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Wave transmission in finite dissipative nonlinear periodic structures Yousefzadeh, Behrooz 2016

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Wave Transmission inFinite Dissipative Nonlinear Periodic StructuresbyBehrooz YousefzadehM.A.Sc., The University of British Columbia, 2010B.Sc., University of Tehran, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University Of British Columbia(Vancouver)December 2016c© Behrooz Yousefzadeh, 2016AbstractSpatially periodic structures exhibit intriguing dynamic characteristics, contribut-ing to their growing applications as phononic crystals, acoustic metamaterials andlightweight lattice materials. A striking feature, employed in many engineeringapplications, is their filtering effect, whereby waves can propagate only in spe-cific frequency intervals known as pass bands. Other frequency components (stopbands) are spatially attenuated as they propagate through the structure.This thesis studies nonlinear wave transmission in periodic structures of finiteextent in the presence of dissipative forces and externally induced nonlinear forces.Perfectly periodic structures with identical units are considered, as well as nearlyperiodic structures with small deviations from periodicity extended throughout thestructure.At high amplitudes of motion, nonlinear forces gain significance, generatingqualitatively new dynamic phenomena such as supratransmission. Supratransmis-sion is an instability-driven transmission mechanism that occurs when a periodicstructure is driven harmonically at one end with a frequency within its stop band.The ensuing enhanced transmission contrasts the vibration isolation characteristicof the same structure operating in the linear regime.In the context of engineering applications, three factors play a significant role:dissipative forces, symmetry-breaking imperfections induced by manufacturingconstraints (disorder) and the finite size of the structure. This thesis systemati-cally investigates the influence of these parameters on supratransmission in a one-dimensional periodic structure, studying the competition between the effects ofdispersion, dissipation, nonlinearity and disorder-borne wave localization (Ander-son localization).iiWe identify the mechanism underlying supratransmission using direct numer-ical simulations and numerical continuation. Based on this insight, we obtain an-alytical expressions for the onset of supratransmission for weakly coupled struc-tures using asymptotic analysis. Particularly, we highlight the non-trivial effects ofdamping on supratransmission in finite structures. We demonstrate that, regardlessof the type of nonlinearity, dissipative forces can delay the onset of supratransmis-sion, and high levels of damping can eliminate it.Given that the spectral contents of transmitted energies fall within the passband, we expect a competition between supratransmission and Anderson localiza-tion. Using direct numerical simulations and continuation techniques, we demon-strate that disorder reduces the transmitted wave energy in the ensemble-averagesense. However, the average force threshold required to trigger supratransmissionremains unchanged.iiiPrefaceParts of the research described in this thesis have been published in peer-reviewedjournals and conference proceedings. These publications report work carried outduring my Ph.D. research under the supervision of Dr. A. Srikantha Phani. Weare the only authors of all the publications and presentations. I was responsible forareas of concept formation, data collection and analysis, and manuscript composi-tion. A.S. Phani was the supervisory author on the entire project and was involvedthroughout the work in concept formation and manuscript composition.Portions of Chapters 2 and 3 are published in Journal of Sound and Vibra-tion [133]. Parts of the same work, at various stages of completion, were also pre-sented at the following conferences: (i) Conference on Nonlinear Vibrations, Lo-calization and Energy Transfer, June 2014, Istanbul, Turkey; (ii) International Sym-posium on Optomechatronic Technologies, Novembver 2014, Seattle, WA [132].A version of Chapter 4 is published in Journal of Sound and Vibration [134].Portions of the same work, at various stages of completion, were also presentedat the following conferences: (i) ASME Applied Mechanics and Materials Con-ference, June 2015, Seattle, WA; (ii) SIAM Conference on Nonlinear Waves andCoherent Structures, August 2016, Philadelphia, PA; (iii) International Congressof Theoretical and Applied Mechanics, August 2016, Montreal, QC.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Periodic Structures in Engineering . . . . . . . . . . . . . . . . . 11.2 Factors Influencing the Dynamic Response of Periodic Structures . 31.3 Energy Transmission in Nonlinear Periodic Structures . . . . . . . 51.3.1 Nonlinear Structures with Exact Periodicity . . . . . . . . 61.3.2 Nonlinear Structures with Disorder . . . . . . . . . . . . 91.4 Research Objectives and Methodology . . . . . . . . . . . . . . . 121.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Supratransmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 A Periodic Structure with Tunable Nonlinearity . . . . . . . . . . 162.1.1 Governing Equations for the Periodic Structure . . . . . . 182.1.2 Tunability of the Nonlinear Forces . . . . . . . . . . . . . 202.2 Band Structure of Periodic Systems . . . . . . . . . . . . . . . . 222.2.1 Forced Response of Linear Structures . . . . . . . . . . . 222.2.2 Free Wave Propagation in Linear and Nonlinear Structures 232.3 Energy Transmission via Harmonic Excitation within a Stop Band 26v2.4 Computing the Supratransmission Threshold . . . . . . . . . . . . 332.4.1 Nonlinear Response Manifold (NLRM) . . . . . . . . . . 332.4.2 Resonances with the Shifted Pass Band . . . . . . . . . . 362.5 Analytical Prediction of the Onset of Supratransmission . . . . . . 382.5.1 Analysis Based on Local Nonlinear Dynamics . . . . . . 382.5.2 Comparison with Numerical Computations . . . . . . . . 412.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 423 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1 Type and Strength of Nonlinearity . . . . . . . . . . . . . . . . . 443.2 Number of Units . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Strength of Coupling . . . . . . . . . . . . . . . . . . . . . . . . 523.5 Forcing Amplitude: Hysteresis . . . . . . . . . . . . . . . . . . . 553.6 Forcing Frequency . . . . . . . . . . . . . . . . . . . . . . . . . 563.7 Strong Propagation Regime . . . . . . . . . . . . . . . . . . . . . 583.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 594 Disorder Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1 Disorder-Borne Energy Localization in Linear Structures . . . . . 614.1.1 Localization Occurring within a Pass Band . . . . . . . . 634.1.2 Quantifying the Degree of Localization . . . . . . . . . . 644.1.3 Influence of Disorder on the Band Structure . . . . . . . . 684.2 Combined Effects of Damping and Disorder Near a Pass Band . . 694.3 Supratransmission in Disordered Nonlinear Periodic Structures . . 744.3.1 Loss of Stability Leads to Enhanced Transmission . . . . 754.3.2 Onset of Transmission Remains Unchanged on Average . 764.3.3 Energy Profiles of Transmitted Waves . . . . . . . . . . . 784.3.4 Average Frequency Spectra Above the Threshold . . . . . 794.3.5 Prediction of Transmitted Energies Based on Linear Theory 824.4 Prediction of the Onset of Supratransmission in Disordered Structures 834.4.1 Analytical Estimate of the Force Threshold . . . . . . . . 834.4.2 Dependence of Threshold Curves on Nonlinearity . . . . . 86vi4.4.3 Dependence of Locally-Nonlinear Behavior on Coupling . 894.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 915 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . 945.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 101Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A Numerical Continuation . . . . . . . . . . . . . . . . . . . . . . . . . 115B Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B.1 Derivation of the Transfer Matrix Formulation . . . . . . . . . . . 118B.2 Derivation of the Onset of Supratransmission . . . . . . . . . . . 119viiList of FiguresFigure 2.1 A schematic of the proposed periodic structure . . . . . . . . 17Figure 2.2 Tunability of the normalized restoring force . . . . . . . . . . 22Figure 2.3 Decay exponents for an infinitely long exactly periodic linearstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.4 Tunability of linear dispersion curves . . . . . . . . . . . . . 25Figure 2.5 The supratransmission phenomenon for the hardening system 30Figure 2.6 The supratransmission phenomenon for the softening system . 31Figure 2.7 Poincare´ maps of the response above the onset of supratrans-mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.8 Normalized amplitude profiles above the onset of supratrans-mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.9 The NLRM for the structure with softening nonlinearity . . . 34Figure 2.10 The NLRM and threshold curve for the structure with harden-ing nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.11 Comparison of the threshold curve with the shifted pass band . 37Figure 2.12 Computed versus predicted threshold curves for a weakly cou-pled structure . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.1 Threshold curves for different values of N (free-free boundaryconditions) . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.2 Threshold curves for different values of N (fixed-fixed bound-ary conditions) . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.3 Influence of damping on the NLRM in a hardening structure . 48Figure 3.4 Influence of damping on threshold curves in a hardening structure 50viiiFigure 3.5 Influence of damping on the onset of supratransmission . . . . 51Figure 3.6 Influence of the strength of coupling on threshold curves . . . 54Figure 3.7 The hysteresis phenomenon for the hardening structure . . . . 56Figure 3.8 Influence of forcing frequency on enhances nonlinear energytransmission . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 4.1 Influence of disorder on response localization within the passband . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 4.2 Comparison of different decay exponents . . . . . . . . . . . 65Figure 4.3 Influence of N on the decay exponent . . . . . . . . . . . . . 67Figure 4.4 Influence of disorder on spatial localization of mode shapes . 68Figure 4.5 Influence of disorder on decay exponent and natural frequen-cies of the linear structure . . . . . . . . . . . . . . . . . . . 69Figure 4.6 Threshold curve for the hardening ordered structure with N = 10 70Figure 4.7 Influence of damping on supratransmission in the vicinity of apass band . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 4.8 Influence of damping on the NLRM in the vicinity of a pass band 72Figure 4.9 Influence of disorder on the NLRM in the vicinity of a pass band 74Figure 4.10 Influence of disorder on the onset of supratransmission at indi-vidual realizations . . . . . . . . . . . . . . . . . . . . . . . 76Figure 4.11 Influence of disorder on the supratransmission force threshold 77Figure 4.12 Influence of disorder on the threshold curve . . . . . . . . . . 78Figure 4.13 Average energy profiles below and above the onset of supra-transmission . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.14 Average frequency spectra of the driven unit . . . . . . . . . . 80Figure 4.15 Average frequency spectra of the end unit . . . . . . . . . . . 80Figure 4.16 Influence of disorder on normalized transmitted energy ratios . 83Figure 4.17 The schematics of two adjacent units in a mono-coupled system. 84Figure 4.18 Dependence of threshold curves on the nonlinear force in asoftening structure . . . . . . . . . . . . . . . . . . . . . . . 87Figure 4.19 Dependence of threshold curves on the nonlinear force in ahardening structure . . . . . . . . . . . . . . . . . . . . . . . 88ixFigure 4.20 Dependence of the locally-nonlinear behavior on strength ofcoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure B.1 The schematic of a unit cell . . . . . . . . . . . . . . . . . . 118xChapter 1Introduction1.1 Periodic Structures in EngineeringAny structure with some form of spatial periodicity, either in its constituent ma-terial properties, internal micro-architecture or boundary conditions is a periodicstructure. In engineering applications, spatially periodic micro-architecture hasbeen utilized in the design of cellular solids to develop lattice materials for multi-functional applications, where a mechanical function such as stiffness or strengthis combined with some other property such as thermal insulation [33, 130]. Exam-ples include lightweight structural applications with superior thermal insulation orsuperior impact and blast resistance [120], as well as ultralight materials (densitybelow 10 mg/cm3) with superior specific stiffness and strength [118].Spatially periodic structures exhibit intriguing wave transmission characteris-tics. Of particular interest in many engineering applications are the wave-filteringfeatures of periodic structures in the frequency (temporal) domain and the wavenum-ber (spatial) domain. In the frequency domain [8], a periodic structure allows selec-tive transmission of waves over certain frequency intervals known as pass bands.All other frequency components (belonging to stop bands) are spatially attenuatedas waves propagate through the periodic structure. This filtering is attributed toBragg scattering, the destructive interference of waves that are periodically scat-tered as they propagate through the structure. A consequence of this filtering isthat vibration energy is localized to the source of excitation in stop bands. In the1wavenumber domain [72], periodic structures exhibit directional wave propagationcharacteristics. The band-pass filtering property of periodic structures has led todevelopment of mechanical filters in electronics [58] and microelectromechanicalsystems for radio frequency applications [27]. The directional dependency of wavepropagation enables steering and focusing of waves, with applications in energyharvesting [11] among others.The wave-filtering characteristics and other unique properties of periodic struc-tures, such as negative group velocity, have spurred an array of applications forthese structures as phononic crystals and acoustic or elastic metamaterials – see [19,24, 76] for recent monographs and reviews on this subject. Phononic crystalsare composites with periodically distributed inclusions of high impedance contrastwith the matrix (host) material. This periodic configuration allows for manipula-tion of wave propagation characteristics of the elastic medium, with applicationsin wave-filtering, wave-guiding and energy harvesting – see [55, 104] for recentreviews on this topic.Bragg scattering is effective when the wavelength of the incident wave is onthe same order as the characteristic spatial periodicity length scale of the struc-ture. Thus, exploiting passive vibration isolation induced by Bragg scattering ischallenging at low frequencies. By low we denote frequencies below the funda-mental resonant frequency of the unit cell under free boundary conditions. Thislimitation can be overcome by coupling a periodic structure to local resonators andinertial amplifiers [34, 131] to gain additional control over the wave propagationcharacteristics. The interaction between the host dispersive medium and the dy-namic properties of the local resonators allows the possibility to realize sub-Braggstop bands. Periodic structures featuring the local-resonance mechanism are calledacoustic or elastic metamaterials. Examples of such ‘meta’ characteristics includesurpassing the mass-law limit of sound transmission [75], albeit in a narrow fre-quency band, and cloaking of acoustic waves [101]. See [20, 42, 79] for recentreviews of some other applications.There are many other examples of periodic structures in engineering applica-tions. In the context of aerospace structures, a wide body of literature alreadyexists on the dynamic response of periodic structures such as rib-stiffened pan-els, plates used in the aircraft fuselage and tail, and sandwich panels with periodic2cores [77, 90]. Another example is bladed disk assemblies in turbomachinery,which are periodic structures with cyclic symmetry. Certain features of their vi-bration characteristics, such as fatigue failure due to mistuning, can be explainedbased on the dynamics of periodic structures [30, 102].To conclude this section, we point to the fundamental contributions to waves inperiodic structures by condensed matter and solid state physicists, particularly inthe context of phonon and electron transport in solids [4, 8], inspired by pioneeringworks of Rayleigh [111] in the context of vibration response of a string (Melde’sexperiment) subjected to parametric excitation via variable tension.1.2 Factors Influencing the Dynamic Response ofPeriodic StructuresAlthough periodic structures have been studied extensively in solid-state physics,transfer of that body of knowledge to problems relevant in engineering applicationsis not a trivial task. As cases in point, the governing equations and boundary con-ditions in engineering can be different from those in other fields (nonlinear forces,in particular); damping often has a significant contribution in engineering whileit is normally absent in solid-state physics; and the typical assumption of infinitesystems in solid-state physics is often unrealistic in engineering applications. Ac-counting for these differences are therefore crucial in making the results directlyapplicable to engineering structures.While the ultimate interest in solid-state physics is in three-dimensional sys-tems, engineering problems often provide the luxury of structures in which peri-odicity is in one or two dimensions only [49]; e.g. bladed disk assemblies in tur-bomachinery and sandwich structures with periodic cores. Apart from this factor,there are common features among periodic structures relevant in engineering appli-cations that set them apart from their classical counterparts in solid-state physics.These features can be divided into four main categories: (i) energy dissipation, (ii)finite length of the structure (iii) deviations from exact periodicity, and (iv) non-linearity. Qualitatively new phenomena may emerge when one or more of theseconditions are present. These deviations often result in localization or confine-ment of macroscopic deformations or high sensitivity of the structure to changes in3parameters.(i) Damping: Energy dissipation, however small, is intrinsic to engineeringstructures and may have significant influence on the dynamic response of a peri-odic structure. Addition of dissipative forces to a periodic structure results in anew spatial attenuation mechanism for the traveling waves. For a linear viscousdissipative force, the corresponding decay rate is exponential and uniform, andcould influence all frequencies. In addition, dissipative forces (if they are strongenough) can result in appearance of spatial stop bands, corresponding to prohibitedwavenumbers. For more details on this aspect see [54, 105].(ii) Finite length: In many applications of periodic structures, the number ofrepeating unit cells is not large enough to warrant an infinite-length approximation.Finite length of a structure prudces reflection of waves from its boundaries, leadingto formation of standing waves with restricted wavenumbers [119]. The precisedispersive characteristics of finite periodic structures also depend on the type ofthe boundary conditions (e.g. fixed, guided or free) and can be analysed usingphase closure principle [89, 119].(iii) Disorder: Engineered periodic materials and structures possess inher-ent imperfections imposed by manufacturing constraints. Such small symmetry-breaking imperfections can lead to significant qualitative changes in the globaldynamic response of the structure under certain conditions. These deviations fromexact periodicity could be a result of either defect or disorder. A defect is nor-mally a large deviation from periodicity that is concentrated at a certain location(or locations) within the structure, such as a crack. In a defective structure, the re-sponse localizes to the vicinity of the defect [92], typically enticing further damageto the structure. Disorder usually refers to small deviations from exact periodicityspread throughout the structure, such as the mistuning in a bladed disk assembly.Disorder results in spatial confinement of the dynamic response near the source ofexcitation, a phenomenon known as Anderson localization. This phenomenon isreviewed in more detail in Section 4.1.(iv) Nonlinearity: Nonlinearities arising from large deformations (geometricnonlinearity), constitutive laws of the materials (material nonlinearity), boundaryconditions or external forces are important considerations in many engineering ap-plications. In the presence of nonlinearity, mechanical systems often exhibit dra-4matic changes in their response caused by a small change of some parameter orsmall-amplitude disturbances. Such sudden changes are often accompanied by alarge amplitude discontinuity in the dynamic response of the structure, often lead-ing to a rich bifurcation structure. In aerospace engineering, for example, therehas been a trend to design slender and lightweight structures to decrease powerconsumption and increase performance. Accordingly, and from a practical view-point, there is an increasing demand for understanding the nonlinear behaviourof these structures and mitigating the corresponding vibration problems [129]. Asimilar situation exists in steel bridges and large marine structures [95]. In MEMSapplications, it is also likely to trigger nonlinear phenomena due to coexistence ofmore than one source of nonlinearity [113]; for example, many microstructures un-dergo large deformations due to their high mechanical compliance; parallel-plateelectrostatic forces that are very commonly used for actuation and detection areinherently nonlinear; and squeeze-film damping, the most common and dominantdamping mechanism, is inherently nonlinear. Accordingly, there has been a re-cent surge of interest in studying nonlinear (periodic) arrays of micromechanicaloscillators [116].1.3 Energy Transmission in Nonlinear PeriodicStructuresThis thesis is on energy transmission through nonlinear periodic structures withdisorder. When the amplitudes of motion are small enough that nonlinear forcesare negligible, wave transmission in periodic structures is dominated by three fac-tors: dispersion, dissipation and disorder. Dispersion is responsible for the theband-pass filtering feature of periodic structures, leading to the formation of passbands and stop bands. If we add damping to a periodic structure, there will be anew attenuation mechanism acting on traveling waves via dissipation of energy.Disorder may lead to confinement of energy to a small spatial region within thestructure (Anderson localization). If there is an external force acting in a spatiallylocalized region of the structure, then energy is confined to that region. In contrastto this point excitation case, predicting the region of localization is not as straight-forward in the distributed excitation case, as in base excitation of a structure or5convected boundary layer excitation of an aircraft panel.Nonlinearities are typically avoided in the design and control of engineeringsystems. Nevertheless, exploiting the nonlinear response of structures can revealhidden design opportunities and accommodate bolder design goals. The reasonis that qualitatively new phenomena emerge when the requirement of linearity isrelaxed. In this section, we review some of the literature pertaining to nonlinearperiodic structures with and without disorder.Throughout the thesis, we distinguish between periodic structures with exactperiodicity and those with disorder by referring to them as ordered and disorderedstructures, respectively.1.3.1 Nonlinear Structures with Exact PeriodicityIt has been known for a long time, that localized wave packets (called solitons)exist in continuous nonlinear systems [22, 112]. Solitons maintain their shapewhen traveling at a constant speed, due to the balance between the nonlinear anddispersive properties of the continuum. In addition, when two solitons pass througheach other, they retain their shapes and only a phase change occurs between them.Continuous systems possess translational invariance in all directions, whereasdiscrete periodic structures are translationally invariant in certain directions only.Solitons cannot propagate in a discrete system due to this lack of an arbitrary trans-lational invariance [31]. On the other hand, discreteness allows the existence ofother types of localized waves that do not exist in a continuous system. Theselocalized waves can propagate in a discrete system in the same manner a solitonpropagates in a continuum. In contrast to solitons, however, collisions betweensuch excitations result in energy transfer between them, with the more localizedexcitations gaining energy from the less localized ones [116]. These spatially lo-calized, time-periodic and stable excitations in perfectly periodic, nonlinear, dis-crete systems are known as discrete breathers or intrinsic localized modes. Unlikesolitons, discrete breathers could be either stationary (“pinned” to a specific spatiallocation) or mobile (traveling through the structure).Two requirements are indispensable for discrete breathers to exist in a peri-odic structure [10]: discreteness and nonlinearity. Discreteness ensures that the6pass bands of the structure are bounded and do not extend to infinity like that ofa continuum. Nonlinearity, on the other hand, allows propagating modes to havea fundamental frequency outside the pass bands. Therefore, the frequency of os-cillations of discrete breathers lies inside the stop bands of the periodic structure.We emphasize that, strictly speaking, this discussion pertains primarily to the freeresponse of an undamped nonlinear periodic structure.Discrete breathers have been experimentally observed in many physical sys-tems – see [32] for a detailed review of some applications. When damping forcesare present (which is the case in engineering problems), an external harmonicforce is required to sustain the discrete breathers in experiments. This is normallyachieved through excitation of the base of the structure. As a case in point [25,116], stationary and moving discrete breathers have been observed in microme-chanical cantilever arrays. Such results are very important in paving the way forsensor applications in MEMS devices. Other mechanical experiments with har-monic base excitation include [29, 56, 63, 123, 124].Discrete breathers provide a route to energy localization in nonlinear periodicstructures. They are primarily investigated as either (i) the periodic orbits of anundamped (Hamiltonian) periodic structure or (ii) the limit cycles of a dampedperiodic structure subject to external harmonic base (uniform) excitation. Apartfrom these, there are two other classes of problems that are important in nonlinearperiodic structures: (iii) pulse propagation and (iv) response to localized harmonicexcitation. Although these four problems are intimately related to each other, adirect connection does not exist between them. This is because the principle ofsuperposition does not generally hold in nonlinear systems.The problem of pulse propagation in nonlinear periodic structures has been atopic of research for a long time. The majority of this literature, originating withinthe physics community, focuses on conservative and infinitely long systems. Inthat context, the main goal is to understand the spreading process of initially lo-calized wave packets (an initial value problem); e.g. see [45] for a review of thistopic. Within the nonlinear mechanics literature, many of the contributions on thistopic are in the context of one-dimensional granular crystals [110]. Nonlinearityin these structures arises from the coupling contact forces between adjacent units(inter-site nonlinearity). Furthermore, there have been detailed theoretical studies7on the spectral and temporal characteristics of wave packet propagation throughthese structures [37, 38, 43]. Refer to [55, Section 4.2] for a review of other the-oretical studies on nonlinear periodic structures pertaining to phononic structuresand materials.In this thesis, we focus on the problem of nonlinear wave transmission due toexternal point harmonic excitation applied to a single unit of the periodic structure.The starting point in this context is understanding the influence of nonlinearity onthe dispersive characteristics of the periodic structure. While the dispersion prop-erties (locations of the pass bands and stop bands) of a periodic structure are fixedin the linear operating regime, they become dependent on the amplitude of motionin the nonlinear regime. This is because nonlinearity makes the resonant frequen-cies of vibrations dependent on the total energy of the vibrating system, which isoften characterized by the amplitudes of motion. As a result, the location and/orwidth of a pass band may vary based on the amplitudes of motion. This charac-teristic also depends on the nature of the nonlinear forces. For early works on thistopic in the engineering literature refer to [14, 128]. Nonlinearity, therefore, canoffer tunable filtering properties and opportunities to enhance the performance oflinear periodic structures. See [97] for an example of such a tunable filter and [96]for an extension of this work to a two-dimensional problem. The same concept ofamplitude-tunable filtering properties also applies to pass bands produced by localresonators within the unit cell of a periodic structure [74, 83]. This concept can beused in design of acoustic metamaterials with tunable properties.In the situation where an external harmonic excitation is applied to one unit ofa nonlinear periodic structure, nonlinearity offers a route to achieve enhanced en-ergy transmission. This may happen due to instability of periodic solutions througheither nonlinear resonances [78] or a saddle-node bifurcation [82, 122]. Nonlin-ear resonances occur due to internal or combination resonances with the externalharmonic excitation [98] and may occur for frequencies inside or outside a passband. The former mechanism, called supratransmission, occurs when the drivingfrequency lies within a stop band. For small values of the driving force, the am-plitudes of motion are small and decay exponentially away from the driven end ofthe structure; i.e. energy transmission is not possible at low amplitudes of the driv-ing force. When the driving force exceeds a certain threshold, energy transmission8becomes possible even though the forcing frequency remains within the stop bandof the periodic structure. Supratransmission was first studied in discrete periodicstructures by Geniet and Le´on [39, 40].The nonlinear supratransmission phenomenon emanates from the physics com-munity. Accordingly, the majority of the existing studies on supratransmissionconsider non-dissipative and infinitely long systems [39, 62, 80, 82, 122]. In com-parison, supratransmission in finite dissipative periodic structures have receivedlittle attention [16, 51, 61, 66]. Most notably, the hysteresis loops associated withthe saddle-node bifurcation at the the onset of supratransmission have been ex-plored [51], and devices have been designed based on the sensitivity of the struc-ture near the point of instability [16]. Also, supratransmission has been utilizedin a defective granular chain to develop an acoustic switch [7]. Supratransmissionhas been observed in experiments on macro-scale mechanical systems [40, 51].Nevertheless, there are no systematic studies on the influence of different systemparameters (damping, strength of coupling, type of nonlinearity, etc.) on this phe-nomenon in finite structures. In particular, there is limited information about theinfluence of damping and strength of coupling on supratransmission [80, 82]. Wenote that these studies are purely numerical.1.3.2 Nonlinear Structures with DisorderNonlinearities can be introduced in an otherwise linear system to achieve quali-tatively new dynamic phenomena. In this approach, nonlinearity can be regardedas a local disorder (similar to a defect). As a case in point, Cho et al. [18] haveintentionally introduced strong stiffness nonlinearity in a microcantilever systemby a nanotube coupling. Further, by changing the placement of the nanotube, theyhave been able to realize coupling forces with either softening or hardening typesof nonlinearity. This led to a new tunable broadband resonator design [17].A very interesting result of adding a single nonlinear attachment to a linearmechanical structure is that the vibration energy of the linear structure can be ir-reversibly and passively transferred to the nonlinear attachment [115, 127]. Suchnonlinear attachments act as local energy sinks and can be used in the design ofbroadband vibration absorbers. Locating the local nonlinear component in a linear9periodic structure can also have practical significance in structural fault diagno-sis [103]; in this context, the appearance of nonlinearity indicates a defect in theperiodic structure.In the examples given above, a single nonlinear elements in an otherwise linearperiodic structure breaks the periodicity of the entire structure. An opposite con-figuration is a nonlinear periodic structure with one or more defects [85]. Usingthe terminology introduced in Section 1.2, either of these configurations results ina defective periodic structure. In contrast, our interest in this work is in the caseof a nonlinear periodic structure with disorder: deviations from periodicity spreadthroughout the structure. Within this framework, we are particularly interested instudying the scenarios in which there is a competition between supratransmissionand Anderson localization.Anderson localization manifests its significance within the pass band of thestructure. At these frequencies, the (average) response of a disordered periodicstructure decays exponentially away from the source of excitation. This is in con-trast to the behavior of the same structure in the absence of disorder. This localiza-tion can have important consequences for the energies transmitted via the supra-transmission mechanism. The reason is that the frequency components of thesenonlinearly transmitted waves lie within the linear pass band of the structure [82].It is therefore natural to expect a competition between nonlinearity and disorder(supratransmission and Anderson localization) with regards to the transmitted en-ergies above the supratransmission threshold.There is a myriad of papers on the interplay between nonlinearity and disor-der in periodic structures. The majority of this literature, similar to the literatureon ordered nonlinear periodic structures, originates from the physics communityand focuses on pulse propagation through conservative and infinitely long systems.Recent reviews of this literature can be found in [73, 94].Within the nonlinear mechanics literature, some of the earlier works on non-linear disordered structures include [28, 64, 117]. These studies deal with the freeresponse of conservative systems; thus, their results are not applicable to the case ofsupratransmission. Many of the remaining studies are concerned with the evolutionof pulses as they propagate through nonlinear disordered structures [2, 52, 84, 109],particularly for granular chains [2, 84, 109]. The specific decay characteristics of10pulses (exponential or algebraic) is found to depend on the relative strengths ofnonlinearity and disorder, as well as the specific form of nonlinear forces. Unfor-tunately, these results are not directly applicable to the case of continuous waveexcitation in supratransmission because superposition does not generally hold innonlinear systems.Wave propagation due to continuous harmonic excitation has been studied instrings loaded with masses, both experimentally and numerically. It is reported inthe experimental work [44, 53] that for forcing frequencies within the pass band,increasing the driving amplitude leads to a decrease in the transmitted energy. Thereported results, though, are obtained for only one realization of disorder. Be-sides, the response regime is confined to driving amplitudes below the onset ofanharmonic motion, which would be below the onset of supratransmission1. Inthe numerical work [114], transmitted energies were found to generally increasewith the amplitudes of incident waves. Although anharmonic regimes were ob-served at high intensities, the corresponding increases in transmitted energies aresurprisingly small in these regions (especially for an undamped structure). Thisis uncharacteristic of supratransmission in which case an energy increase of a feworders of magnitude is expected (see Section 2.3).We are aware of two main studies that directly address the phenomenon of en-hanced energy transmission in nonlinear disordered structures due to continuousharmonic excitation [57, 125]. These studies report the existence of transmissionthresholds for undamped structures of long [57] and (relatively) short [125] lengths.In both studies, the excitation frequencies are limited to the pass bands of the corre-sponding ordered structures (this is the frequency range in which disorder prohibitsenergy transmission from a linear perspective). In [125], the statistical influenceof disorder on the average response is studied for different structure lengths. It isfurther shown that the average threshold amplitude decreases with the number ofunits according to a power law. In the limit of an infinitely long structure, though,the average transmission threshold is expected to remain finite [57].Despite the detailed studies in [57, 125], the case of supratransmission (exci-tation within a stop band) remains unexplored in disordered periodic structures.1The nature of the post-threshold response is non-periodic in supratransmission. This is explainedin Section 2.3.11So is the influence of system parameters (damping, in specific) on the mechanismleading to supratransmission. In the two studies [57, 125] mentioned above, en-ergy dissipation occurs at the boundaries of the periodic structures, but there is nointernal energy loss within the periodic structures themselves. Hence, the influenceof damping forces remains to be investigated. Damping and finite-size effects arenecessary to make the results applicable to engineering structures.1.4 Research Objectives and MethodologyThe goal of this thesis is to study the supratransmission phenomenon in a settingthat is applicable to engineering problems. From the perspective of engineeringapplications, it is essential to incorporate three features in this study: (i) dissipativeforces; (ii) small deviations from exact periodicity distributed throughout the struc-ture (disorder); (iii) the finite size of the structure. This thesis systematically inves-tigates the influence of these parameters on the supratransmission phenomenon ina one-dimensional periodic structure. Within this setting, we address the competi-tion that takes place among the effects of dispersion, dissipation, nonlinearity anddisorder-borne localization (Anderson localization).Dispersion effects are inherently present in periodic structures and are respon-sible for the formation of stop bands. Dissipation effects are unavoidable in engi-neering structures and normally influence all frequencies. Finite-size effects leadto formation of standing waves and restrict the allowable wavenumbers for propa-gating waves. Nonlinearity counters these effects, most importantly those of dis-persion, to give rise to supratransmission, which is an instability-induced transmis-sion mechanism within a stop band. Finally, disorder effects become significantbecause the spectral contents of the waves transmitted above the supratransmis-sion threshold fall within the pass band. This is the same frequency range whereAnderson localization dominates dispersion effects, thus leading to vibration lo-calization near the source of excitation. In this case, our goal is to investigatethe statistical influence of linear disorder on (i) the supratransmission thresholdswithin a stop band, (ii) transmitted energies above the supratransmission threshold,and (iii) the spectrum of the nonlinearly transmitted waves. While some portionsof this problem may only be tackled numerically, such as (ii) and (iii), we aim to12develop analytical estimates to be used in (i) in order to complement the numericalapproach .To this end, we study energy transmission through a discrete nonlinear periodicstructure, subjected to continuous harmonic point excitation at one end. We pro-pose a macro-mechanical periodic structure in this study that consists of coupledsuspended cantilever beams. Within each unit cell, the linear restoring force ofeach beam is combined with a strong nonlinear magnetic force to produce on-sitenonlinearity. The magnetic force can be tuned, thereby providing control over thestrength of nonlinearity, as well as its type (softening or hardening). The idea ofcombining a strong magnetic force with the linear restoring force of a cantileverhas been previously used in the literature, most notably by Den Hartog [23, Sec.8.10], Moon and Holmes [93] and Kimura and Hikihara [63], among others. Theproposed setup can be used to realize both ordered and disordered periodic struc-tures.Using the proposed mechanical structure, we review the mechanism responsi-ble for the onset of nonlinear energy transmission. We use the appropriate compu-tational methodology for calculation of the supratransmission thresholds and de-velop analytical estimates for predicting the onset of supratransmission. We studythe influence of various system parameters on the supratransmission mechanism:type and strength of nonlinearity, number of units, damping, strength of coupling,forcing amplitude and frequency, and disorder. In disordered structures, we inves-tigate the statistical behavior of the structure in an ensemble-average sense.It is noted that the analytical approximations are developed using asymptoticanalysis, which assumes that the nonlinear forces are comparable to linear forces.As we will explain in detail, this assumption may only be valid up to the onset ofsupratransmission for certain range of system parameters. The analytical approx-imations are not valid above the onset of supratransmission (where the responseis no longer periodic) and accordingly cannot describe the response of the struc-ture in that regime. Only direct numerical simulations can be used above the onsetof supratransmission. Further details on the analyses are provided in Sections 2.5and 4.4.This thesis does not contain an experimental component. As outlined in Sec-tion 5.3, the long-term perspective of this research project (beyond the present13thesis) is to build the mechanical setup proposed here in order to validate the theo-retical findings in this thesis.1.5 Thesis OutlineThe rest of this work is organized in four main chapters.We start in Chapter 2 by presenting the proposed mechanical setup. We de-rive the governing equations of motion and explain the controls provided by themagnetic forces: (i) strength of nonlinearity, (ii) type of nonlinearity, (iii) disor-der. In this chapter, we focus on an ordered periodic structure (no defect or dis-order). The periodic structure is very short (6 unit cells), has light damping andweak coupling forces between adjacent units. We explain the supratransmissionphenomenon and identify its underlying mechanism using direct numerical sim-ulations and numerical continuation techniques. The onset of supratransmissionis computed numerically by constructing the nonlinear response manifold of thesystem. The mechanism for nonlinear energy propagation is explained and its rel-evance to the resonances of the driving force with the shifted pass band of thestructure is discussed. We then derive analytical expressions for predicting theonset of transmission based on the local nonlinear dynamics of the driven unit.In Chapter 3, we investigate the influence of various system parameters onsupratransmission in ordered structures. The following parameters are considered:type and strength of nonlinearity, number of units, damping and strength of cou-pling. We highlight the importance of forcing frequency by contrasting supratrans-mission with the case of harmonic excitation within a pass band. We also brieflyaddress the hysteresis phenomenon accompanying supratransmission, as well asbroadband energy propagation at very high driving amplitudes.In Chapter 4, we study the influence of disorder on supratransmission. Westart by an overview of the key features of the linear response of disordered struc-tures (Anderson localization). We consider a periodic structure with 10 units inthis chapter. The choice of other system parameters, such as strength of disorder,are motivated in this context. We highlight the non-trivial effects of damping anddisorder on the transmission mechanism. We then discuss the statistical effects oflinear disorder on supratransmission. The changes in threshold curves, transmitted14energies and transmitted wave spectra are discussed in an ensemble-average sense.Finally, we present an analytical formula for predicting the onset of supratransmis-sion in ordered and disordered structures, and investigate its range of validity. Indoing so, we investigate the influence of nonlinearity and strength of coupling onthe average transmission threshold curves.We conclude in Chapter 5 by summarizing the contributions and limitations ofthis work and proving suggestions for future research on this topic.15Chapter 2SupratransmissionIn this chapter, we review the supratransmission phenomenon in a short, dampedperiodic structure. We identify the instability mechanism underlying supratrans-mission and clarify its relation to resonances of the driving force with the shiftedpass bands of the structure. Approximate analytical expressions are derived forpredicting the onset of supratransmission using asymptotic analysis, and validatedagainst results from numerical computations.2.1 A Periodic Structure with Tunable NonlinearityFigure 2.1 shows the proposed periodic structure consisting of N repeating units.Each unit is made of a thin, suspended cantilever beam of length ` with a tip mass(a permanent magnet). Two electromagnets (hatched rectangles in Figure 2.1) arefixed to the ground at a vertical distance h below the tip mass, and interact withthe permanent magnet (black rectangles). The electromagnets are symmetricallyplaced from the beam axis at a horizontal distance d. This symmetric arrangementensures that the vertical position of the cantilever is the equilibrium configuration.Direct current (DC) is passed through each electromagnet such that they have thesame polarity facing the beam. The magnetic forces between the permanent mag-net at the tip and the two electromagnets provide tunable nonlinear restoring forcefor each beam. The first beam is excited with a harmonic force f˜ (t˜ ) = F˜ cos(ω f t˜ ),where F˜ is the magnitude of the applied force, ω f is the driving frequency and t˜16NS…L…coupling rod N NS Sd dhunit cellelectromagnetspermanent magnetsFigure 2.1: A schematic of the periodic structure made of N unit cells. Therepeating unit is indicated by the dashed box. The external harmonicforce, f˜ (t˜ ) = F˜ cos(ω f t˜ ), is applied to the first unit only. Two electro-magnets, operated by direct currents, are fixed to the ground under thebeam in each unit. The currents are chosen such that the electromagnetshave the same polarity facing the beam. We fix h = d/10 throughoutthis work.is time. A coupling rod of length L couples the displacements of adjacent beams.The spacing between adjacent beams (L) is large compared to the separation be-tween the magnets (2d) in order to avoid magnetic interference effects between themagnetic fields in adjacent units.The proposed mechanical setup has features that make it a good candidate forour study on nonlinear periodic structures. It can be easily manufactured com-pared to its counterparts, such as arrays of micromechanical resonators [116]. Theinstrumentation required for performing the tests are often available to researchersin mechanical vibrations (electrodynamic shaker and accelerometer). Furthermore,the tunable magnetic force makes it possible to control the form of nonlinearity, afeature that can be used to replicate different types of nonlinear forces and therebydifferent types of nonlinear periodic structures. This cannot be achieved with sim-pler setups such as arrays of coupled pendulums [40]. The limitation here is theform of the nonlinear force (described in Section 2.1.1). As cases in point, the re-sults obtained in this thesis may not be readily extended to systems with nonlinearfriction forces or saturable nonlinear forces without further investigation. As will17be demonstrated in this chapter, the nonlinear force provided via the electromag-netic interaction is appropriate for generating the supratransmission phenomenon.The strength of the coupling force between adjacent units may also be adjusted byplacing the coupling rod at different heights along the cantilevers. The setup mayalso be used to realize disordered periodic structures. This can be done by adjust-ing the currents going through the electromagnets, as described in the followingsection.2.1.1 Governing Equations for the Periodic StructureThe chain of coupled cantilever beams in Figure 2.1 is a continuous system withinfinite degrees of freedom, with each individual cantilever beam having infinitenumber of vibration modes. Therefore, there will be infinitely many pass bands,extending to infinite frequency. In the first pass band, all cantilever beams vibratein their fundamental mode with different phases. In the second pass band, thebeams vibrate in their second deformation mode shape with different phases, andso on. We focus on energy transmission in frequency ranges within the first passband of the structure so that a dynamical representation with N degrees of freedomis appropriate. Thus the governing system of partial differential equations reducesto a set of coupled ordinary differential equations. The equation governing thevibrations of a unit cell (before coupling to adjacent units) can be expressed asfollows:¨˜un+2ζ˜ ˙˜un+ω20 u˜n+ F˜M,n = 0 (2.1)where overdot denotes time derivative, u˜n (t˜ ) is the lateral deflection of the tip ofbeam n, ζ˜ is the coefficient of viscous damping, ω0 is the first natural frequencyof the beam when the electromagnets are removed and F˜M,n is the horizontal com-ponent of the magnetic force acting on the tip of the beam resulting from the in-teraction of the permanent magnet and two electromagnets. The axial componentof the force is ignored because the coupling between axial and lateral vibrationsof the beams are negligible. If the displacements of the tip of the beam are smallcompared to its length, equal currents pass through the two electromagnets withinthe unit cell, and the magnets are modeled as magnetic poles, then the magnetic18force F˜M,n can be written acording to Coulomb’s law:F˜M,n = µ˜n(d+ u˜n)((d+ u˜n)2+h2)3/2 − µ˜n(d− u˜n)((d− u˜n)2+h2)3/2(2.2)This is the horizontal component of the magnetic force acting on the nth beam.The constants µ˜n depend on the strengths of the magnetic poles and contain thenon-geometric dependencies of the magnetic force. The magnetic force can betuned, thus providing control over the strength of nonlinearity, as well as its type(softening or hardening). This will be explained in detail in Section 2.1.2.To normalize the governing equations, we use ω0 for time and d for displace-ments. We consider a linear coupling force between adjacent units due to the cou-pling rod. Including this force and dividing all terms by ω20 d, we arrive at thenon-dimensional form of the governing equations (2.1) for N unitsu¨n+2ζ u˙n+un+ kc∆2(un)+FM,n = fn cos(Ωt), 1≤ n≤ N (2.3)Here, un ≡ u˜n/d is the normalized displacement of each unit, kc represents thenormalized coupling force between adjacent units and ∆2(un) = 2un−un+1−un−1everywhere except at the boundaries, where ∆2(u1) = u1−u2 and ∆2(uN) = uN −uN−1 (free boundary conditions). The normalized external force is given by fn = Ffor n = 1 and zero