fs3 fc0 sc0 ls7 ws0">and deriveda set of modulation equations using an aver-</div><div class="t m0 x5 h6 y7d ff1 fs3 fc0 sc0 ls4 ws0">aging method and showed that nonlinear attachments</div><div class="t m0 x5 h6 y7e ff1 fs3 fc0 sc0 ls7 ws0">if designed properly, can act as passive energy sinks</div><div class="t m0 x5 h6 y7f ff1 fs3 fc0 sc0 ls8 ws0">of spurious vibrations. Vakakis et al. in [8] studied the</div><div class="t m0 x5 h6 y80 ff1 fs3 fc0 sc0 ls7 ws0">dynamics of linear discrete systems connected to local</div><div class="t m0 x5 h6 y81 ff1 fs3 fc0 sc0 ls7 ws0">essentially nonlinear end attachments and observedthat</div><div class="t m0 x5 h6 y82 ff1 fs3 fc0 sc0 ls8 ws0">if the system parameters are chosen properly and if the</div><div class="t m0 x5 h6 y83 ff1 fs3 fc0 sc0 ls8 ws0">external excitation is sufﬁciently strong, irreversible,</div><div class="t m0 x5 h6 y84 ff1 fs3 fc0 sc0 ls7 ws0">passive transfer of energy occurs from the chain to the</div><div class="t m0 x5 h6 y85 ff1 fs3 fc0 sc0 ls7 ws0">attachment. Another evidence of a strongly nonlinear</div><div class="t m0 x5 h6 y86 ff1 fs3 fc0 sc0 ls7 ws0">oscillator acting as a passive energy sink can be found</div><div class="t m0 x5 h6 y87 ff1 fs3 fc0 sc0 ls8 ws0">in the work of Manevitch et al. [9], where the model</div><div class="t m0 x5 h6 y88 ff1 fs3 fc0 sc0 ls8 ws0">considered was a semi-inﬁnite chain of coupled lin-</div><div class="t m0 x5 h6 y89 ff1 fs3 fc0 sc0 ls8 ws0">ear, grounded oscillators, weakly coupled to a strongly</div><div class="t m0 x5 h6 y8a ff1 fs3 fc0 sc0 ls7 ws0">nonlinear oscillator at its free end. An important ﬁnd-</div><div class="t m0 x5 h6 y8b ff1 fs3 fc0 sc0 ls4 ws0">ing of this work was that energy pumping can occur</div><div class="t m0 x5 h6 y8c ff1 fs3 fc0 sc0 ls7 ws0">even when there is no damping in the system. In another</div><div class="t m0 x5 h6 y8d ff1 fs3 fc0 sc0 ls10 ws0">work [10] on provingthe vibration isolating capabilities</div><div class="t m0 x5 h6 y8e ff1 fs3 fc0 sc0 ls8 ws0">of a single degree of freedom nonlinear oscillator with</div><div class="t m0 x5 h6 y8f ff1 fs3 fc0 sc0 ls8 ws0">high-static–low-dynamic stiffness, A. Carrella and co-</div><div class="t m0 x5 h6 y90 ff1 fs3 fc0 sc0 ls7 ws0">workers considered a lumped mass model supported by</div><div class="t m0 x5 h6 y91 ff1 fs3 fc0 sc0 ls7 ws0">three springs to the base and compared the force and</div><div class="t m0 x5 h6 y92 ff1 fs3 fc0 sc0 ls8 ws0">displacement transmissibility by ﬁnding closed-form</div><div class="t m0 x5 h6 y93 ff1 fs3 fc0 sc0 ls8 ws0">solutions, considering a harmonic force on the mass</div><div class="t m0 x5 h6 y94 ff1 fs3 fc0 sc0 ls4 ws0">and a harmonic motion of the base in each case sepa-</div><div class="t m0 x5 h6 y95 ff1 fs3 fc0 sc0 ls7 ws0">rately. Even in the cases of asymmetrical loading con-</div><div class="t m0 x5 h6 y96 ff1 fs3 fc0 sc0 ls7 ws0">ditions, besides energy harvesting, the effectiveness of</div><div class="t m0 x5 h6 y97 ff1 fs3 fc0 sc0 ls8 ws0">a nonlinear oscillator with high-static–low-dynamic-</div><div class="t m0 x5 h6 y98 ff1 fs3 fc0 sc0 ls8 ws0">stiffness as a vibration isolator compared to a linear</div><div class="t m0 x5 h6 y99 ff1 fs3 fc0 sc0 ls7 ws0">oscillator has been proved in [11] by A. Abolfathi et.</div><div class="t m0 x5 h6 y9a ff1 fs3 fc0 sc0 ls8 ws0">al. Very recently, in a series of papers, Karlicic et al.</div><div class="t m0 x5 h6 y9b ff1 fs3 fc0 sc0 ls3 ws0">[12–15] employed the incremental harmonic balance</div><div class="t m0 x1 h8 y33 ff6 fs5 fc0 sc0 ls3 ws0">123</div><div class="t m0 x7 h9 y34 ff7 fs6 fc3 sc0 ls3 ws0">Content courtesy of Springer Nature, terms of use apply. Rights reserved.</div></div><div class="pi" data-data='{"ctm":[1.754739,0.000000,0.000000,1.754739,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/638a1c5a9f0e6d0d655fab5c/bg3.jpg"><div class="t m0 x1 h2 y35 ff1 fs0 fc0 sc0 ls9 ws0">Propagation characteristics of an elastic</div><div class="t m0 x1 h6 y36 ff1 fs3 fc0 sc0 ls7 ws0">(IHB) method in the context of nonlinear piezoelectric</div><div class="t m0 x1 h6 y37 ff1 fs3 fc0 sc0 ls7 ws0">energy harvesting. Also, a recent paper [16]byMQNiu</div><div class="t m0 x1 h6 y38 ff1 fs3 fc0 sc0 ls7 ws0">and LQ Chen studies the optimization of a quasi-zero</div><div class="t m0 x1 h6 y39 ff1 fs3 fc0 sc0 ls3 ws0">stiffness isolator via oblique beams.</div><div class="t m0 x9 h6 y3a ff1 fs3 fc0 sc0 ls7 ws0">Evidence of the use of a geometrically and essen-</div><div class="t m0 x1 h6 y3b ff1 fs3 fc0 sc0 ls8 ws0">tially nonlinear snap-through truss to effectively absorb</div><div class="t m0 x1 h6 y3c ff1 fs3 fc0 sc0 ls8 ws0">vibrations of the linear oscillator can also be found in</div><div class="t m0 x1 h6 y3d ff1 fs3 fc0 sc0 ls8 ws0">the existing literature. Avramov and Mikhlin [ 17 ]used</div><div class="t m0 x1 h6 y3e ff1 fs3 fc0 sc0 ls8 ws0">the nonlinear normal mode (NNM) approach to investi-</div><div class="t m0 x1 h6 y3f ff1 fs3 fc0 sc0 ls8 ws0">gate the phenomenon of elastic oscillations absorption</div><div class="t m0 x1 h6 y40 ff1 fs3 fc0 sc0 ls8 ws0">by a snap-through truss by considering a single degree</div><div class="t m0 x1 h6 y41 ff1 fs3 fc0 sc0 ls7 ws0">of freedom linear oscillator to represent an elastic con-</div><div class="t m0 x1 h6 y42 ff1 fs3 fc0 sc0 ls7 ws0">tinuum and a nonlinear absorber with three equilib-</div><div class="t m0 x1 h6 y43 ff1 fs3 fc0 sc0 ls8 ws0">rium positions (snap-through truss) attached to the lin-</div><div class="t m0 x1 h6 y44 ff1 fs3 fc0 sc0 ls8 ws0">ear oscillator. Considering the same system as in [17],</div><div class="t m0 x1 h6 y45 ff1 fs3 fc0 sc0 ls7 ws0">the authors in [18] studied the forced oscillations of the</div><div class="t m0 x1 h6 y46 ff1 fs3 fc0 sc0 ls3 ws0">snap-through truss close to its stable equilibrium posi-</div><div class="t m0 x1 h6 y47 ff1 fs3 fc0 sc0 ls8 ws0">tion by the classical multiple scales method. In their</div><div class="t m0 x1 h6 y48 ff1 fs3 fc0 sc0 ls8 ws0">next work on snap-through truss [19], Avramov and</div><div class="t m0 x1 h6 y49 ff1 fs3 fc0 sc0 ls3 ws0">Mikhlin investigated the possibility of absorption of</div><div class="t m0 x1 h6 y4a ff1 fs3 fc0 sc0 ls8 ws0">the forced oscillations underwent by a linear elastic sys-</div><div class="t m0 x1 h6 y4b ff1 fs3 fc0 sc0 ls8 ws0">tem using a snap-through truss. They used a combina-</div><div class="t m0 x1 h6 y4c ff1 fs3 fc0 sc0 ls7 ws0">tion of the nonlinear normal vibrations modes method,</div><div class="t m0 x1 h6 y4d ff1 fs3 fc0 sc0 ls7 ws0">the Rauscher approach and the asymptotic analysis to</div><div class="t m0 x1 h6 y4e ff1 fs3 fc0 sc0 ls8 ws0">study the forced oscillations of the essentially nonlin-</div><div class="t m0 x1 h6 y4f ff1 fs3 fc0 sc0 ls7 ws0">ear two degree of freedom system. More work on snap-</div><div class="t m0 x1 h6 y50 ff1 fs3 fc0 sc0 ls7 ws0">through motions of a system can be found in [20] where</div><div class="t m0 x1 h6 y51 ff1 fs3 fc0 sc0 ls7 ws0">the authors have studied the nonlinear normal mode</div><div class="t m0 x1 h6 y52 ff1 fs3 fc0 sc0 ls7 ws0">(NNM) of the snap-through motions of a shallow arch.</div><div class="t m0 x1 h6 y53 ff1 fs3 fc0 sc0 ls7 ws0">The stability of the snap-through motions was studied</div><div class="t m0 x1 h6 y54 ff1 fs3 fc0 sc0 ls7 ws0">by using the Ince algebraization and the method of Hill</div><div class="t m0 x1 h6 y55 ff1 fs3 fc0 sc0 ls8 ws0">determinants. In the context of studying the dynamics</div><div class="t m0 x1 h6 y56 ff1 fs3 fc0 sc0 ls7 ws0">of a single degree of freedom linear oscillator subjected</div><div class="t m0 x1 h6 y57 ff1 fs3 fc0 sc0 ls7 ws0">to harmonic forcing and coupled to a geometrically</div><div class="t m0 x1 h6 y58 ff1 fs3 fc0 sc0 ls4 ws0">nonlinear light end attachment by complexiﬁcation-</div><div class="t m0 x1 h6 y59 ff1 fs3 fc0 sc0 ls7 ws0">averaging method, there is a work done in a recent paper</div><div class="t m0 x1 h6 y9c ff1 fs3 fc0 sc0 ls3 ws0">[21] by Yang Liu et. al. Very recently there have been a</div><div class="t m0 x1 h6 y9d ff1 fs3 fc0 sc0 ls4 ws0">couple of works [22,23], based on modal interactions,</div><div class="t m0 x1 h6 y9e ff1 fs3 fc0 sc0 ls7 ws0">by A. Mojahed and co-workers on the effectiveness of a</div><div class="t m0 x1 h6 y9f ff1 fs3 fc0 sc0 ls8 ws0">geometrically nonlinear oscillator in absorbing vibra-</div><div class="t m0 x1 h6 ya0 ff1 fs3 fc0 sc0 ls8 ws0">tions, mitigating shocks and transferring energy in a</div><div class="t m0 x1 h6 ya1 ff1 fs3 fc0 sc0 ls4 ws0">targeted manner.