elsewhere, with F ≡ F˜/dω20 as the normalized forcing ampli-tude, Ω ≡ ω f /ω0 as the normalized forcing frequency and t ≡ ω0 t˜ as normalizedtime. FM,n represents the only nonlinear force in the model and is given byFM,n = µn((1+un)((1+un)2+ r2)3/2− (1−un)((1−un)2+ r2)3/2)(2.4)where r ≡ h/d is a fixed parameter and µn ≡ µ˜n/d3ω20 is a control parameter thatcan be changed. In the case of an ordered periodic structure, we have µn = µ0for all n. For a disordered periodic structure, µn = µ0 + δµn where |δµn| < |µ0|.Hereafter, all parameters used in the thesis, including time, are non-dimensional.Expanding (2.4) about its trivial equilibrium point, un = 0, we obtainFM,n ≈−µn(a1un+a3u3n+ · · ·)≡ k1un+ k3u3n+ · · · (2.5)19where a1 and a3 only depend on the (fixed) parameter r as follows:a1 =2(2− r2)(1+ r2)5/2(2.6a)a3 =8−24r2+3r4(1+ r2)9/2(2.6b)The constants k j in (2.5) are defined as k j ≡−µna j for j = 1,3,5, . . . .We set r = 0.1 throughout the work. Unless otherwise specified, we use kc =0.05(1+ k1) as the default value for the strength of coupling between adjacentunits – the need for weak coupling will be motivated in Section 4.1. Furthermore,we consider lightly damped structures. The values of µn change throughout thework to realize different nonlinear forces, and are reported in each case. The im-portance of the type of nonlinearity (determined by the sign of µn) is addressedin Section 3.1. The remaining two parameters are F and Ω, which are typicallyfree parameters. Throughout this chapter, we consider a periodic structure with sixunits, N = 6.2.1.2 Tunability of the Nonlinear ForcesThe source of nonlinearity in the proposed periodic structure is FM,n, which actsas an on-site nonlinear force for each unit. The type and strength of nonlinearitycan be tuned by changing the control parameter µn. In particular, the strengthof nonlinear terms increases when we increase the magnitude of µn, as evidentfrom (2.4). Moreover, the sign of µn determines whether nonlinearity is of thehardening or softening type. If µn < 0, then k3 > 0 in (2.5) and we have a hardeningnonlinearity. Likewise, a softening nonlinearity can be realized by setting µn > 0,thus obtaining k3 < 0 according to (2.5).In order to show the tunability of nonlinear forces, we consider the normalizedrestoring force acting on a beam in a single unit. Denoted by Pn, the normalizedrestoring force is the sum of a normalized elastic force (un) and a normalized mag-netic force (FM,n),Pn = un+FM,n (2.7)The linear natural frequency of a unit is the first derivative of the restoring force at20the origin,ω21 ≡(∂Pn∂un)un=0= 1+ k1 (2.8)Note that the linear part of the magnetic force (i.e. k1un) has an influence on thenatural frequencies of the system. The sign of k1 depends on µn according to (2.5),such that the first natural frequency of the periodic structure (ω1) is below 1 whenµn > 0 (softening nonlinearity) and above 1 when µn < 0 (hardening nonlinearity).The nonlinear force FM,n can be adjusted within each unit by changing the cur-rents passing through the electromagnets, In, thereby varying µn in (2.4). The signof In depends on the direction of the current passing through the electromagnetswithin that unit. Note that FM,n also depends on the geometrical configuration ofthe setup through parameter r, though we fix r = 0.1 in this work and only changeµn.Following Kimura and Hikihara [63], we have assumed µn = ξ0 + ξ1In for alln, where ξ0 corresponds to the normalized magnetic strength of the ferromagneticcore of the electromagnets, and ξ0 = 0.0151 and ξ1 = 0.0029. Due to the pres-ence of ferromagnetic cores (ξ0 6= 0), the magnetic coefficients µn do not dependsymmetrically on the currents; i.e. µn(In) 6= µn(−In). If the electromagnets hadnon-ferromagnetic cores, then we would have ξ0 = 0.Figure 2.2 shows the variation of Pn as a function of un for a single unit. Wecan see that when both electromagnets are removed (µn = 0), the restoring force islinear as expected. By placing the electromagnets with the polarity shown in Fig-ure 2.1 and setting In = 15 mA, we get a nonlinear restoring force of the softeningtype. A hardening nonlinear force is realized when the direction of the currentspassing through both electromagnets is reversed (In = −15 mA). Notice that therestoring forces for the hardening and softening systems are not symmetric withrespect to the restoring force of the linear system. This is due to the presence offerromagnetic cores in the electromagnets, as explained in the previous paragraph.Because of the same reason, the case of In = 0 does not correspond to µ0 = 0 andis therefore different from the linear case (Pn is not shown for In = 0 here).210 0.1 0.2 0.3 0.4 0.500.10.20.30.40.50.6Restoring force, PnDeflection, un  No electromagnet (µn = 0)In = +15 mA (softening)In = −15 mA (hardening)Figure 2.2: The dependence of the normalized restoring force, Pn = un+FM,n,on the currents passing through the two electromagnets, In. Notice thatthe restoring forces for the hardening (blue dashed curve) and softening(red dash-dotted curve) systems are not symmetric with respect to therestoring force of the linear system (black solid line). This is due to thepresence of ferromagnetic cores in the electromagnets.2.2 Band Structure of Periodic Systems2.2.1 Forced Response of Linear StructuresFor small-amplitude vibrations, we can linearize (2.3) to getu¨n+2ζ u˙n+ω21 un+ kc∆2(un) = fn cos(Ωt) , fn = 0 for 2≤ n≤ N (2.9)where ω1 is defined in (2.8). The solutions to the linear system (2.9) can be writtenas followsun(t) = u1(t)e−iκ(n−1)e−γ0(n−1) (2.10)where i =√−1, κ is normalized wavenumber, u1(t) denotes the response of thefirst unit as a function of time, and γ0 ≥ 0 is a real-valued decay exponent. u1(t) =U1 exp(iΩt), in which U1 is the complex-valued amplitude of motion. When γ0 > 0,amplitudes of vibration attenuate exponentially through the periodic strcture andenergy propagation is not possible over long distances. We get complete transmis-sion when γ0 = 0. Substituting solution (2.10) into the governing equations (2.9),22we arrive at the followingcosh(γ0)cos(κ) = 1+(ω21 −Ω2)/(2kc) (2.11a)sinh(γ0)sin(κ) = (2ζΩ)/(2kc) (2.11b)In the absence of damping (ζ = 0), we have γ0 = 0 (complete transmission) forω21 ≤Ω≤ω21 +4kc, which is the pass band of the structure. Outside this frequencyrange, γ0 > 0 and waves will attenuate exponentially away from the source (n= 1).This can be seen in Figure 2.3 where γ0 is plotted as a function of Ω, for differentvalues of damping. The area with a white background indicates frequencies forwhich γ0 = 0 in the undamped system. We can see that the presence of dampingresults in spatial decay of response at all frequencies, even within the pass band.In particular, the influence of damping on the decay exponent is mostly significantwithin the pass band. We note that the analysis in this section is exact for aninfinitely long exactly periodic linear structure.To conclude this section, we point out that viscous damping, if it is highenough, can result in emergence of stop bands in the wavenumber domain; i.e.spatial stop bands corresponding to imaginary wavenumbers. See [54] for moredetails on the influence of high levels of damping on the band structure of (or-dered) linear periodic structures.2.2.2 Free Wave Propagation in Linear and Nonlinear StructuresFor small displacements around the vertical equilibrium position, we can writethe following equations for free vibrations of the unforced, undamped periodicstructureu¨n+ω21 un+ kc(2un−un−1−un+1) = 0 (2.12)where ω1 is defined in (2.8). In an infinitely long periodic structure, the linearequations (2.12) admit plane wave solutionsun(t) =U cos(ωt−qn) (2.13)23Figure 2.3: Decay exponent γ0 for an infinitely long exactly periodic linearstructure. The white and grey backgrounds indicate the pass and stopbands, respectively. When ζ > 0, the decay exponent is always positive,even within the pass band. Anderson localization is relevant for frequen-cies within the pass band, whereas supratransmission occurs within thestop band – we will discuss these phenomena in subsequent chapters.where U is the wave amplitude and q is the normalized wave number, with 0≤ q≤pi . Using this solution, the linear dispersion relation is found to be:ω2 = ω21 +4kc sin2(q/2). (2.14)The above dispersion relation describes the first pass band of the system, whichstarts at a finite frequency, ω1. The upper frequency of the pass band edge (denotedby ωu) is associated with the wave number q = pi . Therefore, from (2.14), we haveω2u = ω21 +4kc (2.15)Notice that the width of the pass band (ω2u −ω21 ) is directly proportional to thestrength of coupling, kc. We can also conclude from (2.14) and (2.8) that the loca-tion of the pass band (but not its width) depends on µn, and thereby on the currentspassing through the electromagnets. This can be verified from Figure 2.4, wherethe dispersion curve (2.14) is plotted for the softening (I = 15 mA) and hardening(I = −15 mA) realizations of the magnetic force. If the electromagnets were re-moved (µn = 0), then the pass band would start from the first natural frequency of240 pi/4 pi/2 3pi/4 pi0.80.911.11.2Wave number, qFrequency, ω  In = −15 mA (hardening)In = +15 mA (softening)Figure 2.4: The dependence of the linear dispersion curve, (2.14), on the cur-rents passing through the electromagnets. The red dashed curve cor-responds to the hardening structure (I = −15 mA) and the blue dash-dotted curve to the softening structure (I = 15 mA). The six dots alongeach curve correspond to the natural frequencies of the finite structurewith free boundary conditions. The horizontal dotted line depicts thefirst natural frequency of the structure when the electromagnets are re-moved. See (2.8) and its foregoing explanation regarding the depen-dence of linear dispersion curves on the type of nonlinearity.the beams (ω1 = 1).The finite periodic structure shown in Figure 2.1 is symmetric, has symmetricunit cells and has open (free-free) boundary conditions. Therefore, the first naturalfrequency of the finite structure coincides with the the lower edge of the pass bandof the infinite system [87]. The remaining natural frequencies of the finite structuredistribute within the pass band of the infinite system with a uniform wave numberspacing of pi/N, where N is the number of the units [119]. As a result, the highestnatural frequency of the finite system increases with the number of units and, in thelimit of infinite units, it reaches ωu. Again, we emphasize that these conclusionspertain to our model of the infinite degrees of freedom system reduced to frequen-cies within the first pass band region. The six computed natural frequencies of thefinite periodic structure studied in this work are shown as dots in Figure 2.4. Asexpected, all of them lie within the first pass band of the infinite system, with thelowest one coinciding with the lower edge of the pass band.25For finite amplitudes of motion, the dispersion relation naturally becomes de-pendent on the amplitude of motion. To obtain the dispersion relation for a weaklynonlinear system (small but finite amplitudes of motion), we need to account forthe dependence of the natural frequencies on the amplitudes of motion; i.e. ω =ω(U) in (2.13). A standard asymptotic analysis, such as the Lindstedt-Poincare´ ormultiple-scale method, can be performed for this purpose – e.g. see [97, 128] forsimilar analyses. To the leading order (i.e. for weak nonlinearity), the amplitude-dependent correction to the linear dispersion relation results in the following dis-persion relationω2 = ω21 +4kc sin2(q/2)+3/4k3U2 (2.16)where U represents the amplitude of motion. According to the amplitude-dependentdispersion relation (2.16), nonlinearity can shift the pass band, but cannot changeits width. This is not surprising because the width of the pass band depends onthe coupling force between the units, which is linear in this case. We note thatthis approximation is valid for finite but small amplitudes of motion. Higher-ordercorrections would be needed to improve the approximation.For the infinite system, we can readily study the edges of the pass band in moredetail. At the lower edge, adjacent units move in phase with each other (q= 0) andwe have un±1 = un. At the upper edge, adjacent units move out of phase (q = pi)and un±1 = −un, as a result. We can therefore decouple the equations at the twoedges of the pass band as follows:u¨n+un+FM,n = 0 (2.17a)u¨n+(1+4kc)un+FM,n = 0 (2.17b)where the natural frequencies of (2.17a) and (2.17b) correspond to the lower andupper edges of the pass band, respectively.2.3 Energy Transmission via Harmonic Excitation withina Stop BandConsider the hardening system with I =−15 mA, which has its upper linear natu-ral frequency at ω6 ≈ 1.139. We choose ζ = 0.004. The forcing frequency is fixed26above the linear pass band of the infinite system; i.e. Ω>ωu ≈ 1.145 – we will seein Section 3.1 that supratransmission occurs above the pass band in a structure withhardening nonlinearity. We choose a small driving amplitude F and numericallyintegrate the governing equations (2.3) for a large number of periods, T = 2pi/Ω,starting from zero initial displacements and velocities. We then repeat this proce-dure for increasing values of F . For a given F , we increase the forcing amplitudesmoothly from zero to F to avoid formation of shocks in the numerical solution.This is done by replacing F with F (1− exp(−t/τ1)), and is particularly importantfor undamped systems. After the initial transient part of the solution is passed, wecalculate the energy in each unit, denoted by En (normalized), as follows:En =1(m2−m1)T∫ m2Tm1T(un(t)F)2dt (2.18)We have used τ1 = 50, m1 = 500 and m2 = 3500 in this chapter. Notice that energyis normalized to compensate for the linear increase in response amplitudes, un(t),due to increase in F .Figure 2.5(a) shows the time-averaged energy in the last unit, E6, as a functionof F , for a fixed forcing frequency of Ω = 1.25 above the linear pass band (hard-ening nonlinearity). We can see a threshold around F = 0.116 above which thereis a sudden large increase in the energy that reaches the end of the structure. Thefrequency components of the response at the first and last units are also shown fortwo cases: (1) F = 0.114, below the transmission threshold in Figure 2.5(b), (2)F = 0.116, above the threshold in Figure 2.5(c). The frequency spectrum for eachoscillator is obtained by taking the Fast Fourier transform (FFT) of the correspond-ing time series after the initial transient has passed; no additional windowing orscaling is used. We can see in Figure 2.5(b) that the driving frequency (Ω= 1.25)is the predominant frequency component throughout the structure below the thresh-old. Therefore, the response of the system remains harmonic below the threhsold.Above the threshold, see Figure 2.5(c), the driven unit moves with a high amplitudeand its response is broadband and highly nonlinear. Figure 2.5(c) shows that thefrequencies of the transmitted waves are predominantly within the linear pass bandof the structure. The amplitude of waves with frequencies within the stop bandattenuate due to dispersion effects. Notice that the amplitudes of waves within the27pass band also decrease due to dissipation.We observe the same phenomenon in the case of a softening system (I =15 mA). In this case, the lower linear natural frequency of the structure is atω1 ≈ 0.88. We fix the forcing frequency below this value, at Ω = 0.85 < ω1, andfollow the same procedure as before to observe supratransmission. Figure 2.6(a)shows E6 as a function of F . We observe the onset of supratransmission occurringbetween F = 0.019 and F = 0.021. We show the frequency components of theresponse at F = 0.019 in Figure 2.6(b). As expected, the response is harmonicwith the same frequency as the driving force (Ω = 0.85). Because this frequencyis within the stop band, we observe a large decrease in the amplitudes of motionas we go from n = 1 to n = 6. At F = 0.021 (above the transmission threshold),we see in Figure 2.6(c) that the response at the first unit is highly nonlinear. As wego through the periodic structure to its other end, the frequency components withinthe linear stop band are highly attenuated by dispersion. Accordingly, the spectrumof the transmitted waves lies mainly within the pass band of the linear structure.Thus supratransmission is a band-limited transmission phenomenon.It is worth mentioning that the nature of the post-threshold response in supra-transmission can vary based on system parameters (driving frequency, type ofnonlinearity, etc). There does not seem to be a reliable methodology that canpredict the nature of the post-threshold response a-priori. We will see in Sec-tion 2.4.1 that supratransmission occurs as long as the post-threshold response isnon-periodic. Accordingly, classification (from a dynamical-systems viewpoint) ofthe post-threshold non-periodic attractors is not of primary interest in this context.To indicate this point, we consider the post-threshold responses of structureswith hardening and softening nonlinearities in Figures 2.5(c) and 2.6(c), respec-tively. The response of the hardening system appears to be chaotic, while that ofthe softening systems appears to be quasi-periodic – a quasi-periodic motion con-sists of two or more incommensurate frequencies. To confirm this observation,we construct the Poincare´ map of the two responses above the supratransmissionthreshold. Because of the external harmonic force applied to the structure, thePoincare´ map can be constructed by (stroboscopically) recording the displacementand velocity of the response at a fixed phase of the driving force – see [70, Ch. 1]for more details. Figure 2.7 shows the projection of this Poincare´ map (obtained28at zero phase) onto the phase space of the last unit, (uN ,vN). The Poincare´ map ofthe response for the hardening system, Figure 2.7(a), consists of a cloud of scat-tered points. This implies that the response is chaotic – note that a rigorous proofof chaos (and its classification) is immaterial in the current discussion. We seein Figure 2.7(b) that the Poincare´ map of the response for the softening systemis a closed curve. This confirms that the post-threshold response of the softeningsystem is quasi-periodic.Figure 2.8 shows the normalized amplitude profiles of the structures, belowand above the supratransmission threshold. For each unit, the amplitude Un iscomputed as half the difference between the maximum and minimum values, Un =(max{un(t)}−min{un(t)})/2, after the initial transient part of the response ispassed. This is equal to the amplitude of the motion in the phase space. For thehardening system, we see in Figure 2.8(a) that the amplitude profile shows expo-nential decay below the threshold. This is a characteristic of the linear behavior ofperiodic structures when the forcing frequency is within the stop band. Above thethreshold, on the other hand, the decay in amplitude is no longer exponential. Thissuggests that above the threshold, the energy injected in the stop band transmits todistant units with less attenuation. Qualitatively, we make the same observation inFigure 2.8(b) for the softening system. It is important to note that attenuation overlarge distances is always present in a damped structure. In this sense, dampingprevails over nonlinearity.290 0.5 1 1.5 210−410−2100U10 0.5 1 1.5 210−610−410−2100U6Frequency0 0.5 1 1.5 210−610−410−2100U10 0.5 1 1.5 210−1010−810−610−4U6Frequency0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.145 0.1510−810−510−2101Forcing amplitude, FTransmitted energy, E6onset of transmission(a)(b) (c)Figure 2.5: The supratransmission mechanism for the hardening system (I =−15 mA). (a) Energy transmitted to the end of the chain, E6, as a func-tion of the driving amplitude, F , at Ω= 1.25. The blue arrow indicatesthe onset of transmission. (b) Frequency components of the first unit(U1) and last unit (U6) for F = 0.114, below the transmission thresholdat Ω = 1.25. (c) Frequency components of the first unit (U1) and lastunit (U6) for F = 0.116, above the transmission threshold at Ω = 1.25.The six vertical lines in (a) and (b) indicate the linear natural frequenciesof the structure.300 0.5 1 1.5 210−810−610−410−2100U10 0.5 1 1.5 210−810−610−410−2100U6Frequency0 0.5 1 1.5 210−810−610−410−2100U10 0.5 1 1.5 210−810−610−410−2100U6Frequency0.005 0.01 0.015 0.02 0.025 0.0310−310−210−1100101102Forcing amplitude, FTransmitted energy,  E6onset of transmission(a)(b) (c)Figure 2.6: The supratransmission mechanism for the softening system (I =15 mA). (a) Energy transmitted to the end of the chain, E6, as a functionof the driving amplitude, F , at Ω = 0.85. The blue arrow indicates theonset of transmission. (b) Frequency components of the first unit (U1)and last unit (U6) for F = 0.020, below the transmission threshold atΩ= 0.85. (c) Frequency components of the first unit (U1) and last unit(U6) for F = 0.021, above the transmission threshold at Ω= 0.85. Thesix vertical lines in (a) and (b) indicate the linear natural frequencies ofthe structure.31−0.5 −0.25 0 0.25 0.5−0.5−0.2500.250.5u6 (t)v6 (t)−0.3 −0.2 −0.1 0 0.1 0.2−0.2−0.100.10.2u6 (t)v6 (t)(a) (b)Figure 2.7: The projection of the Poincare´ map of the response just abovethe transmission threshold onto the phase space of the last unit. (a)the hardening system at F = 0.116, (b) the softening system at F =0.021. Notice that the post-threshold response of the hardening systemis chaotic, while that of the softening system is quasi-periodic.1 2 3 4 5 610−510−410−310−210−1100Unit number, nUn / U1  1 2 3 4 5 610−310−210−1100Unit number, nUn / U1  (a) (b)F = 0.114F = 0.116F = 0.019F = 0.021Figure 2.8: Normalized amplitude profiles (Un/U1) for (a) the hardening sys-tem and (b) the softening system. In each case, the amplitude profilesbelow and above the transmission threshold are shown respectively withblue squares and black circles. The amplitude profiles show exponentialdecay below the threshold, which is a characteristic of linear behavior.322.4 Computing the Supratransmission Threshold2.4.1 Nonlinear Response Manifold (NLRM)In contrast to the existing literature on supratransmission, we are concerned withperiodic structure with damping and finite length. For undamped and infinitely longperiodic structures, Maniadis et al. [82] have observed that as the forcing ampli-tude is increased, transmission starts when the periodic solutions lose their stability.They show that the onset of transmission coincides with the first turning point ofthe nonlinear response manifold (NLRM). This is defined as the manifold of initialconditions with zero initial velocities that give a time-reversible periodic solutionat a given driving amplitude [67, 82]. At the first turning point of the NLRM fora non-dissipative system, the stable periodic solution that continues from the lin-ear regime (passing through the trivial equilibrium) collides with another unstableperiodic solution that continues from a discrete breather (a non-zero solution cor-responding to zero driving amplitude)1. The NLRM possesses a turning point at agiven forcing frequency provided that the free system supports breather solutionsat that frequency [82]. The collision between the continuations of the trivial equi-librium and the discrete breather corresponds to a saddle-node bifurcation, whereone of the Floquet multipliers of the dynamical system exits the unit circle on thepositive real axis. Further technical details about bifurcations of periodic orbits canbe found in [70, Ch. 5].We compute the NLRM using numerical continuation techniques as imple-mented in AUTO [26]. This numerical approach allows us to follow the evolutionof the steady-state periodic response of the structure as a function of the forcingamplitude, without the need to directly integrate the governing equations at eachstep. In order to perform this computation, the governing equations are first recastas a boundary value problem and then continuation methods are used to followthe branches of periodic solutions and their subsequent bifurcations; refer to Ap-pendix A for an explanation of this process. Note that there is no need to imposea phase condition to the NLRM in a damped system (as opposed to Hamiltonian1Discrete breathers are spatially localized, time-periodic solutions in discrete nonlinear periodicsystems; see [5, 10] for more details.330 0.005 0.01 0.015 0.02 0.025 0.0300.10.20.30.40.50.60.70.8Forcing amplitude, FResponse amplitude, U1linear responseperiodic responsenon−periodic responseunstable solution  StableUnstableFigure 2.9: The projection of the NLRM of the softening system onto theU1 − F plane for Ω = 0.85 . The NLRM is shown using the blackcurve. The solid section of this curve represents stable solutions andthe dashed section represents unstable solutions. The grey dash-dottedline shows the linear response of the structure. The results of directnumerical integration (DNI) are shown using the circle markers. Pe-riodic responses are shown by empty markers and aperiodic responsesare shown by filled markers. The blue diamond indicates an unstableperiodic solution at the same F as the first turning point.systems) because the response frequency and phase are fixed by the external force– we have limit cycles. In fact, the NLRM can be redefined in this case as themanifold of initial conditions on a limit cycle at a given driving amplitude. Thisdefinition is trivial, but to keep the connection with the undamped case, we willrefer to the response of the forced and damped system as the NLRM as well.Figure 2.9 shows the NLRM for the softening system at Ω= 0.85, along withresults from direct numerical integration of the governing equations (2.3). The re-sponse is harmonic below the transmission threshold (empty markers) and followsthe linear response for small values of driving amplitude. At the first turning pointof the NLRM, this harmonic response becomes unstable (through a saddle-nodebifurcation) and the response inevitably jumps to another basin of attraction foranother attractor, which is aperiodic (filled markers). This jump is accompanied by340 0.05 0.1 0.15 0.2 0.25 0.3 0.3510−1100101Forcing amplitude, FResponseamplitude,U1  NLRM @ Ω = 1.25NLRM @ Ω = 1.40Threshold curveDNI @ Ω = 1.25DNI @ Ω = 1.40Figure 2.10: Evolution of the response (represented by U1) as a function ofthe driving amplitude for the hardening system. Solid black curve:NLRM at Ω = 1.25; dashed blue curve: NLRM at Ω = 1.40. Thedash-dotted red curve shows the locus of the first turning points ofthe NLRM as Ω varies. The results of direct numerical integration(DNI) at each driving frequency are shown using the markers: trianglesfor Ω = 1.25 and circles for Ω = 1.40. Periodic responses are shownby empty markers and aperiodic responses by filled markers. We seethat the first turning point in each NLRM coincides with the onset ofsupratransmission. At the threshold, the response jumps to a different(non-periodic) branch.a large increase in the energy transmitted to the end of the structure. Note that thereis another periodic solution branch right above the first turning point, indicated inFigure 2.9 by the blue diamond. This solution is unstable. If the upper branch werestable, the solution would have jumped to that branch instead, accompanied by amodest increase in energy transmitted, but not an increase of an order of magni-tude. It is important to note that the instability of this upper solution branch is anecessary requirement for supratransmission. We are not aware of any other workthat highlights this requirement.Figure 2.10 shows the NLRM for the hardening system at two different forcingfrequencies above the linear pass band. Similar to the case of the softening system,the response is harmonic below the transmission threshold (empty markers) andjumps to an aperiodic branch (filled markers) at the first turning point of the NLRM.35Figure 2.10 also shows the transmission threshold curve of the hardening system(red dashed curve). The threshold curve shows the locus of the first turning pointsof different NLRMs as the driving frequency Ω varies. It is worth noting that thethreshold curve is a codimension-two manifold, meaning that it is parameterizedby two free variables (F and Ω in this case). We see its projection onto the U1−Fplane in Figure 2.10. Also, notice that the threshold curve terminates (in a cusp) ata finite value of F . This is due to the presence of damping, and will be discussedin detail in Section 3.3.2.4.2 Resonances with the Shifted Pass BandRepeating the numerical analysis for different values of Ω reveals that as the driv-ing frequency is varied within the linear stop band (away from the edge of the passband), the transmission threshold increases. Moreover, supratransmission occursabove the pass band in the hardening case and below the pass band in the softeningcase – see Section 3.1 for more details. Also, it is known [14, 128] that nonlinearitycan alter the dispersion relation of periodic structures and may shift the location ofthe pass band depending on the total energy level of the system2. Putting all thistogether, one might be inclined to explain the nonlinear supratransmission phe-nomenon occurring due to the resonance of the driving force with the shifted passband of the structure: due to increased amplitudes of motion, the pass band shiftsto lower frequencies in the case of softening nonlinearity and the driving force res-onates with the shifted pass band. The same argument can be made for frequenciesabove the pass band in the case of a hardening nonlinearity.We show here that the true mechanism is indeed the saddle-node bifurcation de-scribed in Section 2.4, and the resonance of the driving force with the shifted passband merely provides a qualitative explanation of the phenomenon. Figure 2.11compares the edge of the pass band with the transmission threshold for the soft-ening system (I = 15 mA). The lower edge of the pass band is computed as thenonlinear normal mode of (2.17a). The transmission threshold is computed for anundamped system (ζ = 0) as described in Section 2.4.1, and corresponds to the2This shift is towards lower frequencies in a structure with softening nonlinearity and towardshigher frequencies in the case of a hardening nonlinearity. We can also see this in the amplitude-dependent dispersion relation in (2.16).360.7 0.725 0.75 0.775 0.8 0.825 0.85 0.875 0.910−210−1100Frequency, ΩResponse amplitude, U1  Pass band, linearPass band edge, nonlinearResponse at threshold (continuation)Response at threshold (DNI)Figure 2.11: Comparison of the transmission threshold and the edge of thepass band for the structure with softening nonlinearity. The verticalline shows the lower edge of the pass band for the linear system (ω1).The dash-dotted blue curve shows the lower edge of the pass band forthe nonlinear system, which is computed as the nonlinear normal modeof (2.17a). The dashed red curve shows the amplitude of the first unit,U1, at the onset of transmission as a function of the driving frequency.The empty diamond markers show U1 at the driving amplitude just be-low the onset of transmission. They are obtained from direct numericalintegration (DNI) of the equations of motion (2.3).point where the periodic solutions lose stability through a saddle-node bifurcation.The onset of transmission is also obtained from direct numerical integration (DNI)of (2.3) at separate driving frequencies. The amplitudes of the first unit, U1, at thedriving amplitudes just below the threshold are shown for each driving frequencywith empty markers. We can see in Figure 2.11 that the results from DNI agreewell with the threshold curve. Notice that the threshold curve and the edge of thepass band show a similar trend: they both increase as we move farther into thelinear stop band. More importantly, however, we see that the two curves do notmatch. This is not a surprising feature after all: the threshold curve traces thelocus of saddle-node bifurcations while the pass band edge is a nonlinear normalmode (backbone curve) of the structure. These two curves are not expected to co-incide. The distinction between these two loci in single forced oscillators is indeedamong the early topics discussed in typical textbooks on nonlinear vibrations suchas [99, 121].In conclusion, Figure 2.11 and the accompanying explanation indicate that the37resonance of the driving force with the shifted pass band is not a correct predictorof the onset of transmission from either a quantitative or phenomenological per-spective. Similar observations are made for damped structures or in the case ofhardening nonlinearity, but the results are not shown here. We will further see inChapter 3 that the onset of supratransmission is a function of damping, strength ofcoupling and other system parameters.2.5 Analytical Prediction of the Onset ofSupratransmission2.5.1 Analysis Based on Local Nonlinear DynamicsTo find the force threshold for the onset of supratransmission, we look for the onsetof instability for the harmonic solutions of the dynamical system. To this end,we first derive the equations governing the evolution of the envelope of harmonicwaves using multiple-scale analysis; see [99, Ch. 6] for details of the methodology.We seek solutions to (2.3) of the following form:un(t1, t2) = εψn(t2)e−iΩt1 + c.c. (2.19)where c.c. denotes the complex conjugate terms and ε is a small parameter. Thereare two time scales, t1 ∝ 1 and t2 ∝ ε2. The strength of coupling and dampingcoefficient are small, kc ∝ ε2 and γ ∝ ε2. Because we are looking for the small-amplitude wave envelope, we take the driving amplitude to be small, F ∝ ε2. Thebalance of equations at O(ε) is trivial. The same applies to the next order due tothe odd nature of the nonlinear force; i.e. only odd orders appear in (2.5). Fromthe balance of equations at O(ε3), we obtain the following equation for ψn:−2iΩψ ′n− iβψn+σ1ψn−3αψ2nψ∗n + kc∆2(ψn) = fn/2 (2.20)where an uppercase asterisk denotes complex conjugate, prime denotes differenti-ation with respect to the slow time scale, t2, and ∆2(.) is the same as in (2.3). The38following new parameters have been introduced in (2.20) for ease of referenceβ ≡ 2ζΩ (2.21a)σ1 ≡ ω21 −Ω2 (2.21b)α ≡ µna3 (2.21c)Also, β =O(ε2), σ1 =O(ε2) and α =O(1). Note that because the periodic systemis very short (a few number of units) and the coupling between adjacent units issmall, a spatial homogenization scheme, as often used in the literature [62, 66,126], is not appropriate here.We know that the amplitudes of motion decay rapidly along the structure awayfrom the driven unit. As a first approximation, therefore, we decouple the firstunit from the rest of the periodic structure, and look in the reduced system forpoints where the stability of motion changes (through a saddle-node bifurcation).It is expected that these points can be used to estimate the onset of instability and,therefore, the supratransmission force threshold for the entire structure. We notethat this approximation may only be valid in the limit of very weak coupling, alsoknown as the anti-continuum limit. See [82] for a similar approach.The dynamic reduction of the periodic structure to a single unit depends on thelocation of the driving frequency relative to the linear pass band. In a system withsoftening type of nonlinearity, supratransmission occurs below the lower edge ofthe pass band. In hardening systems, on the other hand, supratransmission occursabove the upper edge of the pass band. Adjacent units move in phase with eachother at the lower edge and out of phase at the upper edge. Therefore, before trun-cating (2.20) at n = 1, we take ∆2(ψn) = 0 in the softening case and ∆2(ψn) = 4ψnin the hardening case. We note that this argument is valid for an infinite periodicstructure. In a finite structue, as discussed in Section 2.2.2, only one of the edgesof the pass band coincides with those of the infinite structure. As a result, taking∆2(ψn) = 4ψn above the pass band is an approximation in our periodic structurebecause of the free-free boundary conditions.With these considerations in mind, the wave envelope equation (2.20) reduces39to the following:−2iΩψ ′1− iβψ1+σψ1−3αψ21ψ∗1 = F/2 (2.22)where σ represents the distance of the driving frequency from the closest edge ofthe pass band, defined differently based on the type of nonlinearity:σ ≡{ω21 −Ω2 = σ1, softeningω21 +4kc−Ω2 = σ1+4kc, hardening(2.23)We take the solution of (2.22) to be ψ1 = A1 exp(iθ1). Using this solution andlooking for the steady-state response, we arrive at the following equations for A1and θ1:βA1 = (F/2)sinθ1 (2.24a)3αA31+σA1 = (F/2)cosθ1 (2.24b)We eliminate θ1 from (2.24) to get the response curve for ψ1:(βA1)2+(3αA31+σA1)2− (F/2)2 = 0 (2.25)which can be used to plot the approximate NLRM at a given Ω. Notice that (2.25)depends on Ω through the parameter σ , defined in (2.23).Equation (2.25) can be rearranged as a cubic equation in A21. For the NLRM tohave turning points, it is required that this cubic equation have three real roots. Theonset of supratransmission, where there is a turning point (saddle-node bifurcation)in the NLRM, corresponds to points where the cubic equation has three real rootswith two of them being equal [50]. To satisfy this condition, a certain relationshould hold between the coefficients of the cubic equation [1, Ch. 3]. This resultsin the following relation for the value of driving amplitude that corresponds to lossof stability:F2th,1 =−881α(σ(σ2+9β 2)± (σ2−3β 2)3/2)(2.26)The subscript 1 has been used to emphasize that one unit is used in the approx-40imation. Supratransmission is possible (i.e. the NLRM possesses a saddle-nodebifurcation) provided that the right-hand side of (2.26) is real and positive; other-wise, the NLRM does not have a turning point.It is worth emphasizing that (2.26) traces (within the approximation limits) thelocus of the first saddle-node bifurcation of the NLRM. Any other bifurcation ofthe NLRM will go unnoticed by this formulation.2.5.2 Comparison with Numerical ComputationsTo validate the analytical estimate of the supratransmission force threshold, wecompare the threshold curves predicted by (2.26) with numerical results obtainedusing continuation methods. Given that the analysis is performed assuming veryweak coupling between adjacent units, our immediate goal is to show that the re-sults are valid in the parameter range where they were derived. Accordingly, weuse a value of kc = 0.001 for coupling in this section. We will further explore (andupdate) the validity of the locally nonlinear analysis throughout Chapter 3 and inSection 4.4.Figure 2.12 shows the supratransmission threshold curves as a function of σ ,the distance from the pass band edge. The red solid curves are obtained numeri-cally from the governing equations (2.9), and threshold curves predicted by (2.26)are shown using the black dashed curves. Using σ allows us to compare the soften-ing and hardening systems on the same plot. We see for the structure with softeningnonlinearity that the analytical estimate predicts the threshold curve accurately, es-pecially close to the pass band edge (σ = 0) where the force threshold is lower.A similar observation is made for the structure with hardening nonlinearity. Com-pared to the softening system, the onset of transmission for the hardening system ispredicted less accurately. This discrepancy between analytical and computationalresults occurs due to the finite size of the structure, as anticipated during the deriva-tion3. Figure 2.12 also shows that as the forcing frequency moves farther into thestop band (as |σ | increases) the accuracy of the analysis decreases. This is becausethe force threshold increases with |σ |, making contributions from higher nonlinear3Recall that the analysis in Section 2.5.1 is based on an infinite periodic structure. Because weare dealing with a finite structure, one of the lower or upper edges of the pass band is inevitablypredicted inaccurately.41terms more significant.−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.410−310−210−1100Distance from the passband, σTransmission threshold, Fth  softening hardeningAnalytical predictionNumerical continuationFigure 2.12: The supratransmission threshold curves for a weakly coupledstructure as a function of σ , the distance form the pass band edge.σ < 0 for the structure with softening nonlinearity (I = 15 mA) andσ > 0 for the structure with hardening nonlinearity (I = −15 mA).The solid curves are obtained using numerical continuation and dottedcurves are obtained from (2.26) based on the local nonlinear dynamicsof the driven unit. The vertical line denotes the pass band edge, whereσ = 0 for both the hardening and softening structures.In summary, prediction of the onset of supratransmission based on the localnonlinear dynamics of the driven unit can be used to explain the supratransmissionphenomenon for very weak values of coupling. The results from this analysis aremore reliable from a qualitative perspective as opposed to a quantitative point ofview.2.6 Concluding RemarksIn this chapter, we studied the supratransmission phenomenon in a discrete non-linear periodic structure with damping and finite length. This essentially nonlinearphenomenon occurs when the periodic structure is harmonically forced at one endwith a forcing frequency lying within its linear stop band. Beyond a certain thresh-old of forcing amplitude, supratransmission occurs due to loss of stability of theperiodic solutions that are initially localized to the driven unit. At the onset ofsupratransmission, the response of the system moves from the basin of attractionof a limit cycle to that of a non-periodic attractor – this non-periodic attractor may42be either quasi-periodic or chaotic. Supratransmission is accompanied by a verylarge increase (orders of magnitude) in the energy transmitted through the periodicstructure. The frequency spectra of the nonlinearly transmitted waves lie inside thelinear pass band of the periodic structure.We reviewed the instability mechanism leading to supratransmission using di-rect numerical simulations and numerical continuation techniques. We showedthat the onset of supratransmission coincides with the first turning point (saddle-node bifurcation) of the nonlinear response manifold (NLRM) of the damped, finitestructure. This is the same mechanism that leads to supratransmission in infinite-dimensional Hamiltonian systems. We also highlighted that the onset of supra-transmission does not necessarily coincide with the resonance of the driving forcewith the shifted pass band at higher amplitudes of motion.Furthermore, based on the local nonlinear dynamics of the driven unit, we ob-tained closed-form analytical expressions for predicting the onset of supratrans-mission in weakly coupled periodic structures. This approximate formulation isbased on finding the saddle-node bifurcations of the NLRM. For both hardeningand softening types of nonlinearity, we verified the validity of the analytical esti-mates by comparing them with results obtained from numerical continuation for aweakly coupled periodic structure.43Chapter 3Parametric StudySupratransmission is a generic instability-driven transmission phenomenon in dis-crete nonlinear periodic structures. In this chapter, we explore how supratransmis-sion depends on various system parameters. The following parameters are consid-ered: type and strength of nonlinearity, number of units, damping and strength ofcoupling. We highlight the importance of forcing frequency by contrasting supra-transmission with the case of harmonic excitation within a pass band. We alsobriefly address the hysteresis phenomenon accompanying supratransmission, aswell as broadband energy propagation at very high driving amplitudes. The influ-ence of structural irregularities (disorder) is addressed separately in Chapter 4.3.1 Type and Strength of NonlinearityThe most significant role of the type of nonlinearity (softening or hardening) is todetermine on which side of the pass band supratransmission may occur. Supra-transmission may be observed below the pass band for a structure with softeningnonlinearity, and above it for a structure with hardening nonlinearity. The reasonis that the NLRM possesses a turning point (saddle-node bifurcation) only on oneside of the pass band1.We can use the formula (2.26) to determine the dependence of supratransmis-sion on the type of nonlinearity. To exclude the influence of damping, we consider1 We cannot rule out the possibility that the NLRM may lose stability through other bifurcationson the other side of the pass band. We are not aware of any work addressing this aspect.44the undamped structure (β = 0). For the undamped case, (2.26) reduced to thefollowing:Fth,1 =49√σ3−α (3.1)In order to get a positive real value on the right-hand side of (3.1), σ and α shouldhave opposite signs. In the case of softening nonlinearity, we have α < 0 andsupratransmission occurs when σ > 0; i.e. for Ω< ω1, below the linear pass band.Likewise, we have α > 0 for a hardening nonlinear force and supratransmissionmay occur when σ < 0, which is above the pass band. A similar observation hasbeen made based on the symmetries of the the wave envelope equations in [122].Regarding the strength of nonlinearity, we see from (3.1) that increasing |α|reduces the force threshold at the onset of supratransmission for both softening andhardening types of nonlinearity. The same observation can be made using (2.26)for the onset of supratransmission in damped structures.3.2 Number of UnitsThe question we are addressing in this section is whether the number of units (N)has a qualitative effect on the supratransmission phenomenon. To this end, wefocus on the undamped structure (ζ = 0)2. For an undamped structure, we alreadyknow from the literature [39, 82, 122] that supratransmission exists for infinitelylong periodic structures. We showed in Chapter 2 that supratransmission also existsin a very short periodic structure with N = 6. Here, we explore the influence of Non the threshold curves for small and moderate N, with N ∈ {5,10,15,20}. Weconsider both softening and hardening types of nonlinearity, with the nonlinearforce in (2.5) adjusted such that the coefficient of its cubic term is k3 =±0.2. Thedefault value of kc = 0.05 is used for the strength of coupling.Figure 3.1 shows the threshold curves as a function of σ for different values ofN. These threshold curves are obtained using numerical continuation, as explainedin Section 2.4. We see that for both types of nonlinearity, the influence of N is2In the presence of damping, energy dissipation will prevail nonlinearity (as well as disorder) forlarge N. No information thus reaches the end of the chain; i.e. EN from (2.18) is exponentially small.From that perspective, it would not matter whether supratransmission occurs for large N or not.45−0.02 −0.01 0 0.01 0.02 0.03 0.0410−510−410−310−210−1Distance from the pass band, σTransmission threshold, Fth  softening hardeningN = 20N = 15N = 10N =  5Figure 3.1: Threshold curves as a function of σ for different values of N.The periodic structure has free-free boundary conditions. σ < 0 forthe structure with softening nonlinearity (k3 =−0.2) and σ > 0 for thestructure with hardening nonlinearity (k3 = +0.2). For both structures,the transmission threshold Fth increases with N, but only at driving fre-quencies very close to a pass band edge. The hardening system is moresensitive to N than the softening system.significant only in the close vicinity of the edge of the pass band (near σ = 0). Asexpected, this difference decreases by increasing the number of units.Furthermore, we see in Figure 3.1 that the structure with hardening nonlinear-ity is more sensitive to N than the softening structure; in fact, the structure withsoftening nonlinearity is almost insensitive to N. We explain this based on theboundary conditions of the entire periodic structure. Recall from Section 2.1 thatthe periodic structure studied in this work is subject to free boundary conditions atboth ends (Figure 2.1). For the finite structure, this means that the lower edge ofthe pass band does not change with N. Thus, varying N has a minimal effect onthe dynamics of the softening periodic structure below the pass band. To supportour claim, we repeat the computations of Figure 3.1 for the same periodic struc-ture, but with fixed boundary conditions at both ends. We show the results of thesecomputations in Figure 3.2. We see that the dependence of threshold curves on Nis now similar for both types of nonlinearity. For both boundary conditions, theinfluence of N on threshold curves is significant only for very small values of |σ |.In summary, increasing the number of units can increase the force threshold atthe onset of supratransmission at driving frequencies very close to the edge of apass band.46−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.0410−510−410−310−210−1Distance from the pass band, σTransmission threshold, Fth  softening hardeningN = 20N = 15N = 10N =  5Figure 3.2: Threshold curves as a function of σ for different values of N. Theperiodic structure has fixed-fixed boundary conditions; c.f. Figure 3.1.The number of units plays a significant role only in the close vicinity ofthe edge of the pass band (near σ = 0).Note that Figure 3.2 is the only instance where we consider a fixed-fixed bound-ary condition throughout the thesis. In all other analyses, based on the mechanicalstructure of Figure 2.1, we use free-free boundary conditions. In the remainder ofthis chapter, in accordance with Chapter 2, we use a structure with six units.3.3 DampingAs we have already seen in Figure 2.10, the first turning point of the NLRM occursat a higher driving amplitude as the forcing frequency moves farther from the linearpass band. For an undamped structure, the force threshold goes to zero when theforcing frequency reaches the pass band. In other words, the threshold curve has avertical asymptote in the (F −Ω) plane, coinciding the the edge of the pass band(at σ = 0). Therefore the NLRM for an undamped structure has a turning point forall forcing frequencies in the stop band – this only happens on one side of the passband as already explained in Section 3.1.The topology of the NLRM depends on the amount of damping in the structure.Figure 3.3 shows the NLRM atΩ= 1.25 for I =−15 mA (hardening) and differentvalues of damping. We can see that the structure of the undamped NLRM is quali-tatively different from that of a damped NLRM. In the undamped case, the NLRMcrosses the zero-force axis (F = 0) after the first turning point. These crossingpoints correspond to discrete breathers, which are non-zero harmonic solutions in47the unforced undamped periodic structure that are spatially localized to the drivenunit3. The amplitude profiles of the two discrete breathers are shown in Figure 3.3in the insets. The damped NLRMs do not cross the zero-force axis because thedamped structure cannot sustain steady-state motion with non-zero amplitude. Thedamped NLRM will have a second turning point for non-zero values of dampingbelow a certain threshold – see also [81]. The existence of subsequent turningpoints or other bifurcations of the NLRM depends strongly on the nature of thenonlinear force among other parameters.