</div><div class="t m0 x9 h6 ya2 ff1 fs3 fc0 sc0 ls7 ws0">The analysis of a dynamical system comprising of an</div><div class="t m0 x1 h6 ya3 ff1 fs3 fc0 sc0 ls8 ws0">elastic continuum with discrete nonlinearity has been</div><div class="t m0 x1 h6 ya4 ff1 fs3 fc0 sc0 ls8 ws0">a subject of interest in mechanics for quite some time</div><div class="t m0 x1 h6 ya5 ff1 fs3 fc0 sc0 ls4 ws0">now. Nayfeh and Asfar [24] investigated the r esponse of</div><div class="t m0 x1 h6 ya6 ff1 fs3 fc0 sc0 ls8 ws0">a bar constrained by a nonlinear spring with cubic stiff-</div><div class="t m0 x1 h6 ya7 ff1 fs3 fc0 sc0 ls8 ws0">ness nonlinearity to harmonic excitation. By using the</div><div class="t m0 x1 h6 ya8 ff1 fs3 fc0 sc0 ls8 ws0">method of multiple time scales, they found closed-form</div><div class="t m0 x5 h6 ya9 ff1 fs3 fc0 sc0 ls7 ws0">analytical solutions for the response of the bar in the</div><div class="t m0 x5 h6 yaa ff1 fs3 fc0 sc0 ls7 ws0">different cases of resonance. Their work did not address</div><div class="t m0 x5 h6 yab ff1 fs3 fc0 sc0 ls7 ws0">the analysis of a bar coupled to a nonlinear oscillator,</div><div class="t m0 x5 h6 yac ff1 fs3 fc0 sc0 ls7 ws0">but it presented a systematic procedure based on con-</div><div class="t m0 x5 h6 yad ff1 fs3 fc0 sc0 ls7 ws0">ventional perturbation methods for ﬁnding the vibra-</div><div class="t m0 x5 h6 yae ff1 fs3 fc0 sc0 ls3 ws0">tion response of an elastic continuum with a discrete</div><div class="t m0 x5 h6 yaf ff1 fs3 fc0 sc0 ls8 ws0">nonlinearity. Considering the similar model as in [24],</div><div class="t m0 x5 h6 yb0 ff1 fs3 fc0 sc0 ls3 ws0">Lee et al. in [25] checked the validity of the asymptotic</div><div class="t m0 x5 h6 yb1 ff1 fs3 fc0 sc0 ls7 ws0">solution obtained by Nayfeh from the method of mul-</div><div class="t m0 x5 h6 yb2 ff1 fs3 fc0 sc0 ls7 ws0">tiple time scales by using the ﬁnite difference method</div><div class="t m0 x5 h6 yb3 ff1 fs3 fc0 sc0 ls7 ws0">to ﬁnd a numerical solution to the problem and then</div><div class="t m0 x5 h6 yb4 ff1 fs3 fc0 sc0 ls8 ws0">comparing it with the asymptotic solution. One more</div><div class="t m0 x5 h6 yb5 ff1 fs3 fc0 sc0 ls8 ws0">reference work in this context by Özkaya et al. [26]</div><div class="t m0 x5 h6 yb6 ff1 fs3 fc0 sc0 ls7 ws0">deals with the nonlinear vibrations of a beam-mass sys-</div><div class="t m0 x5 h6 yb7 ff1 fs3 fc0 sc0 ls8 ws0">tem by using the perturbation method of multiple time</div><div class="t m0 x5 h6 yb8 ff1 fs3 fc0 sc0 ls8 ws0">scales to ﬁnd the approximate analytical solutions. As</div><div class="t m0 x5 h6 yb9 ff1 fs3 fc0 sc0 ls7 ws0">an example of a work on an elastic continuum inter-</div><div class="t m0 x5 h6 yba ff1 fs3 fc0 sc0 ls8 ws0">acting with an essentially nonlinear end attachment,</div><div class="t m0 x5 h6 ybb ff1 fs3 fc0 sc0 ls4 ws0">we can take into account the paper by Avramov and</div><div class="t m0 x5 h6 ybc ff1 fs3 fc0 sc0 ls4 ws0">Gendelman [27] which focuses on the interaction of an</div><div class="t m0 x5 h6 ybd ff1 fs3 fc0 sc0 ls8 ws0">elastic beam subjected to a transverse periodic force,</div><div class="t m0 x5 h6 ybe ff1 fs3 fc0 sc0 ls8 ws0">with an essentially nonlinear oscillator attached to the</div><div class="t m0 x5 h6 ybf ff1 fs3 fc0 sc0 ls7 ws0">midpoint of the beam (the antinode for the ﬁrst mode</div><div class="t m0 x5 h6 yc0 ff1 fs3 fc0 sc0 ls7 ws0">of beam vibration). It was concluded from this work</div><div class="t m0 x5 h6 yc1 ff1 fs3 fc0 sc0 ls7 ws0">that the nonlinear absorber could be used effectively</div><div class="t m0 x5 h6 yc2 ff1 fs3 fc0 sc0 ls7 ws0">for absorbing the vibrations of the continuous, forced</div><div class="t m0 x5 h6 yc3 ff1 fs3 fc0 sc0 ls8 ws0">system. In the same context, Mikhlin and Reshetnikova</div><div class="t m0 x5 h6 yc4 ff1 fs3 fc0 sc0 ls3 ws0">[28] studied the dynamical interaction of an elastic sys-</div><div class="t m0 x5 h6 yc5 ff1 fs3 fc0 sc0 ls7 ws0">tem and an essentially nonlinear absorber by approxi-</div><div class="t m0 x5 h6 yc6 ff1 fs3 fc0 sc0 ls8 ws0">mating the continuous elastic system by a linear oscilla-</div><div class="t m0 x5 h6 yc7 ff1 fs3 fc0 sc0 ls8 ws0">tor with a relatively big mass and considering an essen-</div><div class="t m0 x5 h6 yc8 ff1 fs3 fc0 sc0 ls8 ws0">tially nonlinear oscillator with a relatively small mass</div><div class="t m0 x5 h6 yc9 ff1 fs3 fc0 sc0 ls8 ws0">to be linearly coupled to the bigger mass. They anal-</div><div class="t m0 x5 h6 yca ff1 fs3 fc0 sc0 ls7 ws0">ysed the nonlinear normal modes of vibration to ﬁnd</div><div class="t m0 x5 h6 ycb ff1 fs3 fc0 sc0 ls8 ws0">that a stable vibration absorption mode exists in a large</div><div class="t m0 x5 h6 ycc ff1 fs3 fc0 sc0 ls7 ws0">region of the system parameters. A paper by Krack et.