−0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.310−1100101Forcing amplitude, FResponse amplitude, U1  ζ =    0ζ = 0.01ζ =  0.1ζ =  0.2Figure 3.3: The NLRM of the hardening system at Ω = 1.25 and differentvalues of damping are shown by the curves without markers. The resultsof direct numerical integration at each damping value are shown usingthe markers: red squares, ζ = 0; black circles, ζ = 0.01; blue diamonds,ζ = 0.1; magenta triangles, ζ = 0.2. Periodic responses are shown byempty markers and non-periodic responses are shown by filled markers.Increasing the damping ratio eventually eliminates the turning point ofthe NLRM. We also observe an intermediate stage (ζ = 0.1) where theonset of supratransmission is delayed until the third turning point of theNLRM. The intersections of the undamped NLRM with the zero-forceaxis, shown by red stars, correspond to discrete breathers. The insetsshow the amplitude profiles of these discrete breathers.3The NLRM of the undamped structure may cross the zero-force axis many times. We showonly two of these crossings in Figure 3.3. See [82] for more details about the zero-crossings and theassociated discrete breathers.48Comparing the NLRMs at non-zero damping values, we see in Figure 3.3 thatas damping increases the first turning point occurs at a higher driving amplitude.Moreover, the second turning point moves to the right (higher driving amplitudes)at a higher rate than the first turning point and the distance between the two turningpoints decreases. Eventually, there exists a threshold for the damping coefficient(ζ = 0.160 in this case) above which the two turning points merge and disappear.The NLRM above this threshold does not have a turning point.There is an intermediate stage during this transition, ζ = 0.1 in Figure 3.3,where the low-amplitude periodic solution jumps to a secondary (stable) periodicbranch before supratransmission occurs. In this case, in contrast to the behaviorof undamped structures, supratransmission does not occur at the first turning pointof the NLRM. Instead, it is delayed until subsequent turning points4. We concludethat damping changes the topology of the NLRM and may effectively delay orcompletely eliminate the onset of supratransmission.We can use (2.26) to study the influence of damping on the transmission thresh-old in more detail. In an undamped system (β = 0), the transmission threshold goesto zero at the edge of the linear pass band; i.e. Fth = 0 if σ = 0. In order to obtaina real value for F1 in the damped case (β 6= 0), it is required that σ2− 3β 2 > 0in (2.26). There is therefore a frequency range in the immediate vicinity of thepass band in which the NLRM does not have a turning point and supratransmissioncannot occur. The critical value σcr at which transmission starts can be obtained bysetting σ2−3β 2 = 0. Recalling the definition of σ from (2.22), this can be writtenin terms of the critical forcing frequency, denoted by Ωcr, as follows:Ωcr =√ω2u +3ζ 2+√3ζ ≈ ωu+√3ζ , α > 0√ω21 +3ζ 2−√3ζ ≈ ω1−√3ζ , α < 0(3.2)where the approximations are made for small damping (ζ 1). We see in (3.2) thatas damping increases, the onset of supratransmission occurs at a forcing frequencyfarther from the pass band. Accordingly, the value of forcing amplitude required at4 In Section 4.2, we further elaborate on this non-trivial effect of damping, where we also high-light the additional role played by disorder in this context.491.12 1.14 1.16 1.18 1.2 1.22 1.24 1.2610−410−310−210−1100Forcing frequency, ΩForce threshold, Fthstop band  Numerical: ζ = 0Numerical: ζ = 0.004Numerical: ζ = 0.010Analytical: ζ = 0Analytical: ζ = 0.004Analytical: ζ = 0.010Eq. (3.3)Figure 3.4: The transmission threshold curves for the structure with harden-ing nonlinearity at different values of damping. Solid curves are ob-tained using numerical continuation and dashed curves are from the an-alytical solution (2.26). Red: ζ = 0; black: ζ = 0.004; blue: ζ = 0.010.The magenta dash-dotted curve, obtained from (3.3), shows the locusof the critical values of F and Ω at the onset of supratransmission. Thevertical grey line shows the upper edge of the pass band (ω6 ≈ 1.139).Ωcr for the onset of supratransmission, denoted by Fcr, can be obtained from (2.26):Fcr =√3281 |α|(Ω2cr−ω2u )3 , α > 0√3281 |α|(ω21 −Ω2cr)3 , α < 0(3.3)This is the minimum force required in a damped periodic structure to trigger thesupratransmission phenomenon. This behavior is in contrast to that of undampedstructures where the minimum force amplitude required approaches zero as theforcing frequency approaches the pass band (notice that Fcr = 0 for ζ = 0).Figure 3.4 shows the threshold curves of the structure with hardening nonlin-earity for different values of damping. The solid curves are obtained using nu-merical continuation, and threshold curves predicted by (2.26) are shown by the50Figure 3.5: The dependence of supratransmission on damping ratio for hard-ening and softening types of nonlinearity. The solid curves, obtainedfrom (3.2), depict the boundaries between regions where supratrans-mission may exist and where supratransmission does not exist. Thegrey area corresponds to the linear pass band of the structure.dashed curves. We see that the threshold force goes to zero at the edge of the linearpass band in the undamped structure (the red solid curve). We also observe a dis-crepancy between numerical computations and analytical predictions (for example,compare the two red curves). This is because the analysis is based on an infiniteperiodic structure, in which the upper edge of the pass band is at ωu. The upperedge of the pass band in the finite system is slightly lower than ωu, as explained inSection 2.2.2. For nonzero values of damping, the minimum forcing amplitude re-quired for supratransmission to occur is nonzero and occurs at a forcing frequencyaway from the linear pass band. The locus of the critical forcing frequency andamplitude at the onset of transmission is given by (3.3), which is shown by thedash-dotted curve in Figure 3.4. This locus is the upper envelope of the thresh-old curves. All transmission threshold curves lie between the undamped thresholdcurve and the locus of the critical value; i.e. (3.1) and (3.3), respectively.Figure 3.5 shows the critical frequency at the onset of supratransmission, Ωcr,as a function of damping ratio, ζ , for both types of nonlinearity. The two solidcurves split the (Ω− ζ ) plane into areas where supratransmission may occur, di-vided by an area where supratransmission does not exist. The boundaries are ob-tained from (3.2). For very low values of damping, the prohibited region is mostly51comprised of the linear pass band. As damping increases, the critical forcing fre-quencies moves away from the edges of the pass bands. It is crucial to note thatderivation of (3.2) is based on the locus of the first turning point (saddle-node bi-furcation) of the nonlinear response manifold – recall the analysis in Section 2.5.1.We cannot definitively rule out the possibility that once the first turning point iseliminated by damping, the NLRM will remain stable for all values of forcing am-plitude. Nevertheless, we have not observed this in our numerical investigations,nor has such behavior been reported in the literature.In summary, damping (i) introduces a threshold on the minimum forcing am-plitude required for supratransmission to occur (ii) may delay the onset of supra-transmission until the third turning point of the NLRM, (iii) may eliminate a rangeof frequencies from the stop band in which supratransmission occurs. Althoughthe predicted critical values of forcing amplitude and frequency are only fairly ac-curate, we can still use (2.26) and (3.3) to qualitatively explain the influence ofdamping on the supratransmission phenomenon.3.4 Strength of CouplingThe strength of coupling, kc, is directly related to the width of the pass band, asshown in (2.15). Increasing the strength of coupling widens the linear pass bandof the structure, in particular by moving the upper edge of the pass band to higherfrequencies. Therefore, if the driving frequency is fixed above the pass band andkc is increased, the linear pass band eventually reaches the driving frequency andenergy transmission occurs through resonance with the linear modes of the struc-ture. This essentially linear phenomenon may occur in a nonlinear system as well,and is not further discussed here. Instead, we study the influence of the strength ofcoupling on the (nonlinear) supratransmission phenomenon.The analysis in Section 2.5.1 is based on reducing the dynamics of the periodicstructure to that of a single unit (the driven unit). That analysis is therefore unableto account for the strength of coupling. We begin this section by updating ourprediction of the onset of instability to account for the strength of coupling. Toimprove our reduced model, we truncate the periodic structure at the second unitinstead of the first one. In this new reduced model, the first unit is taken to be52the same as that of the original periodic structure and the second unit is assumedto be linear. This approximation is appropriate for small but non-zero values ofcoupling; note that (2.26) is only valid for kc = 0.For the reduced system with two degrees of freedom, we can proceed in thesame way as in Section 2.5.1 for the one-unit approximation to obtain the responsecurve. The equation describing the loci of turning points (saddle-node bifurca-tion) of the NLRM of the reduced system can then be obtained in the same waythat (2.26) was derived from (2.25). Thus, the value of force threshold that corre-sponds to loss of stability of periodic solutions is found to be:F2th,2 =−881α(S(S2+9T 2)± (S2−3T 2)3/2)(3.4)where the subscript 2 is used to emphasize that two units are used in the approxi-mation, and S and T are defined as below:S≡ (σ + kc)(1−R) (3.5a)T ≡ β (1+R) (3.5b)R denotes the ratio of the amplitudes of the two units:R≡(A2A1)2=k2cβ 2+(σ + kc)2(3.6)where A1 and A2 are the amplitudes of the driven unit (n = 1) and the linear unit(n = 2), respectively. As expected, we obtain Fth,2 = Fth,1 for kc = 0.Figure 3.6 shows the supratransmission threshold curves at three different strengthsof coupling as a function of σ , the distance from the pass band edge. Using σallows us to compare the softening system (I = 15 mA) and hardening system(I = −15 mA) on the same plot. The solid curves are obtained using numericalcontinuation, as described in Section 2.4. For both the structures with softeningand hardening nonlinearities, we see that increasing the strength of coupling doesnot have a significant influence on the critical frequency at which supratransmissionstarts. The minimum forcing amplitude for the onset of supratransmission, how-ever, increases with the strength of coupling. In fact, the supratransmission thresh-53−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.410−310−210−1100softening hardeningDistance from the passband, σTransmission threshold, Fth  numerical: kc = 0.001numerical: kc = 0.050numerical: kc = 0.100analytical: kc = 0.001analytical: kc = 0.050analytical: kc = 0.100Figure 3.6: The transmission threshold curves at different strengths of cou-pling as a function of σ . σ < 0 for the softening system and σ > 0 forthe hardening system. Three coupling coefficients are used in each case;red: kc = 0.001, black: kc = 0.050, green: kc = 0.100. Solid curves areobtained using numerical continuation and dashed curves are from theanalytical solution (3.4). The vertical line denotes the pass band edge.old is increased at all forcing frequencies if the coupling becomes stronger. This istrue even in an undamped structure for which the threshold curve approaches zeroat the edge of the pass band [82].The dashed curves in Figure 3.6 show predictions of the threshold curves basedon (3.4). We see that the reduced model can capture the behavior of the thresholdcurves only at very weak coupling, κ = 0.001. For higher strengths of coupling,the reduced model can still capture the qualitative behavior of the softening sys-tem, but not that of the hardening system. Specifically, it fails in predicting thecritical forcing frequency and amplitude at the onset of supratransmission in thestructure with hardening nonlinearity. Two reasons contribute to this: (a) reducingthe periodic structure to two units only works in the limit of weak coupling; (b) theupper linear natural frequency of the reduced model (two degrees of freedom) islower than that of the full model (six degrees of freedom). While the former is anintrinsic limitation of the reduced model, the latter can be improved by offsettingthe threshold curves with respect to their corresponding edge of the pass band (notshown here).54In summary, increasing the strength of coupling (i) increases the minimumforce amplitude required for the onset of supratransmission; (ii) does not changethe frequency location of the onset of supratransmission significantly with respectto the nearest edge of the pass band.3.5 Forcing Amplitude: HysteresisAssociated with a saddle-node bifurcation, one normally expects to observe thehysteresis phenomenon. This results from the co-existence of two (or more) stablesolutions near the bifurcation point. As expected, hysteresis can also be observednear the supratransmission threshold.In order to observe the hysteresis effect, a different numerical approach is re-quired than the one used in Section 2.3. The approach we use here is based onsweeping the driving amplitude back and forth near the supratransmission thresh-old; this method can also be used in experiments. We start from a small forcingamplitude below the threshold, F = 0.09 in Figure 3.7. Once the steady state isreached, the forcing amplitude is smoothly increased in small increments, and anew steady state is obtained at each step (black circle markers). Once we passthe supratransmission threshold, we reverse this procedure and decrease the forc-ing amplitude stepwise (blue square markers). Figure 3.7 shows the results ofthis computation for the structure with hardening nonlinearity (I = −15 mA) atΩ= 1.25. We can clearly observe the hysteresis phenomenon.The hysteresis associated with supratransmission has been previously studiedboth theoretically and experimentally; e.g. see [51, 62, 66]. Further discussion ofthe hysteresis effect, such as the influence of sweeping parameters on the observedwidth of the hysteresis gap, is not within the scope of the present work.550.09 0.095 0.1 0.105 0.11 0.115 0.1210−810−510−2101Forcing amplitude, FTransmitted energy, EN  Sweeping upSweeping downFigure 3.7: The hysteresis effect at Ω = 1.25 for the structure with harden-ing nonlinearity. Black circle markers correspond to up-sweep of F andblue square markers correspond to sweeping F down. Empty mark-ers denote periodic solutions and filled markers denote non-periodic re-sponse.3.6 Forcing FrequencyThe driving frequency plays a crucial role in determining the energy transmissioncharacteristics of a nonlinear periodic structure. To highlight this, we comparesupratransmission (driving within a stop band) to the case in which a periodicstructure is driven within its pass band. When the driving frequency lies withinthe pass band, energy transmission occurs in a linear fashion at low driving am-plitudes, with a small decay occurring due to dissipative forces – see Figure 2.3.These linear solutions eventually lose their stability if the driving amplitude is in-creased. This loss of stability is not usually accompanied by a large increase in thetransmitted energy, in strong contrast with what happens for stop band excitation.We show this for a periodic structure with the following parameters: kc = 0.05,ζ = 0.005, k3 =+0.2 (hardening) and N = 10. Figure 3.8 shows the energy trans-mitted through this strcture as a function of F at different forcing frequencies nearthe upper edge of the pass band (ωN ≈ 1.149).In all forcing frequencies shown in Figure 3.8, loss of stability is accompaniedwith an increase in the transmitted energy. At Ω = 1.16, excitation is within thestop band and we see an increase in the transmitted energy over a few orders ofmagnitude (supratransmission). In comparison, the increase in transmitted energies56is not as significant for the driving frequencies within the pass band, namely atΩ= 1.13 and Ω= 1.14. At Ω= 1.15, which is on the edge of the pass band, lossof stability leads to enhanced transmission, but not as significant as what happens athigher forcing frequencies within the stop band. Within the pass band, the locationofΩwith respect to the linear natural frequencies changes the onset of transmission(Fth), yet the qualitative behavior explained here remains the same; i.e. eventualloss of stability and an increase in EN . If the structure were undamped, then theonset of transmission would have approached zero at the linear natural frequenciesof the structure.0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110−510−410−310−210−1100101102Forcing amplitude, FTransmitted energy, EN  Ω = 1.13Ω = 1.14Ω = 1.15Ω = 1.16Figure 3.8: The influence of forcing frequency (Ω) on enhanced nonlinearenergy transmission in a damped periodic structure. Energy at the endof the structure (EN) is plotted as a function of driving amplitude (F)for different forcing frequencies. The upper edge of the pass band isat ωN ≈ 1.149. Ω = 1.13 and Ω = 1.14 are inside the pass band, Ω =1.15 is at the edge of the pass band, and Ω = 1.16 is above the passband. Filled markers indicate periodic responses and empty markersindicate non-periodic responses. In all four cases, loss of stability leadsto an increase in EN . This increase is most significant when excitationis within the stop band (supratransmission).Although loss of stability leads to enhanced transmission within both pass andstop bands, the increase in transmitted energies is much smaller within the passband. One explanation is that the response is already extended throughout thestructure within the pass band, thus waves can reach the end of the structure with-out much attenuation – compare the transmitted energies in Figure 3.8 at low forc-ing amplitudes. Another factor could be the instability mechanism. Linear solu-57tions lose their stability through saddle-node bifurcation at Ω = 1.16 (and drivingfrequencies above it), whereas loss of stability occurs through a Neimark-Sackerbifurcation [70, Ch. 5] at Ω= 1.15. The latter is also the typical instability mech-anism inside the pass band. We have found damping to play a major role in de-termining the behavior of the Floquet multipliers and consequently the mechanismleading to loss of stability within and in the vicinity of a pass band. A detailed anal-ysis of the influence of damping on the bifurcation structure of the NLRM withina pass band is very interesting from a nonlinear-dynamics perspective but lies out-side the framework of this thesis. This is also not a pressing study from a practicalpoint of view because supratransmission is most significant when the driving fre-quency is away from the pass band. The additional role played by disorder in thiscontext will be discussed in Section 4.2.3.7 Strong Propagation RegimeThe nonlinear transmission phenomenon discussed in the preceding sections of thiswork occurs above a certain driving amplitude threshold. If the driving amplitudeis increased further, a second threshold may exist in the case of undamped struc-tures (possibly for driving frequencies on both sides of the pass band) above whichthere is another large increase in the transmitted energy through the chain [82]. Inthis context, the operating range between the first and second thresholds is referredto as the weak propagation regime, and the range above the second threshold isknown as the strong propagation regime. The strong energy propagation is due tolarge-amplitude chaotic motions of the structure, and the transmitted wave is char-acterized by a broad frequency spectrum extending beyond the linear pass band.Also, the increase in the transmitted energy at the second threshold is much largerthan the increase at the first threshold.To the best of our knowledge, the strong propagation regime has only been re-ported in a semi-infinite Hamiltonian system with an on-site Morse potential [82].Further investigation of the strong propagation regime and the possible influenceof damping on the existence of the second threshold is beyond the scope of thepresent work.583.8 Concluding RemarksWe studied the influence of various system parameters on the supratransmissionphenomenon for a perfectly periodic structure. Our main findings are summarizedhere:1. The type of nonlinearity determines on which side of the pass band supra-transmission can occur. This is above the pass band for a hardening systemand below it for a softening system. Increasing the strength of the nonlinearforce lowers the force threshold for the onset of supratransmission.2. Increasing the number of units can shift the threshold curves towards higherforcing thresholds at driving frequencies very close to the edge of a passband.3. In general, damping increases the force required for the onset of supratrans-mission at all forcing frequencies (away from the pass band). More signif-icantly, damping may eliminate supratransmission within a frequency rangein the immediate vicinity of the linear pass band, introducing a non-zerothreshold on the minimum force amplitude required for transmission to oc-cur. Increasing damping widens this frequency range and increases the mini-mum force. Furthermore, damping may delay the onset of supratransmissionby stabilizing the periodic solutions. Supratransmission may occur in thiscase at higher forcing amplitudes.4. Increasing the strength of coupling increases the minimum force required forthe onset of supratransmission for all driving frequencies, but does not alterthe location of the onset of transmission with respect to the nearest edge ofthe pass band.Using the analytical estimates of the supratransmission threshold from Sec-tion 2.5.1, we explained the effects of nonlinearity and damping from a phenomeno-logical viewpoint. Moreover, we updated the existing analysis in order to includethe influence of the strength of coupling on the onset of supratransmission. Theimproved analysis is based on the local nonlinear dynamics of the driven unit, withthe second unit included in the analysis but assumed to be linear (the rest of the59structure is truncated after the second unit). The updated approximate formulationcan account for the strength of coupling, though it is still limited to weak couplingstrengths.We also briefly discussed the hysteresis phenomenon associated with supra-transmission, and explained one methodology for observing it numerically. Fur-thermore, we briefly considered the instability-driven increase in energy transmis-sion when the driving frequency is within the pass band (not supratransmission).We showed that the increase in transmitted energies in this situation is not signifi-cant when compared to the energy increase via supratransmission.60Chapter 4Disorder EffectsWe explore the influence of linear disorder on supratransmission in this chapter. Westart by an overview of the key features of the linear response of disordered struc-tures (Anderson localization), and highlight the combined influence of dampingand disorder on supratransmission near a pass band. We then discuss the statisticaleffects of disorder on the transmission mechanism. In particular, we investigatethe statistical influence of linear disorder on (i) the supratransmission thresholdswithin a stop band, (ii) transmitted energies above the supratransmission threshold,and (iii) the spectrum of the nonlinearly transmitted waves. We then present anapproximate analytical formulation for predicting the onset of supratransmissionin disordered structures and investigate its range of validity.4.1 Disorder-Borne Energy Localization in LinearStructuresWave propagation within a linear periodic structure may be significantly influencedby the presence of small disorder. In a disordered structure, waves scatter dif-ferently at the boundaries between adjacent units because the units are no longeridentical. This can result in spatial decay of the response amplitude at frequencieswithin (and particularly near the edges of) the pass band of the underlying orderedstructure. If we average the response over many different realizations of a pre-scribed disorder, then the response becomes localized to the source of excitation61and decays exponentially away from it. This statistical phenomenon is known asAnderson localization, after the seminal work of Philip W. Anderson in solid-statephysics [3] – see also [47, 71, 86].Disorder in engineering structures usually means small variations in spatiallydistributed (extended) structural parameters such as stiffness, mass, damping, orsupport conditions. These small irregularities can lead to significant qualitativechanges in the global dynamic response. This spatial confinement of energy, orlocalization, is called strong localization [108], and occurs when the strength ofcoupling between adjacent units is weak in comparison to the strength of disorder.In weak localization, the coupling force is strong and damping effects dominateover disorder [12, 108]. Even in undamped structures, weak localization effectsare only significant over very long distances (at least a few hundred units [108]).Given that engineering structures always have damping and do not typically consistof such large number of units, only strong localization is relevant in the majority ofengineering applications. Accordingly, we study a damped finite periodic structurewith weak coupling – we define the strength of coupling in (4.2). We consider alightly damped structure and use a linear viscous damping model with a mass-proportional damping matrix. Refer to [106, 107] for a comprehensive reviewof general non-proportional and non-viscous damping models. An overview ofsimultaneous effects of damping and disorder in linear periodic structures can befound in [12]. See [13] for an extension of this work to structures with finite length.It is important to note that disorder-borne confinement of energy only occurs inan ensemble-average sense and that individual realizations of disorder may behavedifferently. In particular, it is possible that the normal modes of a disordered struc-ture are localized away from the driving point for a particular realization. Theseanomalous realizations can have a significant influence on the average response ofthe ensemble, to the extent that using a linear average may not necessarily givethe typical value (statistical mode) of the ensemble [12, 48]. This is important inthe case of weak localization [13], where localization length scales are large. Fora damped structure, the contributions from anomalous realizations are much lesssignificant because of the uniform decay caused by damping [12, 48].Anderson localization can occur as a result of disorder that is present in eitherthe grounding springs (on-site potential), coupling springs (inter-site potential) or62masses within the periodic structure [65]. It is argued [108] that random massesand grounding springs influence the degree of localization in a similar manner,while random coupling springs have a weaker influence in comparison. Here, weonly consider disorder that is applied to the linear grounding springs of the system.This type of disorder is sufficient to capture strong localization in our system. Inpresence of such disorder, the linear governing equations of (2.9) are replaced withu¨n+2ζ u˙n+ω21 (1+δkn)un+kc∆2(un) = fn cos(Ωt) , fn = 0 for 2≤ n≤N (4.1)where δkn are random numbers with a uniform probability density function. Weassume that δkn are distributed independently around a zero mean, < δkn >= 0,such that |δkn| ≤D. We refer to D as the strength of disorder. We further introducea parameter C that denotes the strength of coupling between unitsC ≡ kc/ω21 (4.2)Our earlier assumption of weak coupling can be expressed as C < 1. The valueof C represents the ratio of coupling to grounding spring stiffness, and does notdepend on the mass (coefficient of u¨n). With the exception of Section 4.4.3, we useC = 0.05 in this chapter, which is the same strength of coupling used in Chapters 2and 3. We use a periodic structure with N = 10 units in this chapter.4.1.1 Localization Occurring within a Pass BandA crucial parameter that determines the degree of localization is D/C, the ratio ofthe strengths of disorder to coupling [47]. As this ratio increases, the degree oflocalization within the pass band increases as well. This is shown in Figure 4.1where we plot the normalized amplitude profile (Un/U1) for different strengthsof disorder. The results for the disordered structures are obtained after averagingover an ensemble of 105 realizations to ensure convergence of the average valuesat all forcing frequencies considered. We see in Figure 4.1 that the response ofthe ordered structure (D/C = 0) is extended through the system, and the smalloverall attenuation in response amplitudes is due to damping. The oscillations inthe response of the ordered structure near the end of the structure are caused by the631 2 3 4 5 6 7 8 9 1010−210−1100nAmplitude profile, | Un / U1 |  D / C = 0D / C = 1D / C = 2Figure 4.1: The influence of disorder on response localization at Ω = 1.12,near the middle of the pass band. The average response becomes lo-calized to the driven unit (n = 1) as the value of D/C increases. Thestrength of coupling is kept constant, C = 0.05. An ensemble of 105realizations are used for each non-zero value of D (namely D = 0.05and D = 0.10). Other system parameters are N = 10, ζ = 0.005 andω21 = 1.05.waves reflected at the free boundary at the end of the structure (n = 10). Addingdisorder results in smaller response amplitude in comparison with the ordered case.As the strength of disorder increases, the response becomes localized to n = 1,where the external force is applied. In addition, disorder effects eventually subduewave reflections at the boundaries, which is why the boundary effects at n = 10become insignificant for high values of D/C.4.1.2 Quantifying the Degree of LocalizationWhen there is exponential spatial decay of the response due to disorder, we expect|Un| ∝ exp(−γnn) (4.3)in an average sense. γn is the localization factor or decay exponent and describesthe average rate of exponential amplitude decay per unit. As the length of theperiodic structure extends to infinity (equivalently, when averaged over many re-alizations), expression (4.3) yields the correct decay rate [49]. Calculation of thedecay exponent merely based on (4.3) involves solving for all response amplitudes641 2 3 4 5 6 7 8 9 1010−1100101Unit number, nAmplitude, | Un / F |(a)1 2 3 4 5 6 7 8 9 1010−1100101Unit number, nAmplitude, | Un / F |(b)  Amplitude profileγn , Eq (4.3)γN , Eq (4.4)γ , Eq (4.5)Figure 4.2: Comparison of the three decay exponents in describing amplitudeprofiles; (a) an extended response, (b) a localized response. The redcircles denote the actual response at each unit, red dash-dotted lines areobtained based on γn (curve fit), black dashed lines are based on γN andblue solid lines are based on γ .and fitting an exponential curve to the data.There exist different approaches for studying the decay exponent analytically,such as using transfer-matrix [108] or receptance-matrix [91] formulations. In theformer case, one could make use of the properties for products of random matri-ces [36] and obtain asymptotic estimates for decay exponents [12]. Alternatively,one could replace (4.3) with|UN | ≡ F exp(−γNN) (4.4)and use the properties of tridiaognal matrices to obtain expressions for γN [46, 49].It is important to note that (4.4) is true for any finite linear system and does notmean that the response is exponentially decaying. Nevertheless, if the response isexponentially decaying (as anticipated in Anderson localization), then the value ofγN obtained from averaging (4.4) represents the average decay rate in the limit of avery long structure (as N→ ∞).Based on the value of γN alone one cannot draw any conclusions about theresponse of the periodic structure. In addition, because we are dealing with rela-65tively short periodic structures in this work (N = 10), the boundary effects will havean influence on γN . These make the interpretation of the decay factor, as definedin (4.4), somewhat ambiguous. An alternative is to redefine the decay exponent byreplacing |UN | in (4.4) with |UN/U1|; i.e. normalizing the response at the end ofthe chain with that of the driven unit. Then (4.4) is replaced by∣∣∣∣UNU1∣∣∣∣≡ e−γ(N−1) (4.5)Notice that the multiplier for the decay exponent is now (N− 1) because a wavegoes through N−1 units from n= 1 to n= N. Based on definition (4.5), we obtainγ > 0 whenever the response has decayed through the structure; i.e. |UN |< |U1|. Anegative value of the decay exponents is obtained whenever |UN | > |U1|; we haveobserved this only at resonance frequencies for structures with few number of unitsand very light damping. We emphasize that definition (4.5) does not imply that thespatial decay envelop is exponential.We use two different amplitude profiles to compare the performance of thethree decay exponents γn, γN and γ in describing amplitude profiles of finite struc-tures. These two amplitude profiles are the (ensemble-) average response of thelinear structure considered in Section 4.1.1 at Ω= 1.12 for D/C = 0 (extended re-sponse) and D/C = 2 (localized response). We reproduce these amplitude profilesin Figure 4.2, along with the estimates obtained based on γn, γN and γ . We see thatthe proposed definition of the decay exponent (γ) describes the actual amplitudeprofile better than the classical definition (γN). The advantage of using γ over adecay exponent based on curve fitting (i.e. γn) is that the former can be used forobtaining analytical estimates of the response. Since we do not make use of theseanalytical expressions in this work, we do not further elaborate on them.Figure 4.3 shows the influence of the number of units on the decay exponent γin an ordered structure. We see that as N increases, the results for a finite systemapproach those for the infinite system based on γ0 defined in (2.11). It is possibleto show analytically for an ordered structure that in fact γ → γ0 as N→ ∞. Basedon the foregoing discussions in this section, we use the decay exponent definedin (4.5) in this work.Another important characteristic of disorder-borne localization is that the mode66Figure 4.3: The dependence of the decay exponent on the number of units,N, for a linear ordered structure. As the size of the structure increases,the decay exponent of the finite structure approaches that of the infinitestructure; i.e. γ → γ0 as N→ ∞. The grey area corresponds to the stopband.shapes of a disordered structure become spatially localized [47, 68]. This can bequantified using the inverse participation ratio (IPR), defined byIPR =∑Nn=1(U2n )2(∑Nn=1U2n)2 (4.6)where Un are the steady-state amplitudes of the n-th mode shape of the system;e.g. see [68] for more details. The IPR is a scalar with a value between 1/N and1. If all the units are moving with the same amplitude (uniform response), thenIPR = 1/N. If only one unit is moving (absolute localization), then IPR = 1.For the same ensemble used for Figure 4.1, we have computed the average IPRfor the first and last mode shapes. We show this in Figure 4.4, along with a typicalmode shape of the ensemble at three values of D/C. We can see that the modeshapes become spatially localized as disorder becomes stronger. One can also usea modal expansion to express the response of the forced structure in terms of itsmode shapes; in this light, the response localization we observed in Figure 4.1 canbe explained based on spatial localization of the mode shapes [49].670 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.10.150.20.250.30.350.40.450.5Average inverse participation ratio , IPRStrength of disorder, D / C  n = 1 (first mode)n = N (last mode)Figure 4.4: The influence of disorder on spatial localization of the modeshapes of the structure. IPR, defined in (4.6), is plotted as a function ofD/C for the first and last mode shapes. The insets show the mode shapeof a typical realization of disorder within the ensemble at D/C = 0,1,2.The result for n = 1 are shown in black and those for n = N are shownin grey.4.1.3 Influence of Disorder on the Band StructureThe frequency of excitation plays a major role in determining the influence ofdisorder on the dynamic response of periodic structures. On average, less energyis transmitted to the end of a disordered structure at frequencies lying within thepass band – recall Figure 4.1. Figure 4.5(a) shows the average decay exponent γas a function of driving frequency Ω for different values of disorder. We can seethat disorder results in increased values of decay exponent within the pass bandof the ordered structure. Within the stop band, however, the decay exponent hasa larger value for the ordered structure. Similar observations have been made ininfinite undamped structures [35]. In addition, Figure 4.5(a) shows that disorder68Figure 4.5: The influence of disorder on the linear response of the structure.(a) The average decay exponent γ , defined in (4.5), is plotted for differ-ent values of disorder. The grey area corresponds to the stop band. (b)The average natural frequencies of the first (ω1) and last (ωN) modesare plotted as a function of D/C. The black circles correspond to ω1and grey squares to ωN . The empty markers correspond to the mini-mum and maximum values of each natural frequency within the ensem-ble. The horizontal lines indicate the natural frequencies of the orderedstructure.has the overall effect of slightly widening the pass band of a periodic structure.As a result, the transmitted energy is much smaller in a disordered structure butcovers a wider frequency range in comparison to an ordered structure. The sameconclusion can be made based on Figure 4.5(b), where we show the first and lastnatural frequencies of the structure as a function of the strength of disorder. Wecan see that as D/C increases, ω1 decreases and ωN increases on average.4.2 Combined Effects of Damping and Disorder Near aPass BandWe explained in Section 3.3 that, in general, the force threshold at the onset ofsupratransmission (Fth) increases if the value of damping is increased. Close tothe edge of a pass band, however, damping can play a more significant role. Toillustrate this, we compare the transmitted energies in a damped (ζ = 0.005) and anundamped (ζ = 0) ordered structure with N = 10. Figure 4.6 shows the thresholdcurve for the damped structure studied in this chapter in the absence of disorder.69Given the low value of damping, we expect Fth to be very similar for the dampedand undamped structures considered here. Thus, Figure 4.6 gives a good estimatefor values of Fth in the undamped structure.1.15 1.2 1.25 1.3 1.35 1.410−210−1100Forcing frequency, ΩThreshold force, Fthstop bandFigure 4.6: The threshold curve for the ordered structure (D = 0), showingthe dependence of the driving amplitude at the onset of transmission,Fth, as a function of the driving frequency, Ω. Other system parametersinclude N = 10, ζ = 0.005, C = 0.05 and k3 = 0.2. The upper edgeof the pass band is at ωN ≈ 1.149. As explained in Section 3.3, thethreshold curve terminates near the pass band in a cusp.Figure 4.7 shows EN as a function of F at different values of Ω close to thepass band – recall that the upper edge of the pass band is located at ωN ≈ 1.149. InFigure 4.7, we see that atΩ= 1.16 the onset of supratransmission occurs near Fth≈0.05 for both the damped and undamped structures. The transmitted energies abovethe threshold are lower for the damped structure, which can be easily attributed toenergy loss through damping. At Ω = 1.18, we see that supratransmission occursat the expected value of Fth ≈ 0.08 for the undamped structure. In contrast, thedamped structure has a relatively insignificant increase in EN between F = 0.075and F = 0.100 (indicated as weak jump in Figure 4.7). The response of the dampedstructure is periodic below and above F = 0.08, though with different amplitudes.Above F = 0.350, supratransmission occurs in the damped structure as well. Asimilar difference between the damped and undamped structures is observed atΩ=1.22. At Ω = 1.26 (and driving frequencies above it), supratransmission occursaround the same forcing thresholds for the damped and undamped structures, asinitially expected.To understand the unexpected behavior of the damped structure when 1.18 .700 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410−1510−1210−910−610−3100103Forcing amplitude, FTransmitted energy, EN  weak jumpweak jumpΩ=1.16 , ζ=0.005Ω=1.18 , ζ=0.005Ω=1.22 , ζ=0.005Ω=1.26 , ζ=0.005Ω=1.16 , ζ=0.000Ω=1.18 , ζ=0.000Ω=1.22 , ζ=0.000Ω=1.26 , ζ=0.000Figure 4.7: The influence of damping on supratransmission in the vicinity ofa pass band for an ordered structure. Transmitted energy (EN) is plot-ted as a function of driving amplitude (F) for different driving frequen-cies (Ω). The upper edge of the pass band is at ωN ≈ 1.149. Filledmarkers indicate periodic response and empty markers indicate non-periodic response. Because damping is small, the threshold force (Fth)of the undamped structure is expected to be very close to that of thedamped structure. At Ω= 1.16 and Ω= 1.26, supratransmission occursat the expected force threshold based on Figure 4.6. At Ω = 1.18 andΩ= 1.22, the response of the damped structure jumps to another stableperiodic branch at the expected value of F (indicated as ‘weak jump’).In contrast, supratransmission occurs for the undamped structure at theexpected value of F .Ω . 1.26, we consider the evolution of the periodic solutions of the damped andundamped structures at different values of Ω as a function of F . This informationcan be obtained from the nonlinear response manifold (NLRM), as discussed inSection 2.4. Figure 4.8 shows the projection of the NLRMs on the UN −F planefor the same four values of Ω that are used in Figure 4.7.At Ω = 1.16, shown in Figure 4.8(a), we see that the two NLRMs start fromthe origin and follow the linear solution (i.e. UN ∝ F) for small values of F . Atthe first turning point (TP1), the periodic solutions lose their stability through asaddle-node bifurcation. Because neighboring periodic solutions either do not ex-71−0.02 0 0.02 0.04 0.06 0.08 0.100.0050.010.0150.02Forcing amplitude, FUN  ζ=0.005ζ=    0−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40123456x 10−5Forcing amplitude, FUN  ζ=0.005ζ=    0−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.402468x 10−7Forcing amplitude, FUN  ζ=0.005ζ=    0−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4012345x 10−8Forcing amplitude, FUN  Ω = 1.16(a)Ω = 1.18(b)Ω = 1.22(c)Ω = 1.26(d)ζ=0.005ζ=    0Figure 4.8: The influence of damping on the NLRM of the structure at dif-ferent forcing frequencies: (a) Ω = 1.16, (b) Ω = 1.18, (c) Ω = 1.22,(d) Ω= 1.26. In each plot, the projection of the NLRM is plotted in theUN −F plane for ζ = 0.005 (black curve) and ζ = 0 (grey curve). Foreach NLRM, the thick solid sections represent stable solutions and thindash-dotted portions represent unstable solutions.ist or are unstable at this point, the solution jumps to a non-periodic branch. Thisis accompanied by the large increase in transmitted energies shown in Figure 4.7.After TP1, the undamped response normally crosses the zero-force axis (F = 0)multiple times. These zero-crossings correspond to non-zero time-periodic solu-tions in the undamped system that are spatially localized to the driven unit – thesesolutions are called discrete breathers (DB). See [82] for more details about thezero-crossings and the associated DBs. For a damped structure, it is important tonote that the NLRM does not cross the zero-force axis because the structure can nolonger sustain steady-state motion with non-zero amplitude.At Ω = 1.18, shown in Figure 4.8(b), we see that another stable periodic so-72lution exists at TP1 for the damped structure. As a result, the solution jumps tothat solution branch when F is increased beyond its value at TP1. The response atthis upper periodic branch is dominantly harmonic (with frequency Ω) and has ahigher value of EN than the linear solution – see Figure 4.7. Nevertheless, supra-transmission does not occur until the third turning point (TP3) of the NLRM aroundF ≈ 0.375. For the undamped structure, other solutions at TP1 are unstable andthe solution jumps to a non-periodic branch.At Ω= 1.22, shown in Figure 4.8(c), the situation is similar to what happens atΩ = 1.18. The main difference is the range over which the upper periodic branchis stable: compared to Ω = 1.18, TP3 occurs at a lower value of F . Although theundamped NLRM has stable portions, we have not observed any situation in whichthe stable branch extends to TP1 – the same observation is made in [82]. Thus,supratransmission occurs at TP1 for the undamped structure. By Ω= 1.26, shownin Figure 4.8(d), the damped NLRM has changed such that TP3 occurs again at alower value of F than TP1. Accordingly, supratransmission occurs at TP1 for boththe damped and undamped structures.Figure 4.8 clearly shows that the stability of the upper branch of NLRM atthe TP1 depends on the existence of damping. We have found that disorder canalso play a role here. We show this in Figure 4.9 for two different realizationsof disorder at Ω = 1.22. We see in Figure 4.9(a) that increasing the strength ofdisorder has a stabilizing effect on the upper branch of periodic solutions. Weobserve the opposite effect for the other realization in Figure 4.9(b).Investigating the necessary/sufficient conditions for the stability of the uppersolution branch at the first turning point of the NLRM sets forth a very interestingproblem. Knowing the location of TP3 or subsequent turning points would notsuffice for this purpose because the upper branch can change stability betweenturning points (via Neimark-Sacker bifurcation), as seen in Figures 4.8(b-d). Asystematic investigation of the influence of damping and disorder on the stabilityof the upper branch at TP1 can be done using numerical continuation. This study,however, falls outside the scope of our present work – it is also of less relevance inapplications based on supratransmission because supratransmission is significant atfrequencies away from a pass band. Consequently, we will only consider drivingfrequencies for which supratransmission occurs via jump to an anharmonic branch730 0.1 0.2 0.3 0.400.10.20.30.40.50.60.70.80.911.1Forcing amplitude, FResponse amplitude, U1  D / C = 0D / C = 1D / C = 20 0.1 0.2 0.3 0.400.10.20.30.40.50.60.70.80.911.1Forcing amplitude, FResponse amplitude, U1  (a) (b)D / C = 0D / C = 1D / C = 2Figure 4.9: The influence of disorder on the NLRM of the damped structureat Ω = 1.22. Two different realizations of disorder are considered inpanels (a) and (b), with increasing the strengths of disorder (D/C). Theresults for D/C = 0 are reproduced from Figure 4.8(c). Notice thatincreasing D/C has opposing effects on the stability of the upper branchof periodic solutions in (a) and (b).at the first turning point of the nonlinear response manifold [57, 82]. In this light,we will consider Ω≥ 1.25 in Section 4.3.4.3 Supratransmission in Disordered Nonlinear PeriodicStructuresWe investigate the influence of disorder on supratransmission in this section. Withthe exception of Section 4.3.1, the results presented here are pertinent in an ensemble-average sense, meaning that they describe the behavior of a typical disorderedstructure; i.e. the statistical mode of the ensemble.Similar to Section 4.1.1, we consider disorder as random linear spring con-stants taken from a uniform distribution, with −D ≤ δkn ≤ D. We consider a dis-order range of 0≤ D/C ≤ 2. This range of deviation from periodicity captures theexpected physical effects of disorder, as explained in Section 4.1. This providessufficient justification for stopping at D/C = 2. Keeping C = 0.05 constant, we74vary the strength of disorder by changing D. Thus, we are studying a disorderedperiodic structure with weak coupling and weak disorder (0≤ D≤ 10%). Strictlyspeaking, a disorder value of 50% (for example) would not violate any physicallaw, but at that point we would be getting closer to a random structure as opposedto a disordered periodic structure.Throughout this section, the nonlinear force in (2.5) is adjusted such that thecoefficient of its cubic term is k3 = +0.2 (hardening nonlinearity). Accordingly,we only consider forcing frequencies above the pass band because supratransmis-sion occurs above the pass band in a structure with hardening nonlinearity (recallSection 3.1). The upper edge of the pass band is located at ωN ≈ 1.149 in thiscase. As explained in Section 4.2, we only consider Ω ≥ 1.25 to focus on supra-transmission occurring at the first turning point of the nonlinear response manifold.Qualitatively similar results are expected if a softening nonlinearity is chosen andforcing frequencies are below the pass band. We will consider both hardening andsoftening types of nonlinearity in Section 4.4.2.4.3.1 Loss of Stability Leads to Enhanced TransmissionFor a given forcing frequency, the onset of transmission in a disordered structuremay occur at a different value of the driving amplitude when compared with thecorresponding ordered structure. The change in threshold force amplitude, andwhether it occurs or not, depends on the particular realization that is being consid-ered. We have shown this in Figure 4.10 for two different realizations of disorderwith D/C = 2.Figure 4.10(a) shows the evolution of the periodic solutions as a function ofthe driving amplitude, F . For low driving amplitudes (near the origin), the re-sponse amplitude increases linearly with F , as expected from linear theory. As Fincreases, there is a turning point in the response (saddle-node bifurcation). Thesolution then jumps to a non-periodic branch. Depending on the realization beingconsidered, the onset of transmission may be lower or higher than the onset for theordered structure. Results from direct numerical integration of (2.3) are shown inFigure 4.10(b). We can see that there is a significant decrease in the decay exponentthat occurs at a value of F consistent with the turning points in Figure 4.10(a).750 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450123456x 10−9Response amplitude, UN  (a)D/C = 0D/C = 2D/C = 20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4500.751.52.25Forcing amplitude, FDecay exponent, γ(b)Figure 4.10: The influence of disorder on the onset of supratransmission atΩ = 1.30. (a) the locus of periodic solutions in the UN − F plane,the first turning point corresponding to loss of stability; (b) decay ex-ponent as a function of F obtained from direct numerical integration(DNI) of the governing equations. Solid red curve corresponds to theperiodic structure (D = 0), black dashed curve and blue dash-dottedcurve correspond to two different realizations of the disordered struc-ture, both with D/C = 2. For both realizations, ∑δkn ≈ 0. Apart fromthe presence of disorder, all other system parameters are the same as inSection 4.2. The two plots (a) and (b) have the same horizontal axis.4.3.2 Onset of Transmission Remains Unchanged on AverageKnowing that the onset of transmission occurs at different forcing amplitudes de-pending on the specific realization of a given disorder, we want to know the averagevalue of the onset of transmission in an ensemble-average sense. To find the an-swer, we have computed the exact numerical value of the threshold force Fth(Ω)using numerical continuation. Figure 4.11(a) shows the values of Fth at Ω = 1.30for an ensemble of 1500 realizations with D/C = 2. Comparing the cumulativeaverage with the total average indicates that the results have converged. The rel-760.