</div><div class="t m0 x5 h6 ycd ff1 fs3 fc0 sc0 lsb ws0">al [29] also studies the nonlinear modal interactions in</div><div class="t m0 x5 h6 yce ff1 fs3 fc0 sc0 ls8 ws0">the presence of a friction-joint coupling between two</div><div class="t m0 x5 h6 ycf ff1 fs3 fc0 sc0 ls7 ws0">Euler–Bernoulli beams in order to ﬁnd the efﬁcacy of</div><div class="t m0 x5 h6 yd0 ff1 fs3 fc0 sc0 ls8 ws0">friction damping in the presence of such interactions.</div><div class="t m0 x12 h6 yd1 ff1 fs3 fc0 sc0 ls7 ws0">As we can see, the effectiveness of a nonlinear oscil-</div><div class="t m0 x5 h6 yd2 ff1 fs3 fc0 sc0 ls8 ws0">lator as an irreversible, passive, nonlinear energy sink</div><div class="t m0 x5 h6 yd3 ff1 fs3 fc0 sc0 ls7 ws0">and standard procedures to ﬁnd analytical solutions for</div><div class="t m0 x5 h6 yd4 ff1 fs3 fc0 sc0 ls8 ws0">the class of free/forced vibration problems involving</div><div class="t m0 x5 h6 yd5 ff1 fs3 fc0 sc0 ls8 ws0">an elastic continuum with discrete nonlinearity can be</div><div class="t m0 x5 h6 yd6 ff1 fs3 fc0 sc0 ls8 ws0">found extensively in the existing literature. In the con-</div><div class="t m0 x5 h6 yd7 ff1 fs3 fc0 sc0 ls8 ws0">text of wave propagation in an elastic continuum with</div><div class="t m0 x5 h6 yd8 ff1 fs3 fc0 sc0 ls7 ws0">a discrete nonlinear end attachment, Gendelman and</div><div class="t m0 x5 h6 yd9 ff1 fs3 fc0 sc0 ls7 ws0">Manevitch [</div><div class="t m0 x13 h6 yda ff1 fs3 fc2 sc0 lsb ws0">30] considered rectangular wave propaga-</div><div class="t m0 x6 h8 y33 ff6 fs5 fc0 sc0 ls3 ws0">123</div><div class="t m0 x7 h9 y34 ff7 fs6 fc3 sc0 ls3 ws0">Content courtesy of Springer Nature, terms of use apply. Rights reserved.</div></div><div class="pi" data-data='{"ctm":[1.754739,0.000000,0.000000,1.754739,0.000000,0.000000]}'></div></div>

<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/638a1c5a9f0e6d0d655fab5c/bg4.jpg"><div class="t m0 x8 h2 y35 ff1 fs0 fc0 sc0 ls0 ws0">A. Bhattacharyya, S. Adhikari</div><div class="t m0 x1 h6 y36 ff1 fs3 fc0 sc0 ls8 ws0">tion in an ideal elastic string attached to a strongly</div><div class="t m0 x1 h6 y37 ff1 fs3 fc0 sc0 ls7 ws0">nonlinear oscillator through a linear spring, subjected</div><div class="t m0 x1 h6 y38 ff1 fs3 fc0 sc0 ls7 ws0">to an external excitation in the form of a rectangular</div><div class="t m0 x1 h6 y39 ff1 fs3 fc0 sc0 ls8 ws0">pulse applied to the string at inﬁnity. Considering the</div><div class="t m0 x1 h6 y3a ff1 fs3 fc0 sc0 ls8 ws0">oscillator mass to be negligible and the stiffness of the</div><div class="t m0 x1 h6 y3b ff1 fs3 fc0 sc0 ls8 ws0">coupling linear spring to be very large, the approximate</div><div class="t m0 x1 h6 y3c ff1 fs3 fc0 sc0 ls7 ws0">analytical solutions were derived for the response of the</div><div class="t m0 x1 h6 y3d ff1 fs3 fc0 sc0 ls7 ws0">nonlinear oscillator. The conclusions drawn from this</div><div class="t m0 x1 h6 y3e ff1 fs3 fc0 sc0 ls8 ws0">work were that this kind of system can be successfully</div><div class="t m0 x1 h6 y3f ff1 fs3 fc0 sc0 ls7 ws0">described by a combination of asymptotic methods and</div><div class="t m0 x1 h6 y40 ff1 fs3 fc0 sc0 ls7 ws0">that energy pumping is possible from the elastic con-</div><div class="t m0 x1 h6 y41 ff1 fs3 fc0 sc0 ls7 ws0">tinuum to the nonlinear oscillator. We can ﬁnd more</div><div class="t m0 x1 h6 y42 ff1 fs3 fc0 sc0 ls8 ws0">evidence of the use of asymptotic analysis for studying</div><div class="t m0 x1 h6 y43 ff1 fs3 fc0 sc0 ls8 ws0">the wave propagation behaviour in strongly/essentially</div><div class="t m0 x1 h6 y44 ff1 fs3 fc0 sc0 ls7 ws0">nonlinear dynamical systems, composed of discrete</div><div class="t m0 x1 h6 y45 ff1 fs3 fc0 sc0 ls7 ws0">particles forming a dimer chain, in a recent work of</div><div class="t m0 x1 h6 y46 ff1 fs3 fc0 sc0 ls7 ws0">Z. Ahsan and K. R. Jayaprakash [31]. The paper uses</div><div class="t m0 x1 h6 y47 ff1 fs3 fc0 sc0 ls8 ws0">the same multiple time scales analysis to study the evo-</div><div class="t m0 x1 h6 y48 ff1 fs3 fc0 sc0 ls8 ws0">lution of solitary waves and primary pulses in granular</div><div class="t m0 x1 h6 y49 ff1 fs3 fc0 sc0 ls8 ws0">dimers with and without velocity proportional founda-</div><div class="t m0 x1 h6 y4a ff1 fs3 fc0 sc0 ls8 ws0">tion damping. In two more works [32,33], Jayaprakash</div><div class="t m0 x1 h6 y4b ff1 fs3 fc0 sc0 ls8 ws0">et al. studied the travelling waves in one-dimensional</div><div class="t m0 x1 h6 y4c ff1 fs3 fc0 sc0 ls8 ws0">nonlinear trimer granular lattice by employing numeri-</div><div class="t m0 x1 h6 y4d ff1 fs3 fc0 sc0 ls4 ws0">cal methods and the recently developed analytical pro-</div><div class="t m0 x1 h6 y4e ff1 fs3 fc0 sc0 ls8 ws0">cedure based on the singular, multi-scale perturbation</div><div class="t m0 x1 h6 y4f ff1 fs3 fc0 sc0 ls7 ws0">analysis. In another work on nonlinear trimer model</div><div class="t m0 x1 h6 y50 ff1 fs3 fc0 sc0 ls3 ws0">[34], V. Kislovsky et al. studied, both analytically and</div><div class="t m0 x1 h6 y51 ff1 fs3 fc0 sc0 ls7 ws0">numerically, the resonant mechanism governing the</div><div class="t m0 x1 h6 y52 ff1 fs3 fc0 sc0 ls8 ws0">inter-state transitions of locally excited regimes in a</div><div class="t m0 x1 h6 y53 ff1 fs3 fc0 sc0 ls7 ws0">chain of three coupled anharmonic oscillators subject</div><div class="t m0 x1 h6 y54 ff1 fs3 fc0 sc0 ls8 ws0">to localized excitation. In the context of elastic, soli-</div><div class="t m0 x1 h6 y55 ff1 fs3 fc0 sc0 ls8 ws0">tary wave scattering at softening-hardening interfaces</div><div class="t m0 x1 h6 y56 ff1 fs3 fc0 sc0 ls8 ws0">in strongly nonlinear metamaterials, there is a paper by</div><div class="t m0 x1 h6 y57 ff1 fs3 fc0 sc0 ls8 ws0">F. Fraternali et al. [35] which studies the wave propaga-</div><div class="t m0 x1 h6 y58 ff1 fs3 fc0 sc0 ls7 ws0">tion characteristics in the medium under impact load-</div><div class="t m0 x1 h6 y59 ff1 fs3 fc0 sc0 ls4 ws0">ing.</div><div class="t m0 x9 h6 y9c ff1 fs3 fc0 sc0 ls3 ws0">Based on the literature survey presented so far, it</div><div class="t m0 x1 h6 y9d ff1 fs3 fc0 sc0 ls4 ws0">can be said that the phenomenon of harmonic wave</div><div class="t m0 x1 h6 y9e ff1 fs3 fc0 sc0 ls7 ws0">propagation in an un-approximated elastic bar coupled</div><div class="t m0 x1 h6 y9f ff1 fs3 fc0 sc0 ls7 ws0">to a discrete nonlinear oscillator had not been dealt</div><div class="t m0 x1 h6 ya0 ff1 fs3 fc0 sc0 ls3 ws0">with previously in detail. So, there is a dearth of lit-</div><div class="t m0 x1 h6 ya1 ff1 fs3 fc0 sc0 ls8 ws0">erature when it comes to ﬁnding a closed-form ana-</div><div class="t m0 x1 h6 ya2 ff1 fs3 fc0 sc0 ls7 ws0">lytical solution for this kind of a wave propagation</div><div class="t m0 x1 h6 ya3 ff1 fs3 fc0 sc0 ls7 ws0">problem. The problem approached in [30]wassome-</div><div class="t m0 x1 h6 ya4 ff1 fs3 fc0 sc0 ls8 ws0">what similar to the work presented in this paper. But</div><div class="t m0 x1 h6 ya5 ff1 fs3 fc0 sc0 ls7 ws0">the results were based on some obvious approxima-</div><div class="t m0 x1 h6 ya6 ff1 fs3 fc0 sc0 ls7 ws0">tions which render them invalid in a more generalized</div><div class="t m0 x1 h6 ya7 ff1 fs3 fc0 sc0 ls8 ws0">case. Moreover, the nonlinearity did not arise due to</div><div class="t m0 x1 h6 ya8 ff1 fs3 fc0 sc0 ls7 ws0">the geometry/conﬁguration of the system and also the</div><div class="t m0 x5 h6 ya9 ff1 fs3 fc0 sc0 ls7 ws0">problem was not an essentially nonlinear one. Also,</div><div class="t m0 x5 h6 yaa ff1 fs3 fc0 sc0 ls8 ws0">the work did not give closed-form solutions for the</div><div class="t m0 x5 h6 yab ff1 fs3 fc0 sc0 ls7 ws0">reﬂected pulse and did not attempt to solve the problem</div><div class="t m0 x5 h6 yac ff1 fs3 fc0 sc0 ls7 ws0">by using conventional perturbation techniques. It was</div><div class="t m0 x5 h6 yad ff1 fs3 fc0 sc0 ls8 ws0">mostly aimed at proving the effectiveness of the nonlin-</div><div class="t m0 x5 h6 yae ff1 fs3 fc0 sc0 ls4 ws0">ear oscillator as an energy sink and at ﬁnding the con-</div><div class="t m0 x5 h6 yaf ff1 fs3 fc0 sc0 ls7 ws0">ditions favourable for maximum energy pumping from</div><div class="t m0 