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34051015Threshold force, FthRelative frequency ( % ) 0 500 1000 15000.190.230.270.310.35Realization indexThreshold force, Fth  (a)(b)Individual realizationAverageCumulative averageFigure 4.11: The influence of disorder on the supratransmission force thresh-old Fth atΩ= 1.30 for an ensemble of 1500 realizations with D/C= 2.The value of Fth for each realization is obtained by numerical contin-uation. (a) Individual values of Fth for each realization are shown bydots, the cumulative average is shown by the black solid curve and thetotal average is shown by the red dashed line. (b) The relative fre-quency of occurrence of Fth is shown by solid back lines. The value ofFth for the ordered system is shown using the vertical dash-dotted line.The horizontal dashed line indicates relative frequency of 10%. Theresults suggest that on average the onset of transmission is the samefor ordered and disordered system.ative frequency of occurrence of Fth within the same ensemble is shown in Fig-ure 4.11(b). The average value of Fth for the disordered system was found to be thesame as that of the ordered system. Moreover, we can see in Figure 4.11(b) thatindividual values of Fth for the disordered system are spread uniformly around theonset of transmission for the ordered system. We have made similar observationsat other values of Ω as well (not shown here).Figure 4.12 shows the average values of Fth and their standard deviations forD/C = 2 at different values of Ω away from the linear pass band. The averagevalues show that, in an ensemble-average sense, the onset of supratransmission is771.15 1.2 1.25 1.3 1.35 1.4 1.45 1.510−210−1100Forcing Frequency, ΩThreshold force, Fthstop bandFigure 4.12: The influence of disorder on the average threshold force for thestructure with hardening nonlinearity. The red solid curve shows thethreshold curve for an ordered structure (cf. Figure 4.6). At each forc-ing frequency, the filled circles show the average value of Fth and theempty circles show the corresponding value of standard deviation. Wesee that the average threshold curve for a disordered system is the sameas the threshold curve for an ordered system. The standard deviationof Fth decreases as we move away from the pass band.the same in ordered and disordered systems. For a fixed strength of disorder, weobserve that the standard deviation of Fth decreases as Ω moves away from thepass band. This implies that as Ω moves farther into the stop band, dispersion(due to Bragg scattering) is more dominant than disorder effects. This is in factsimilar to how disorder affects the linear response of the system. As we showedin Section 4.1.3, the most significant influence of disorder occurs at frequencieswithin and near the pass band of the system. We found similar results for D/C = 1.4.3.3 Energy Profiles of Transmitted WavesWe explore how disorder changes the average response above the transmissionthreshold for two strengths of disorder, D/C = 1 and D/C = 2. For each disorderstrength, we consider an ensemble of 1500 disordered structures; this ensemble islarge enough that energies converge to their average values. For every realizationwithin an ensemble, we compute the energy for each unit (En) at a forcing ampli-tude 5% above the onset of transmission – En is defined in (2.18). Average energiesat each strength of disorder are then obtained by averaging energies over the entireensemble. The results presented here are for Ω= 1.30.781 2 3 4 5 6 7 8 9 1010−1710−1410−1110−810−510−2101Unit number, nEnergy, En  D / C = 0D / C = 1D / C = 21 2 3 4 5 6 7 8 9 1010−210−1100101Unit number, nEnergy, En  (a) (b)D / C = 0D / C = 1D / C = 2Figure 4.13: Average energy profiles at Ω= 1.30 for (a) the nonlinear struc-ture above the onset of supratransmission, (b) the linear structure. No-tice the difference between the vertical axes in (a) and (b). Above thethreshold, disorder results in localization of energy to the driven unit.Figure 4.13(a) shows the average energy profiles for the nonlinear system abovethe onset of transmission. We see that energies decay exponentially through thestructure, particularly between n = 2 and n = N− 1 (away from the boundaries).Moreover, energy becomes more localized to the driven unit as the strength of dis-order is increased. These energy profiles are drastically different from the energyprofiles below the threshold, where the response is very similar to the linear re-sponse shown in Figure 4.13(b) – given that the response below the threshold isvery similar to the linear response, we have used the linear energy profiles in Fig-ure 4.13(b). We notice in Figure 4.13 that disorder effects are more significantabove the transmission threshold (Figure 4.13(a)) than below it (Figure 4.13(b)).We explain this in more detail in the following section.4.3.4 Average Frequency Spectra Above the ThresholdFor the system parameters used in this work, we have found that the post-thresholdbranch within the stop band is chaotic, with a frequency spectrum similar to Fig-ure 2.5(c). To understand the average influence of disorder on the transmitted79Figure 4.14: Average frequency spectra of the driven unit for the ordered anddisordered structures. The common peak at 1.30 corresponds to theforcing frequency (Ω = 1.30). The grey area denotes the linear stopband.Figure 4.15: Average transmitted spectra (at n = N) for the ordered and dis-ordered structures; (a) average spectra based on complex-valued am-plitudes, (b) average squared spectra in the vicinity of linear pass band.The average transmitted spectra lie within the linear pass band. Lessenergy is transmitted above the supratransmission threshold as we in-crease the strength of disorder. The grey area denotes the linear stopband.waves above the threshold, we compare average frequency spectra of the first andlast units in the structure. These results are obtained for the same ensembles as inSection 4.3.3. For disordered systems, the frequency spectrum of each individualrealization is obtained as explained in Section 2.3. Before averaging the frequencyspectra within each ensemble, the individual complex-valued spectra for each re-80alization is normalized with its forcing amplitude. This is because the thresholdforce is different for each individual realization (recall Figure 4.10).Figure 4.14 shows the average frequency spectra of the driven unit (n = 1)for the ordered and disordered structures. The three spectra have a common pro-nounced peak at 1.30, which corresponds to the forcing frequency (Ω= 1.30). Thisis the only dominant frequency component for the two disordered structures. Atother frequencies, it is not easy to distinguish between the two disordered spectra,though they both have much lower amplitudes than those of the ordered structure.Based on the dominant peak in the average spectra of the disordered structures, it istempting to infer that the response of the driven unit is harmonic on average. Thisis not correct, however; the frequency component for a given realization is similarto that of the ordered structure in terms of overall magnitude. When averaging overan entire ensemble, the phases are incoherent for any given frequency component(other than 1.30). As a result, the average spectrum of an ensemble will have amuch lower amplitude than its individual realizations.Figure 4.15(a) shows the average frequency spectra at the end of the structure(n = N) for the ordered and disordered structures. The most notable feature ofthe transmitted spectra is that frequency components within the linear stop band(the area with grey background) have significantly decreased compared to n = 1;cf. Figure 4.14. The three spectra are therefore similar in the sense that theycontain frequencies predominantly within the linear pass band. As the strength ofdisorder is increased, the amplitudes at different frequency components decrease,most significantly within the stop band. Overall, there is less energy transmitted tothe end of the structure as disorder strength increases. This is consistent with thedecrease in EN from Figure 4.13.Once we realize that frequency components of the transmitted waves lie withinand near the pass band, we can explain, albeit qualitatively, certain aspects of dis-order effects from a linear perspective as well. Firstly, we expect from linear theorythat within the stop band disorder localizes energy near the source of excitation andthat less energy is transmitted through the structure as the strength of disorder in-creases. This is consistent with the average energy profiles in Figure 4.13(a). Italso explains why within the stop band disorder has a more significant influence onthe response of the structure above the transmission threshold than below it. Below81the threshold, the structure behaves linearly and is therefore barely influenced bydisorder. Above the threshold, on the other hand, the transmitted waves lie withinthe pass band, where disorder affects the results most significantly.Secondly, we showed for a linear system that the transmitted energy covers arelatively wider frequency range as disorder increases (see Figure 4.5). The datain Figure 4.15(a) is not conclusive in this regard though. To investigate this pointfurther, we have computed the average squared spectra for each ensemble: foreach realization within an ensemble, the absolute value of the frequency spectrumis squared, then the arithmetic mean over all realizations is used as the averagesquared spectrum of the ensemble – notice that energy is related to squared ampli-tudes. We have shown the squared spectra in Figure 4.15(b) for frequency compo-nents close to the pass band. We see that as disorder increases, there is less energyin the transmitted waves. Also, energy transmission occurs over a slightly widerfrequency range for stronger disorder strengths. This widening of the transmis-sion band is similar to linear systems, but less pronounced. Notice, however, thatthis widening occurs at very low amplitudes and can therefore pose challenges toexperiments.4.3.5 Prediction of Transmitted Energies Based on Linear TheoryAs we discussed in Sections 4.3.3 and 4.3.4, the average behavior of the trans-mitted waves above the threshold is reminiscent of the average linear behavior ofdisordered structures. In this light, we ask whether the average transmitted ener-gies above the threshold can be predicted based on linear theory. We introduce thetransmitted energy ratio ase = e(D/C)≡ < EN (D/C)>< E1 (D/C)>(4.7)At a given strength of disorder, e describes the ratio of average transmittedenergy to energy in the driven unit. We show the normalized transmitted energyratios e/e(0) for both the linear and nonlinear structures in Figure 4.16. For eachsystem, the transmitted energy ratio is normalized to its value for an ordered struc-ture, e(0). For the linear system, we considered the frequency range between 0 and2 for energy calculation; widening this frequency range did not change the results.820 1 200.20.40.60.81Strength of disorder, D / CEnergy ratio, e / e(0)  10.580.1610.360.04NonlinearLinearFigure 4.16: Normalized transmitted energy ratios, e/e(0), for different val-ues of disorder. The numerical value of each bar is shown above it.These results indicate that linear theory cannot be used for makingquantitative predictions of disorder effects above the threshold.Although linear theory can be used to make qualitative predictions of the averagebehavior of the response above the supratransmission threshold, the results in Fig-ure 4.16 suggest that they are not appropriate for making quantitative predictions.4.4 Prediction of the Onset of Supratransmission inDisordered Structures4.4.1 Analytical Estimate of the Force ThresholdWe make two assumptions to develop a theoretical framework for predicting theonset of transmission. Results from direct numerical simulations suggest that, tosome extent, the periodic structure (i) behaves linearly below the threshold; i.e.for F < Fth, (ii) behaves nonlinearly predominantly at the first unit. These effectsare observed because of the weak coupling between units. Based on these sim-plifications, we linearize the system for n ≥ 2; i.e. treat the nonlinearity locally.Moreover, we only keep the cubic term of FM at n = 1; this is done to keep theanalysis tractable by pencil and paper. We call this model the semi-linear system.A similar approach, in terms of treating the nonlinearity locally, is used in [9] inthe time domain for modeling the response of drill strings. The analysis presentedhere is performed in the frequency domain and consequently applies only to the83steady-state response of the system.In the linear part of the semi-linear model (2 ≤ n ≤ N), we use a transfer-matrix formulation to find the response of the system. We then use a harmonicapproximation of the nonlinear response and find an expression for the onset ofinstability. We only present the final results from this analysis here. Refer toAppendices B.1 and B.2 for derivations.Figure 4.17: The schematics of two adjacent units in a mono-coupled system.Figure 4.17 shows a schematic representation of two adjacent units in the as-sembled periodic structure. On either side of each unit (left and right), there is adisplacement and force. We want to relate the force and displacement on the leftside of the second unit (n = 2) to the force and displacement on the right side ofthe last unit (n = N). Between adjacent units, we can write{uRfR}(n+1)= [T (n+1) ]{uRfR}(n)(4.8)where [T (n+1) ] depends on the properties of the (n+ 1)-th unit. The (complex-valued) amplitudes of the displacement and force on the left side of each unit aredenoted by uL and fL, where a time dependence of exp(iΩt) is assumed. For eachlinear unit, we have[T (n) ] =[1+σn/kc 1/kcσn 1](4.9)whereσn ≡ ω2n +2iζΩ−Ω2 (4.10)Moving along the finite chain from the last unit all the way back to the first unit we84can write {uRfR}(N)= Ttotal{uRfR}(1)=[T11 T12T21 T22]{uRfR}(1)(4.11)where Ttotal = [T ]N−1 for the ordered system and Ttotal = [T (N) ]× ...× [T (2) ] inthe disordered case. Moving from the right to left side of the nonlinear unit, asexplained in Appendix B.2, the forcing amplitude at the onset of transmission, Fth,can be written asF2th = 18α2(p±√q3)(4.12)where p and q depend on the components of Ttotal and [T (1) ]. Supratransmissionoccurs when the right-hand side of (4.12) is real and positive. In particular, we needq > 0; this gives the critical frequency at which the enhanced nonlinear transmis-sion starts. Threshold curves predicted by (4.12) are exact for a periodic structure(ordered or disordered) that has a cubic nonlinearity at n = 1 and is otherwise lin-ear. We have verified this by comparing analytical predictions to exact numericalcomputation of the threshold curves. The two results match very well. We havenot included this comparison for brevity.Although our derivation was based on a periodic structure with on-site nonlin-earity, we expect it to be valid for structures with inter-site nonlinearity as well, pro-vided that ‘strain variables’ (relative displacements of adjacent units) are used. Theanalysis here is very similar to the analyses performed in Sections 2.5.1 and 3.4.The main difference between the present derivation and the analyses in previouschapters is that the former can include any number of linear units. Indeed, one canretrive the results from (2.26) and (3.4) by, respectively, setting N = 1 and N = 2in the formulation presented above.Finally, it is worth mentioning that (4.12) traces (within the approximationlimits) the locus of the first saddle-node bifurcation of the NLRM. Any other bifur-cation of the NLRM will go unnoticed by this formulation. It is possible (thoughpainstaking) to extend the analytical results such that the third saddle-node bifurca-tion of the NLRM can be traced as well. Nevertheless, this might not be worthwhilebecause of the complications explained in Section 4.2.854.4.2 Dependence of Threshold Curves on NonlinearityThe main limitations of the current analysis are in (i) confining the nonlinearity tothe first (driven) unit and (ii) ignoring the higher-order nonlinear terms in FM. Theassumption of local nonlinearity is expected to hold for weak coupling (i.e. smallC) and away from the linear pass band, in particular. Keeping the cubic term isexpected to work for weak nonlinearity (i.e. small |u(1)L |2). In this section, we keepthe strength of coupling unchanged (C = 0.05) and explore how the two aspects ofnonlinearity mentioned above change the threshold curves in an ordered structure.The influence of coupling strength will be explored in Section 4.4.3. The sameconclusions that we draw for an ordered structure apply to individual realizationsof disordered structures as well. In an ensemble-average sense, we showed inSection 4.3.2 that transmission thresholds do not change away from the pass band.Thus, we expect the results in this section to carry over to average properties ofdisordered systems as well.We compare the threshold curves in the following four nonlinear systems:1. Full nonlinearity, global: the system defined in (2.3).2. Cubic nonlinearity, global: the system defined in (2.3), FM,n truncated atcubic term.3. Full nonlinearity, local: the system defined in (2.3), nonlinear terms of FM,nignored for 2≤ n≤ N.4. Cubic nonlinearity, local: the system defined in (2.3), FM,n truncated at cubicterm, nonlinear terms of FM,n ignored for 2≤ n≤ N.We consider both softening and hardening systems. We choose system parameterssuch that the nonlinear terms in the softening and hardening systems have the samemagnitude but opposite signs. The results presented in this section are obtainedusing numerical continuation.Figure 4.18 shows the threshold curves for the four nonlinear systems for a soft-ening system with cubic coefficient k3 = −0.2. Comparing systems with globalnonlinearity to those with local nonlinearity, we see that treating the nonlinear-ity locally results in a slight overestimation of the threshold force. Nevertheless,860.6 0.65 0.7 0.75 0.8 0.85 0.9 0.9510−210−1100Forcing frequency, ΩThreshold force, Fthstop band  0.935 0.94 0.9455678910x 10−3stop bandFull nonlinearity, globalCubic nonlinearity, globalFull nonlinearity, localCubic nonlinearity, localFigure 4.18: Threshold curves for different nonlinear forces in a softeningstructures with k3 = −0.2. Solid curves correspond to systems inwhich all units are nonlinear (global nonlinearity), dashed curves tosystems in which only the first unit is nonlinear (local nonlinearity).Red curves are used when the nonlinear term FM is used as defined in(2.4). Cyan curves are used when FM is truncated at its cubic term.The edge of the pass band is denoted by the horizontal arrows. Theinset shows frequencies close to the pass band edge.this difference is only noticeable at frequencies where the threshold curve starts,which is very close to the pass band (this point is a cusp in the F −Ω plane – seeFigure 4.6). The reason for the good agreement between locally- and globally-nonlinear systems is that the coupling between units is very weak (C  1). AsΩ moves farther into the stop band, we see that a larger force Fth is required totrigger instability. As a result of this, the amplitude of vibrations in the driven unit(i.e. |u(1)L |) becomes larger at the onset of instability. This makes the contributionfrom higher-order nonlinear terms more significant; therefore, the approximation intruncating FM,n at its cubic term becomes less accurate. This is why the differencebetween threshold curves with full nonlinearity and cubic nonlinearity increases aswe move away from the pass band. The same reasoning explains why the systemwith cubic nonlinearity overestimates the threshold force.Truncating the nonlinearity at its cubic term or restricting it to n = 1 has the871.15 1.2 1.25 1.3 1.35 1.410−210−1100Forcing frequency, ΩThreshold force, Fthstop band  1.15 1.155 1.16 1.165 1.170.030.040.050.060.070.08stop bandFull nonlinearity, globalCubic nonlinearity, globalFull nonlinearity, localCubic nonlinearity, localFigure 4.19: Threshold curves for different nonlinear forces in a hardeningstructures with k3 = +0.2. Solid curves correspond to systems inwhich all units are nonlinear (global nonlinearity), dashed curves tosystems in which only the first unit is nonlinear (local nonlinearity).Red curves are used when the nonlinear term FM is used as defined in(2.4). Cyan curves are used when FM is truncated at its cubic term.The edge of the pass band is denoted by the horizontal arrows. Theinset shows frequencies close to the pass band edge.same qualitative effect on threshold curves in hardening systems as it does in soft-ening systems. We can see this in Figure 4.19, which shows the four thresholdcurves for a hardening system with cubic coefficient k3 = 0.2. Apart from thesesimilarities, we see two main differences between threshold curves in hardeningand softening systems. Firstly, restricting nonlinearity to the first unit extends thethreshold curve to the pass band for a hardening system (compare solid curvesto dashed curves in Figure 4.19). To explain this, we recall from Section 4.3.1that threshold curves trace the locus of saddle-node bifurcations, and from Section3.6 that the solutions within the pass band usually lose stability through a Neimark-Sacker bifurcation. When the threshold curve of a locally-nonlinear system contin-ues inside the pass band, it means that the corresponding linear solutions lose theirstability via saddle-node bifurcation. Thus, treating the nonlinearity locally canpredict an incorrect instability mechanism for a globally-nonlinear system close to88(and within) the pass band. Further discussion of the exact bifurcation structure inthe vicinity of the pass band, and its dependence on damping, is beyond the scopeof this work. This is also the frequency range where supratransmission is of lessinterest, as discussed in Section 4.2.Secondly, approximating the nonlinear term as cubic gives a more accurate es-timation of the threshold curves in the softening system than the hardening system.We can see this by comparing threshold curves in Figures 4.18 and 4.19. For agiven distance from pass band, notice that the values of threshold force Fth arehigher in the hardening system than the corresponding ones in the softening sys-tem. Thus, just before the onset of instability, the amplitudes of motion are higherin the hardening system. As a result, the higher-order nonlinear terms are moreimportant in determining the onset of transmission in the hardening structure thanin the softening one. We have confirmed this by computing the threshold curves forsystems in which FM,n is truncated at its quintic term (not shown). The thresholdcurves for the systems with quintic nonlinearity were much closer to the thresholdcurves of the fully nonlinear system.4.4.3 Dependence of Locally-Nonlinear Behavior on CouplingAs already stated, the basis for treating the nonlinearity locally is weak strengthof coupling between units. This approximation is expected to lose accuracy as thestrength of coupling increases. To investigate this, we compute the threshold curvesat different values of C for two hardening structures with full nonlinearity; i.e.having the complete nonlinear form of FM,n from (2.4). In one of them nonlinearityis treated locally, while in the other one all units are nonlinear (globally nonlinear).We compare these threshold curves in Figure 4.20 as a function of the distancefrom pass band, ∆Ω. This is because increasing C moves the pass band edge tohigher frequencies. Thus, we define∆Ω≡Ω−ωN (4.13)where ωN = ωN(C) is the largest linear natural frequency of the structure.We see in Figure 4.20 that the threshold curves extend to the pass band for thelocally nonlinear structures (see the discussion in Section 4.4.2). There is therefore890 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110−210−1100Distance from pass band edge,  ∆ΩThreshold force, Fth  increasing Clocally nonlinearglobally nonlinearC = 0.05C =  0.1C = 0.15C =  0.2C = 0.25Figure 4.20: The influence of coupling strength on threshold curves for thesystem with full nonlinearity. Solid curves correspond to the globallynonlinear case and dashed curves to the locally nonlinear case. Thehorizontal axis shows the distance from pass band edge, where ∆Ωis defined in (4.13). The arrow indicates the direction of increasingcoupling strength. As the strength of coupling increases, restricting thenonlinear forces to n = 1 is inaccurate over a larger frequency range.The red curves (C = 0.05) are reproduced from Figure 4.19.a frequency range in the vicinity of the pass band where using a locally-nonlinearmodel predicts an incorrect instability mechanism; this frequency range increaseswith the strength of coupling. Furthermore, the assumption of local nonlinearitybecomes more inaccurate as C increases, and overestimates the threshold curvesover a larger frequency range. For frequencies far from the pass band, restrictingnonlinearity to the driven unit gives an accurate prediction of the onset of supra-transmission. The reason is that, in this frequency range, the linear response ishighly localized to the driven unit and dispersion (Bragg scattering) dominates overnonlinear forces. As C increases, the assumption of local nonlinearity is accurateover a smaller frequency range.904.5 Concluding RemarksWe studied the interaction among the effects of dispersion, dissipation, disorderand nonlinearity in the context of supratransmission. We considered a dampednonlinear periodic structure of finite length with weakly coupled units. Disorderwas introduced as small variations in the on-site stiffness parameters of the struc-ture, drawn from a uniform statistical distribution.We showed that although individual realizations of a disordered structure havedifferent onsets of supratransmission, the threshold curve is robust to disorder whenaveraged over an entire ensemble. For harmonic excitation away from the passband edge, increasing the strength of disorder has negligible influence on transmit-ted energies below the onset of supratransmission. In contrast, we found averagetransmitted energies to decrease with disorder above the transmission threshold.This happens because the average frequency spectra of the nonlinearly transmittedwaves lie within the linear pass band of the structure. This is the frequency rangewhere where disorder is known to localize the response to the driven unit (An-derson localization). Overall, as the forcing frequency moves away from the passband, dispersion effects become dominant and the influence of disorder decreases.We provided approximate analytical expressions for predicting the onset ofsupratransmission in weakly coupled structures. This formulation is exact for adisordered periodic structure that has cubic nonlinearity in its driven unit and islinear otherwise. We further studied the range of validity of the analysis by study-ing the dependence of threshold curves on nonlinearity and strength of coupling.For forcing frequencies away from the edge of a pass band, where the linear so-lution is highly localized, we found that the nonlinear forces are confined to thedriven unit. In this frequency range, truncating the nonlinear forces at their cubicterms results in overestimation of the onset of supratransmission. However, usingthe complete form of the nonlinear forces gives a very accurate prediction of thethreshold curves. Closer to the pass band edge, the linear response is no longerhighly localized and the nonlinearity spreads to other units. Strong coupling in-validates our analysis, but is of less interest in the context of disordered periodicstructures (Anderson localization).We highlighted a non-trivial influence of damping and disorder on the onset91of supratransmission. Damping may bring about other stable periodic solutions atthe first turning point of the NLRM. In this scenario, supratransmission does notoccur at the first turning point of the NLRM, as normally expected. We showed thatdisorder may either facilitate or inhibit this feature, depending on the realizationand strength of disorder. We further observed that supratransmission could stilltake place at a subsequent (the third) turning point of the NLRM.92Chapter 5ConclusionWhen a nonlinear periodic structure is harmonically forced at one end with a forc-ing frequency within its stop band, wave