x5 h6 yb0 ff1 fs3 fc0 sc0 ls8 ws0">the elastic continuum to the nonlinear oscillator. The</div><div class="t m0 x5 h6 yb1 ff1 fs3 fc0 sc0 ls3 ws0">works covered in [31–33] use the conventional pertur-</div><div class="t m0 x5 h6 yb2 ff1 fs3 fc0 sc0 ls7 ws0">bation analysis to address the wave propagation phe-</div><div class="t m0 x5 h6 yb3 ff1 fs3 fc0 sc0 ls7 ws0">nomenon in nonlinear granular dimer/ trimer chains,</div><div class="t m0 x5 h6 yb4 ff1 fs3 fc0 sc0 ls8 ws0">but not in an un-approximated elastic continuum with a</div><div class="t m0 x5 h6 yb5 ff1 fs3 fc0 sc0 ls7 ws0">discrete nonlinear end attachment. On the contrary, this</div><div class="t m0 x5 h6 yb6 ff1 fs3 fc0 sc0 ls8 ws0">paper mostly aims at using classical perturbation tech-</div><div class="t m0 x5 h6 yb7 ff1 fs3 fc0 sc0 ls7 ws0">niques (multiple time scales) and the harmonic balance</div><div class="t m0 x5 h6 yb8 ff1 fs3 fc0 sc0 ls8 ws0">method to ﬁnd closed-form solutions to the reﬂected</div><div class="t m0 x5 h6 yb9 ff1 fs3 fc0 sc0 ls8 ws0">pulse in an un-approximated elastic bar/rod at the non-</div><div class="t m0 x5 h6 yba ff1 fs3 fc0 sc0 ls7 ws0">linear interface and the dynamic response of the essen-</div><div class="t m0 x5 h6 ybb ff1 fs3 fc0 sc0 ls8 ws0">tially nonlinear oscillator at different levels of approx-</div><div class="t m0 x5 h6 ybc ff1 fs3 fc0 sc0 ls7 ws0">imation.</div><div class="t m0 x12 h6 ybd ff1 fs3 fc0 sc0 ls8 ws0">The problem studied in this paper considers an elas-</div><div class="t m0 x5 h6 ybe ff1 fs3 fc0 sc0 ls8 ws0">tic bar coupled to a snap-through truss/oscillator at one</div><div class="t m0 x5 h6 ybf ff1 fs3 fc0 sc0 ls7 ws0">end. The free end of the bar is subjected to a sinusoidal</div><div class="t m0 x5 h6 yc0 ff1 fs3 fc0 sc0 ls7 ws0">displacement function. Examples of previous research</div><div class="t m0 x5 h6 yc1 ff1 fs3 fc0 sc0 ls8 ws0">works focusing on the analysis of systems comprising</div><div class="t m0 x5 h6 yc2 ff1 fs3 fc0 sc0 ls8 ws0">of a snap-through truss have been presented in this sec-</div><div class="t m0 x5 h6 yc3 ff1 fs3 fc0 sc0 ls7 ws0">tion. A snap-through truss can be mathematically mod-</div><div class="t m0 x5 h6 yc4 ff1 fs3 fc0 sc0 ls3 ws0">elled as a single degree of freedom spring-mass oscil-</div><div class="t m0 x5 h6 yc5 ff1 fs3 fc0 sc0 ls3 ws0">lator with three equilibrium positions arising due to its</div><div class="t m0 x5 h6 yc6 ff1 fs3 fc0 sc0 ls7 ws0">geometric conﬁguration and motion. Usually, the phase</div><div class="t m0 x5 h6 yc7 ff1 fs3 fc0 sc0 ls7 ws0">plane plot of such an oscillator shows homoclinic orbits</div><div class="t m0 x5 h6 yc8 ff1 fs3 fc0 sc0 ls8 ws0">with two centres as the two stable equilibrium points</div><div class="t m0 x5 h6 yc9 ff1 fs3 fc0 sc0 ls7 ws0">and one saddle as the unstable equilibrium point in the</div><div class="t m0 x5 h6 yca ff1 fs3 fc0 sc0 ls4 ws0">middle. When we couple such a snap-through element</div><div class="t m0 x5 h6 ycb ff1 fs3 fc0 sc0 ls7 ws0">with an elastic bar by means of linear springs and con-</div><div class="t m0 x5 h6 ycc ff1 fs3 fc0 sc0 ls8 ws0">strain the motion of the snap-through element in a direc-</div><div class="t m0 x5 h6 ycd ff1 fs3 fc0 sc0 ls7 ws0">tion perpendicular to the longitudinal axis of the bar, we</div><div class="t m0 x5 h6 yce ff1 fs3 fc0 sc0 ls8 ws0">get an essentially nonlinear system with the nonlinear-</div><div class="t m0 x5 h6 ycf ff1 fs3 fc0 sc0 ls8 ws0">ity arising due to the geometric conﬁguration. It cannot</div><div class="t m0 x5 h6 yd0 ff1 fs3 fc0 sc0 ls7 ws0">be solved by conventional analytical methods without</div><div class="t m0 x5 h6 yd1 ff1 fs3 fc0 sc0 ls8 ws0">a basic assumption which is small amplitudes of the</div><div class="t m0 x5 h6 yd2 ff1 fs3 fc0 sc0 ls8 ws0">excitation pulse. With this necessary assumption and</div><div class="t m0 x5 h6 yd3 ff1 fs3 fc0 sc0 ls7 ws0">our proposed asymptotic methodology, approximate</div><div class="t m0 x5 h6 yd4 ff1 fs3 fc0 sc0 ls7 ws0">closed-form analytical solutions have been obtained</div><div class="t m0 x5 h6 yd5 ff1 fs3 fc0 sc0 ls7 ws0">for the response of the bar as well as the snap-through</div><div class="t m0 x5 h6 yd6 ff1 fs3 fc0 sc0 ls8 ws0">oscillator. The asymptotic results have been compared</div><div class="t m0 x5 h6 yd7 ff1 fs3 fc0 sc0 ls7 ws0">to numerical simulations of the actual system, obtained</div><div class="t m0 x5 h6 yd8 ff1 fs3 fc0 sc0 ls8 ws0">computationally by using an iterative ﬁnite difference</div><div class="t m0 x5 h6 yd9 ff1 fs3 fc0 sc0 ls7 ws0">scheme and it has been shown that they match signiﬁ-</div><div class="t m0 x1 h8 y33 ff6 fs5 fc0 sc0 ls3 ws0">123</div><div class="t m0 x7 h9 y34 ff7 fs6 fc3 sc0 ls3 ws0">Content courtesy of Springer Nature, terms of use apply. Rights reserved.</div></div><div class="pi" data-data='{"ctm":[1.754739,0.000000,0.000000,1.754739,0.000000,0.000000]}'></div></div>

<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/638a1c5a9f0e6d0d655fab5c/bg5.jpg"><div class="t m0 x1 h2 y35 ff1 fs0 fc0 sc0 ls9 ws0">Propagation characteristics of an elastic</div><div class="t m0 x1 h6 y36 ff1 fs3 fc0 sc0 ls7 ws0">cantly. It has also been shown in this paper that if the bar</div><div class="t m0 x1 h6 y37 ff1 fs3 fc0 sc0 ls7 ws0">displacement at the nonlinear interface is approximated</div><div class="t m0 x1 h6 y38 ff1 fs3 fc0 sc0 ls8 ws0">by a generalized sinusoidal function of time which is</div><div class="t m0 x1 h6 y39 ff1 fs3 fc0 sc0 ls7 ws0">the superposition of the incident and reﬂected waves,</div><div class="t m0 x1 h6 y3a ff1 fs3 fc0 sc0 ls7 ws0">we can perform a Floquet analysis for the snap-through</div><div class="t m0 x1 h6 y3b ff1 fs3 fc0 sc0 ls8 ws0">oscillator in the case of sufﬁciently large amplitudes</div><div class="t m0 x1 h6 y3c ff1 fs3 fc0 sc0 ls8 ws0">of the excitation pulse. The stability/Floquet analysis</div><div class="t m0 x1 h6 y3d ff1 fs3 fc0 sc0 ls7 ws0">generates a Mathieu plot of excitation frequency vs</div><div class="t m0 x1 h6 y3e ff1 fs3 fc0 sc0 ls8 ws0">approximated excitation amplitude showing the stable</div><div class="t m0 x1 h6 y3f ff1 fs3 fc0 sc0 ls7 ws0">and unstable regions for the dynamic response of the</div><div class="t m0 x1 h6 y40 ff1 fs3 fc0 sc0 ls8 ws0">snap-through oscillator. The stability diagram can be</div><div class="t m0 x1 h6 y41 ff1 fs3 fc0 sc0 ls8 ws0">useful while determining important system excitation</div><div class="t m0 x1 h6 y42 ff1 fs3 fc0 sc0 ls7 ws0">parameters such as amplitude and frequency for which</div><div class="t m0 x1 h6 y43 ff1 fs3 fc0 sc0 ls7 ws0">the dynamic response of the oscillator becomes unsta-</div><div class="t m0 x1 h6 y44 ff1 fs3 fc0 sc0 ls7 ws0">ble/divergent and the oscillator exhibits a snap-through</div><div class="t m0 x1 h6 y45 ff1 fs3 fc0 sc0 ls7 ws0">motion.</div><div class="t m0 x1 h5 ydb ff2 fs3 fc0 sc0 ls7 ws0">2 Problem formulation</div><div class="t m0 x1 h6 ydc ff1 fs3 fc0 sc0 ls4 ws0">2.1 The model</div><div class="t m0 x1 h6 ydd ff1 fs3 fc0 sc0 ls7 ws0">Let us consider a homogeneous, isotropic, elastic bar</div><div class="t m0 x1 h6 yde ff1 fs3 fc0 sc0 ls8 ws0">of mass</div><div class="t m0 x14 hc ydf ff5 fs3 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x15 h6 yde ff8 fs3 fc0 sc0 ls3 ws0">M, length</div><div class="t m0 x16 hc ydf ff5 fs3 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x17 h6 yde ff8 fs3 fc0 sc0 ls3 ws0">L and cross-sectional area A, which</div><div class="t m0 x1 h6 ye0 ff1 fs3 fc0 sc0 ls7 ws0">is free at one end and coupled to a snap-through oscil-</div><div class="t m0 x1 h6 ye1 ff1 fs3 fc0 sc0 ls8 ws0">lator of mass m through a linear spring of stiffness con-</div><div class="t m0 x1 h6 ye2 ff1 fs3 fc0 sc0 ls8 ws0">stant k as shown in Fig. 1. The snap-through oscillator</div><div class="t m0 x1 h6 ye3 ff1 fs3 fc0 sc0 ls7 ws0">is connected to a rigid support through another linear</div><div class="t m0 x1 h6 ye4 ff1 fs3 fc0 sc0 ls8 ws0">spring of stiffness constant k. The motion of the snap-</div><div class="t m0 x1 h6 ye5 ff1 fs3 fc0 sc0 ls8 ws0">through oscillator is constrained to take place in the</div><div class="t m0 x1 h6 ye6 ff1 fs3 fc0 sc0 ls7 ws0">vertical direction only. The force–deﬂection relation-</div><div class="t m0 x1 h6 ye7 ff1 fs3 fc0 sc0 ls8 ws0">ship of the linear springs is given by F = kδ, where</div><div class="t m0 x1 h6 ye8 ff9 fs3 fc0 sc0 ls3 ws0">δ is the deﬂection in the springs and F is the restoring</div><div class="t m0 x1 h6 ye9 ff1 fs3 fc0 sc0 ls7 ws0">force. The natural un-deﬂected length of the springs is</div><div class="t m0 x1 h6 yea ff8 fs3 fc0 sc0 ls3 ws0">L as shown in the initial conﬁguration. For the purpose</div><div class="t m0 x5 h6 yeb ff1 fs3 fc0 sc0 ls7 ws0">of keeping the calculations simple, we have assumed</div><div class="t m0 x5 h6 yec ff1 fs3 fc0 sc0 ls7 ws0">that both the springs have the same stiffness coefﬁcient</div><div class="t m0 x5 h6 yed ff1 fs3 fc0 sc0 ls3 ws0">(k) and same natural length (L).