transmission can still occur if the forcingamplitude is beyond a certain threshold. This is a generic instability-driven trans-mission phenomenon in discrete nonlinear periodic structures, known as supra-transmission.This thesis has addressed the nonlinear phenomenon of supratransmission ina discrete periodic structure using computational and analytical techniques. Wereported several novel findings in this context, specifically targeted to periodicstructures that are relevant in engineering applications: periodic structures withdamping, finite length and disorder (small deviations from exact periodicity spreadthroughout the structure). The knowledge generated in this thesis contributes tothe understanding of high-amplitude energy transmission characteristics of peri-odic structures, and to design of new devices and materials with almost-periodicmicro-architecture to operate in the nonlinear regime.We summarize our findings in Section 5.1 and provide a synopsis of our anal-yses. In Section 5.2, we discuss the limitations of the methodologies used in thisthesis. We provide suggestions for future research in this area in Section 5.3.935.1 Summary of ContributionsWe proposed a macro-mechanical periodic structure that consists of coupled sus-pended cantilever beams (Figure 2.1). Within each unit cell, the linear restoringforce of the beam is combined locally with a strong nonlinear magnetic force toproduce on-site nonlinearity. The magnetic force can be tuned, thus providing con-trol over the strength of nonlinearity, as well as its type (softening or hardening).We carried out our entire investigations based on a mathematical model of thissetup that was developed in Section 2.1.Using this platform, we performed a systematic study of the supratransmissionphenomenon. The nonlinear periodic structures considered throughout the thesishave finite length, light damping and weak coupling forces between adjacent units.We summarize our contributions under three main categories:I. Underlying mechanismWe studied supratransmission using direct numerical simulations and numericalcontinuation techniques (Sections 2.3 and 2.4). We identified the instability mech-anism underlying supratransmission as a saddle-node bifurcation in the nonlin-ear response manifold (NLRM) of the damped, finite structure. This is the samemechanism responsible for supratransmission in infinite-dimensional Hamiltoniansystems. Using continuation methods, the NLRM is constructed numerically andthreshold curves are computed as the loci of the first saddle-node bifurcation pointsas a function of the forcing frequency.The supratransmission phenomenon may be explained, in a qualitative fash-ion, as the resonance of the driving force with the shifted pass bands of the periodicstructure. The behavior of the threshold curve also supports this explanation: as theforcing frequency moves farther from the pass band, a higher force amplitude is re-quired for the onset of supratransmission. We showed in Section 2.4.2 that the truemechanism underlying supratransmission is indeed the instability of periodic solu-tions via a saddle-node bifurcation. We concluded that the resonance of the drivingforce with the shifted pass band is not an accurate predictor of supratransmissionfrom either a phenomenological or quantitative perspective.For forcing amplitudes below the supratransmission threshold, the response of94the structure lies within the basin of attraction of a limit cycle (a periodic attractor).This solution branch continues to the trivial (static) equilibrium of the structure. Atthe onset of supratransmission, the response of the system jumps from this basinof attraction to that of a non-periodic (either a quasi-periodic or chaotic) attractor.This change in the basin of attraction is accompanied by a large increase (ordersof magnitude) in the transmitted energy; i.e. supratransmission. If another peri-odic attractor (limit cycle) existed at the saddle-node bifurcation point, then theresponse would have jumped to that solution branch instead, though the increasein the transmitted energy would be significantly smaller in comparison.The occurrence of supratransmission has two main manifestations in the post-threshold response of the structure: (1) the response of the first unit has a broadbandfrequency spectrum, (2) the response of the last unit has a band-limited frequencyspectrum, concentrated at the pass bands of the linear structure. From this per-spective, supratransmission can be described as a band-limited transmission mech-anism.II. Influence of system parameters (ordered structures)We studied the influence of several system parameters on the supratransmissionphenomenon in ordered structures (Chapter 3). A synopsis of these parametricstudies is provided here:• Forcing frequencyThe forcing frequency plays a significant role in supratransmission. As theforcing frequency moves away from the pass band, dispersion effects become moredominant and a higher force amplitude is required for the onset of supratransmis-sion.The increase in transmitted energies is higher for forcing frequencies that arefarther from the edge of a pass band.• Type and strength of nonlinearityThe type of nonlinearity determines on which side of a pass band supratrans-mission may occur. This is below the pass band in the case of softening nonlinearity95and above it when the nonlinear force is hardening.Increasing the strength of nonlinear forces decreases the force required for theonset of supratransmission.• Number of unitsFor weakly coupled structures, increasing the number of units increases theforce threshold at the onset of supratransmission.For both types of nonlinearity, the influence of the number of units on supra-transmission is significant only for driving frequencies very close to the pass bandedge.• DampingGenerally speaking, damping increases the force required for the onset of supra-transmission. Apart from this, we have identified two other effects that are broughtabout by damping.(i) Damping may eliminate supratransmission within a frequency range in thevicinity of the linear pass band. This occurs because damping can modify thetopology of an NLRM such that it no longer possesses a saddle-node bifurcation.Damping also introduces a threshold on the minimum force amplitude requiredfor supratransmission to occur. This threshold is zero in undamped structures andincreases with damping. Also, increasing the damping ratio widens the frequencyrange over which supratransmission is prohibited by damping.(ii) Damping may delay the onset of supratransmission compared to the ex-pected force threshold of a damped structure (coinciding with the first turning pointof the NLRM, TP1). This could happen provided that damping stabilizes otherbranches of periodic solutions above TP1. In this case, supratransmission cannottake place at TP1 because the response gets trapped to another limit cycle insteadof moving to the basin of attraction of a non-periodic attractor. Supratransmissionmay still take place in this scenario, but at a subsequent bifurcation of the NLRM(e.g. the third turning point).96• Strength of couplingIncreasing the strength of coupling increases the minimum force required for theonset of supratransmission at all driving frequencies.For damped structures with weak coupling, the strength of coupling does notalter the location of the critical frequency at the onset of supratransmission (withrespect to the edge of a pass band).III. Analytical prediction of the onset of supratransmissionUsing numerical simulations, we observed that nonlinear forces are confined to thedriven unit for weakly coupled systems. Relying on this observation, and truncat-ing the nonlinear forces at the cubic terms, we were able to provide three approxi-mate closed-form analytical estimates for predicting the onset of supratransmissionfor weakly coupled periodic structures. These expressions vary in their degree ofcomplexity and range of applicability. Nevertheless, they all rely on estimating thelocus of the first saddle-node bifurcation of the NLRM of the system. The firsttwo analyses are developed based on the equations governing the evolution of theenvelope of harmonic waves through the structure. The third analysis is developedbased on a nonlinear analysis of the first unit coupled with a transfer-matrix for-mulation to account for the (linear) dynamics of the rest of the periodic structure.The first analysis (Section 2.5) is based on the local nonlinear dynamics of thedriven unit, decoupled from the rest of the periodic structure. Accordingly, thisformulation cannot capture the dependence of supratransmission on the strengthof coupling. The results of this analysis are valid for very weak values of cou-pling, near the anti-continuum limit. Nevertheless, this formulation can be used toprovide qualitative explanations of certain features of supratransmission; examplesinclude the influence of nonlinearity and damping (Sections 3.1 and 3.3).The second analysis (Section 3.4) is an improvement of the first one, incor-porating the strength of coupling into the predictions. This analysis is based onthe local nonlinear dynamics of the driven unit coupled to another linearized unit(the rest of the structure is truncated after the second unit). The expression ob-tained from the second analysis can estimate the influence of coupling on thresh-old curves, but cannot be used in a disordered structure. The second formulation97is valid over a wider range of coupling strengths than the first formulation, but it isstill limited to weak strengths of coupling.The third analysis (Section 4.4) is performed for a periodic structure of arbi-trary length, with the nonlinear forces confined to the driven unit and truncated atthe cubic term. Thus, it can be used for estimating the onset of supratransmissionin ordered and disordered periodic structures. We explored the range of validity ofthis formulation by studying the dependence of threshold curves on nonlinearityand strength of coupling. We find that our analytical predictions overestimate theonset of supratransmission, particularly at frequencies close to the pass band.All three analytical approaches rely on weak strength of coupling to ensurethat the nonlinear forces can be confined to the driven unit. Strong coupling inval-idates this assumption, but is of less interest in the context of disordered periodicstructures.Although our analyses were based on a periodic structure with on-site nonlin-earity, we expect them to be generally valid for structures with inter-site nonlin-earity as well, provided that ‘strain variables’ (relative displacements of adjacentunits) are used.IV. Interaction between supratransmission and Anderson localizationWe studied the interaction between supratransmission and Anderson localizationin a weakly coupled disordered periodic structure with damping and finite length.Disorder was introduced as small random perturbations in the stiffness parametersof the structure, drawn from a uniform statistical distribution.We showed that supratransmission persists in the presence of disorder, withindividual realizations of a disordered structure having different force thresholds.The influence of disorder decreases in general as the forcing frequency moves awayfrom the pass band edge, reminiscent of dispersion effects subsuming Andersonlocalization in linear periodic structures.Averaging over an entire ensemble of disordered structures, we found that thethreshold curve is robust to disorder. In other words, the average force thresholdrequired to trigger supratransmission remains unchanged. In contrast, the aver-age transmitted energy above the supratransmission threshold decreases with the98strength of disorder. This happens because the average frequency spectra of thenonlinearly transmitted waves lie within the linear pass band of the structure.5.2 LimitationsNaturally, the methodology adopted in a study includes certain limitations. Wehighlight the following factors as the main limitations of this work.Mathematical modelingThe mathematical modeling of the periodic structure in Figure 2.1 was based onthe assumption that the motion of each beam is described by its fundamental modeshape. This is a reasonable assumption as long as (a) the forcing frequencies areclose to the first pass band of the structure (b) the second pass band is far fromthis frequency range. In the presence of nonlinear forces, however, it is possibleto excite higher frequencies through internal and combination resonances [98]. Inthis situation, the mathematical model developed in Section 2.1.1 would need to beupdated. Ultimately, experimental results can determine whether these effects aresignificant.Another important assumption made in developing the equations of motionwas the modeling of the magnetic forces, FM,n in (2.2). Within each unit cell,we treated the permanent magnet and electromagnets as magnetic poles and usedCoulomb’s law to model the magnetic interaction forces between them. In practice,this force would need to be modeled based on experimental system identification;refer to [59, 100] for comprehensive reviews of relevant experimental methodolo-gies.Throughout this work, dissipative effects were modeled using mass-proportionallinear viscous damping forces. This is a very common assumption in physical mod-eling of periodic structures (even in granular crystals [110]). However, it is con-ceivable that damping in the structure can be non-proportional, for example due toelectrodynamic damping introduced through connecting a shaker to the structure.Again, this aspect would need to be explored experimentally [107].99Weak couplingFor periodic structures with damping and finite length, Anderson localization be-comes relevant at weak strengths of coupling – recall the discussion in Section 4.1.Accordingly, we restricted the scope of this thesis to weakly coupled periodic struc-tures. In the absence of disorder, however, this requirement may be relaxed.Although supratransmission relies on the existence of a stop band, it is notrestricted to weakly coupled structures. In fact, the initial studies on supratrans-mission [39, 40, 60, 62] were performed closer to the continuum limit. Weaklycoupled structures were eventually considered, possibly due to the existence ofdiscrete breathers and specifically the connection between supratransmission andmobile breathers in some parameter ranges [82]. Although supratransmission hasbeen studied for moderate and strong strengths of coupling, these studies mostlypertain to undamped periodic structures of infinite extent. The effects of damp-ing and finite length on supratransmission in (ordered) periodic structures remainunexplored in this range of coupling strengths.Analytical approachThe analytical predictions of the onset of supratransmission were developed basedon estimating the loci of the first turning points (saddle-node bifurcations) of theNLRM. Consequently, any other bifurcation of an NLRM will go unnoticed bythese formulations. For undamped structures, there exists no precedence of supra-transmission occurring through a different mechanism. In damped structures, how-ever, we showed that damping may stabilize another periodic branch at the firstturning point of the NLRM, thereby delaying the onset of supratransmission untilsubsequent bifurcations of the NLRM.It is possible to extend the existing analytical approach to locate the third turn-ing point of the NLRM as well. This becomes possible by truncating the nonlinearforce at the quintic term instead of the cubic term. Note, however, that it mightbe possible for the periodic solutions to lose their stability between the first andthird turning points through a Neimark-Sacker bifurcation. We are not aware of amethodology that can be used to tackle this scenario analytically.In our analyses, we restricted the nonlinear forces to the driven unit. Relaxing100this assumption makes the analytical approach intractable.Disorder effectsThroughout Chapter 4, we used a uniform statistical distribution for realizationof disordered structures. Furthermore, we mostly focused on the average proper-ties of an ensemble. A comprehensive statistical analysis of the threshold curvesand transmitted energies remains to be performed. This analysis could determine,among other things, the relations between the probability density functions andstandard deviations of the disorder parameter and threshold curves. This infor-mation is important from a practical point of view because the statistical mode ofan ensemble may be significantly different from its mean value depending on thecorresponding probability distribution function – this was not the situation encoun-tered for the case studied in this thesis.In this work, we introduced disorder to the on-site stiffness parameter. Froma phenomenological point of view, disorder-borne localization still occurs if disor-der is introduced through other parameters of the system [65, 108]. Nevertheless,it will be useful to obtain a quantitative comparison of disorder effects based ondifferent disorder parameters. Possible additional disorder parameters include cou-pling stiffness, on-site and inter-site damping and on-site modal mass parameters.5.3 Future DirectionsIn addition to the limitations outlined in Section 5.2, a number of other avenuescan be explored for future research on this topic.Experimental realization of supratransmissionAn impactful continuation of this thesis is experimental realization of supratrans-mission in the periodic structure proposed in Section 2.1. This experimental setupcan be readily used to investigate the influence of the type and strength of non-linearity on supratransmission, as well as to investigate the associated hysteresisphenomenon. It may also be used to realize disordered periodic structures by vary-ing the currents passing through the electromagnets in different units.101Band-limited excitationIn supratransmission, external excitation is provided through a harmonic force.This is equivalent to a very sharp frequency spectrum (a Dirac delta function,strictly speaking). In engineering practice, it is not uncommon to encounter a sit-uation where the frequency spectrum of an excitation comprises a narrow band offrequencies. For a given wave form with its bandwidth entirely within the stopband, it would be expected that a phenomenon similar to supratransmission wouldoccur beyond a certain forcing amplitude; i.e. a sudden increase in the transmit-ted energies with the frequency spectra of transmitted waves lying within the passband.Multi-coupled structuresThe periodic structure we studied in this thesis is a mono-coupled structure, mean-ing that the coupling between adjacent units occurs through one coordinate (de-gree of freedom) of the system. Wave propagation in multi-coupled periodic struc-tures is more complicated. Multi-coupled structures carry different types of trav-eling waves, which can convert to each other as they propagate through the struc-ture [15, 88]. 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Phani. Supratransmission in a disorderednonlinear periodic structure. Journal of Sound and Vibration, 380:242–266,2016. → pages iv114Appendix ANumerical ContinuationNumerical continuation is a computational technique that makes it possible to fol-low the evolution of the solutions of systems of ordinary differential equations asa function of a system parameter. This approach was used throughout the thesisfor studying the evolution of the steady state solutions of the structures as a func-tions of forcing amplitude or forcing frequency. In this chapter, we present a verybrief overview of the basic formulation for numerical continuation of periodic or-bits. For comprehensive explanations of numerical continuation techniques, referto [6, 69].Our first task is to set up the governing equations that are second-order in timeas a set of first-order autonomous dynamical system. The first-order formulationmay be obtained by writing the equations of motion in the state space, with dis-placements and velocities as the states. The conversion from a non-autonomoussystem to an autonomous one (i.e. removing the explicit dependence on time onthe right-hand side of the state-space equations) is commonly done by either oftwo methods: (i) introducing time as an additional state, (ii) replacing the time-dependent (harmonic) forcing term with an autonomous system that has a stablelimit cycle and coupling this to the original system. The first method adds oneextra degree of freedom to the dynamical system, while the second approach addstwo extra degrees of freedom.115Following this procedure, the governing equations take the following form:x˙(t) = f (x(t),α) (A.1)where the vector x(t) contains all the states. The scalar parameter α represents thecontinuation parameter, which could be the forcing amplitude, for example.We are seeking solutions of (A.1) that are periodic in time with period T . Be-cause T is not a known parameter a priori (and would normally vary with α), werescale time as t → t/T so that the interval of periodicity is fixed to unity. Withthis, the governing equations transform to the following:x˙(t) = T f (x(t),α) (A.2)in the rescaled time variable.To continue the (steady-state) periodic solutions, we will transform the initial-value problem in (A.2) to a boundary-value problem. The boundary conditions forthis problem is the periodicity condition in the time domain, which can be writtenas follows:x(0) = x(1) (A.3)Recall that we have rescaled time such that the interval of periodicity in (A.2) isunity.Given that we are looking for periodic solutions of an autonomous system, wealso need to impose a phase condition on x to ensure uniqueness of the solutions.Without ensuring the uniqueness of solution, any arbitrary shift in a periodic so-lution would give the same periodic solution. There are various phase conditionsthat can be used. For example, we can specify zero velocity for one of the units.The advantage of this phase condition is that it is realistic from a physical point ofview and realizable in experiments.Thus, numerical continuation of the periodic orbits of the equations of mo-tion transforms to solving the boundary-value problem defined in (A.2) and (A.3),subject to a phase condition to ensure uniqueness of solutions.When using numerical continuation, we need to start with a known solution ofthe system. At each continuation step k, we are seeking the solution (xk,Tk,αk)116based on the known solution at the previous step, (xk−1,Tk−1,αk−1). Also, thesolutions along a periodic branch need to be suitably discretized. This aspect ofcontinuation can be readily dealt with through available software packages such asAUTO [26], Matcont [41] and COCO [21]. We have used AUTO for numericalcontinuation computations in this thesis, which uses the orthogonal collocationmethod.117Appendix BDerivationsB.1 Derivation of the Transfer Matrix FormulationFigure B.1 shows a schematic of a unit cell. The unit cell consists of a unit mass, acoupling spring (kc), a grounding spring (ks), and a damper (2ζ ). Force equilibriumat the left and right ends of the n-th unit, respectively, give the following relationsσnu(n)L + kc(u(n)L −u(n)R ) = f (n)L (B.1a)kc(u(n)R −u(n)L ) = f (n)R (B.1b)where σn is defined in (4.10). From compatibility and force equilibrium betweenFigure B.1: The schematic of a unit cell. We have ks = ω2n for linear units,while ks = ω2n +3/4k3|u(n)L |2 for nonlinear units.118adjacent units we haveu(n+1)L = u(n)R (B.2a)f (n+1)L =− f (n)R (B.2b)We can rearrange (B.1) to have the ‘left’ variables in terms of the ‘right’ variables.Then, using (B.2), we haveu(n)R = (1+σn/kc)u(n−1)R + f(n−1)R /kc (B.3a)f (n)R = σnu(n−1)R + f(n−1)R (B.3b)which gives the transfer matrix in (4.9).B.2 Derivation of the Onset of SupratransmissionApplying Newton’s second law to the right and left nodes of the nonlinear unit atn = 1, respectively, results in the following equationsf (1)R = kc(u(1)R −u(1)L ) (B.4a)f (1)L = (σ1+β )u(1)L + kc(u(1)L −u(1)R ) (B.4b)where we have definedβ = β (uL)≡ 3/4k3|uL|2 (B.5)We rearrange the terms to getu(1)R = (1+(σ1+β )/kc)u(1)L − f (1)L /kc (B.6a)f (1)R = (σ1+β )u(1)L − f (1)L (B.6b)using (B.6) in combination with (4.11), we can relate the response on the rightof n = N to the response on the left of n = 1. The external force applied to thestructure corresponds to f (1)L = F and, without loss of generality, we take F tobe real-valued. To enforce the free boundary on the right end of the structure werequire f (N)R = 0. This, after substituting u(1)R and f(1)R from (B.6) into (4.11), results119in the following(T21(1+(σ +β )/kc)+T22(σ +β ))u(1)L = (T21/kc+T22) f(1)L (B.7)which can be used to find the onset of transmission for the semi-linear system.Settingρ ≡ |u(1)L |2 (B.8)we can obtain the following cubic equation for ρa3ρ3+a2ρ2+a1ρ+a0 = 0 (B.9)wherea3 = (9/16)k23 |T21/kc+T22|2 > 0 (B.10a)a2 = (3/2)k3 Re{(T21/kc+T22)†(σ(T21/kc+T22)+T21)}(B.10b)a1 = |(1+σ/kc)T21+T22|2 (B.10c)a0 =−F2 |T21/kc+T22|2 < 0 (B.10d)Here, the superscript † denotes complex conjugate.Equation (B.9) is a cubic polynomial with real coefficients and, depending onthe relation between its coefficients, may have three real roots or only one. Theonset of transmission is where there are three real roots with two of them beingequal (i.e. multiple real roots). Such points correspond to saddle-node bifurcationsof the Duffing equation [50]. We first re-write (B.9) as followsρ3+ c2ρ2+ c1ρ+ c0 = 0 (B.11)Now we restrict the coefficients of (B.11) such that it has at least two equal realroots by setting [1]c0 =−2(p±√q3)(B.12)120wherep = c32/27− c1c2/6 (B.13a)q = c22/9− c1/3 (B.13b)Using c0 = −(4F/3k3)2, the critical forcing amplitude at the onset of threshold,Fth, can be written as shown in (4.12).In the above analysis, we assumed the structure to have free boundaries at bothends. A very similar formulation applies if the boundary on either end of structureis fixed. In this case, the formula in (4.12) remains valid but the coefficients an in(B.10) need to be updated. For a fixed boundary on the driven end of the structure(left side), we have f (1)L = F − kcu(1)L . To model a fixed boundary on the rightend of the structure, the free boundary condition f (N)R = 0 should be replaced withu(N)R = 0.121

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