</div><div class="t m0 x12 h6 yee ff1 fs3 fc0 sc0 ls7 ws0">The free end of the bar is subjected to a sinusoidal</div><div class="t m0 x5 h6 yef ff1 fs3 fc0 sc0 ls8 ws0">excitation of the form Q(t) =</div><div class="t m0 x18 hc yf0 ff5 fs3 fc0 sc0 ls3 ws0">¯</div><div class="t m0 x19 h6 yef ff8 fs3 fc0 sc0 ls3 ws0">A sin(t). The excita-</div><div class="t m0 x5 h6 yf1 ff1 fs3 fc0 sc0 ls7 ws0">tion is a displacement function, not a force, and it is</div><div class="t m0 x5 h6 yf2 ff1 fs3 fc0 sc0 ls7 ws0">assumed to be imposed on the free end of the bar. But,</div><div class="t m0 x5 h6 yf3 ff1 fs3 fc0 sc0 ls8 ws0">since the bar is a linear and non-dispersive medium, to</div><div class="t m0 x5 h6 yf4 ff1 fs3 fc0 sc0 ls7 ws0">reduce the complexities in algebraic calculations asso-</div><div class="t m0 x5 h6 yf5 ff1 fs3 fc0 sc0 ls8 ws0">ciated with constant phase terms, we shall consider the</div><div class="t m0 x5 h6 yf6 ff1 fs3 fc0 sc0 ls7 ws0">incident wave in our analysis and calculations when it</div><div class="t m0 x5 h6 yf7 ff1 fs3 fc0 sc0 ls8 ws0">strikes the nonlinear interface at x = 0 and consider</div><div class="t m0 x5 h6 yf8 ff1 fs3 fc0 sc0 ls8 ws0">the time for analysis to be starting from that instant.</div><div class="t m0 x5 h6 yf9 ff1 fs3 fc0 sc0 ls8 ws0">This means that a simple pre-calculated phase shift is</div><div class="t m0 x5 h6 yfa ff1 fs3 fc0 sc0 ls7 ws0">performed on the incident wave. All the subsequent</div><div class="t m0 x5 h6 yfb ff1 fs3 fc0 sc0 ls7 ws0">mathematical calculations shall be done including this</div><div class="t m0 x5 h6 yfc ff1 fs3 fc0 sc0 ls7 ws0">phase manipulation in the expressions and in the end</div><div class="t m0 x5 h6 yfd ff1 fs3 fc0 sc0 ls8 ws0">while illustrating the results graphically, we shall add</div><div class="t m0 x5 h6 yfe ff1 fs3 fc0 sc0 ls8 ws0">this phase shift back into our analytical solutions. While</div><div class="t m0 x5 h6 yff ff1 fs3 fc0 sc0 ls4 ws0">deriving the equations of motion and boundary condi-</div><div class="t m0 x5 h6 y100 ff1 fs3 fc0 sc0 ls8 ws0">tions from the Lagrangian of the system initially, it will</div><div class="t m0 x5 h6 y101 ff1 fs3 fc0 sc0 ls7 ws0">be assumed that the displacement at the left end of the</div><div class="t m0 x5 h6 y102 ff1 fs3 fc0 sc0 ls4 ws0">bar be speciﬁed as per our problem deﬁnition. The ini-</div><div class="t m0 x5 h6 y103 ff1 fs3 fc0 sc0 ls7 ws0">tial displacements have been taken to be zero from their</div><div class="t m0 x5 h6 y104 ff1 fs3 fc0 sc0 ls7 ws0">corresponding positions as shown in the schematic of</div><div class="t m0 x5 h6 y105 ff1 fs3 fc0 sc0 ls7 ws0">the model in Fig. 1. u(x , t) and v(t ) are the generalized</div><div class="t m0 x5 h6 y106 ff1 fs3 fc0 sc0 ls7 ws0">displacement functions of the bar and the snap-through</div><div class="t m0 x5 h6 y107 ff1 fs3 fc0 sc0 ls8 ws0">oscillator, respectively, where x is the longitudinal spa-</div><div class="t m0 x5 h6 y108 ff1 fs3 fc0 sc0 ls7 ws0">tial coordinate along the bar and t is time.</div><div class="t m0 x5 h6 y109 ff1 fs3 fc0 sc0 ls7 ws0">2.2 Equations of motion and boundary conditions</div><div class="t m0 x5 h6 y10a ff1 fs3 fc0 sc0 ls7 ws0">The diagram is shown for the stable equilibrium con-</div><div class="t m0 x5 h6 y10b ff1 fs3 fc0 sc0 ls8 ws0">ﬁguration of the snap-through oscillator whose verti-</div><div class="t m0 x1 h2 y10c ff2 fs0 fc0 sc0 ls16 ws0">Fig. 1 Schematic of an</div><div class="t m0 x1 h2 y10d ff1 fs0 fc0 sc0 ls9 ws0">elastic bar coupled with a</div><div class="t m0 x1 h2 y10e ff1 fs0 fc0 sc0 ls5 ws0">snap-through truss</div><div class="t m0 x6 h8 y33 ff6 fs5 fc0 sc0 ls3 ws0">123</div><div class="t m0 x7 h9 y34 ff7 fs6 fc3 sc0 ls3 ws0">Content courtesy of Springer Nature, terms of use apply. Rights reserved.</div></div><div class="pi" data-data='{"ctm":[1.754739,0.000000,0.000000,1.754739,0.000000,0.000000]}'></div></div>