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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Acta mechanica 95 (1992), S. 197-226 
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary The free and forced motions of a nonlinear periodic structure with cyclic symmetry are studied. The structure consists of a number of identical linear flexural members coupled by means of nonlinear stiffnesses of the third degree. It is found that this system can only possessn “similar” nonlinear modes of free oscillation, and that no other modes are possible. Moreover, there exist pairs of nonlinear modes with mutually orthogonal nodal diameters having, in general, distinct “backbone” curves. A multiple-scales averaging analysis is used to study the nonlinear interaction between a pair of modes with orthogonal nodal diameters. As a result of this analysis, it is found that all pairs of nonliner modes along with all their linear combinations are orbitally unstable, and the only possible orbitally stable periodic motions are free travelling waves, that propagate through the structure in the clockwise and anti-clockwise directions. Under harmonic forcing, a bifuraction of a stable branch of forced travelling waves from a branch of forced normal mode motions is detected, and “jump” phenomena between branches of periodic solutions are observed. The analytical results are in agreement with experimental observations of an earlier work, and, in addition, are verified by numerical simulations.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Journal of nonlinear science 5 (1995), S. 485-502 
    ISSN: 1432-1467
    Keywords: Nonlinear mode localization ; motion confinement
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary Steady-state nonlinear motion confinement is experimentally studied in a system of weakly coupled cantilever beams with active stiffness nonlinearities. Quasistatic swept-sine tests are performed by periodically forcing one of the beams at frequencies close to the first two closely spaced modes of the system, and experimental nonlinear frequency response curves for certain nonlinearity levels are generated. Of particular interest is the detection of strongly localized steady-state motions, wherein vibrational energy becomes spatially confined mainly to the directly excited beam. Such motions exist in neighborhoods of strongly localized antiphase nonlinear normal modes (NNMs) which bifurcate from a spatially extended NNM of the system. Steady-state nonlinear motion confinement is an essentially nonlinear phenomenon with no counterpart in linear theory, and can be implemented in vibration and shock isolation designs of mechanical systems.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1573-269X
    Keywords: damped nonlinear systems ; integrability ; Padé approximations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We construct analytical approximations for the transition from strongly nonlinear, early-time oscillations to weakly nonlinear, late-time motions of single degree of freedom, damped, nonlinear oscillators. Two methods are developed. The first relies on (a) derivation of an analytic solution for the initial value problem of an exactly integrable damped system, (b) development of separate early- and late-time approximations to the damped motion using the integrable solution, and (c) patching of the two approximations in the time domain by imposing continuity conditions on the composite solution at the point of matching. The second approach relies on a multiple-scales application of the method of nonsmooth transformations first developed by Pilipchuck, but complemented with a corrected frequency-amplitude relation. This improved relation is obtained by developing two separate frequency-amplitude asymptotic expansions in the frequency-amplitude plane, that are valid for large and small amplitudes, respectively, and then matching them using two-point diagonal Padé approximants. Comparisons between analytical approximations and numerical results validate the two approaches developed
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 13 (1997), S. 327-338 
    ISSN: 1573-269X
    Keywords: Nonlinear localization ; nonlinear differential equations ; Padé approximations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We analyze axisymmetric, spatially localized standing wave solutions with periodic time dependence (breathers) of a nonlinear partial differential equation. This equation is derived in the 'continuum approximation' of the equations of motion governing the anti-phase vibrations of a two-dimensional array of weakly coupled nonlinear oscillators. Following an asymptotic analysis, the leading order approximation of the spatial distribution of the breather is shown to be governed by a two-dimensional nonlinear Schrödinger (NLS) equation with cubic nonlinearities. The homoclinic orbit of the NLS equation is analytically approximated by constructing [2N × 2N] Padé approximants, expressing the Padé coefficients in terms of an initial amplitude condition, and imposing a necessary and sufficient condition to ensure decay of the Padé approximations as the independent variable (radius) tends to infinity. In addition, a convergence study is performed to eliminate 'spurious' solutions of the problem. Computation of this homoclinic orbit enables the analytic approximation of the breather solution.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 15 (1998), S. 245-257 
    ISSN: 1573-269X
    Keywords: Lorenz system ; homoclinic orbits ; Padé approximants
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present an iterative technique to analytically approximate the homoclinic loops of the Lorenz system for σ = 10, b = 8/3 and ρ = ρH = 13.926.... First, the local structure of the homoclinic solution for t → 0 ± and t → ± ∞ is analyzed. Then, global approximants are used to match the local expansions. The matching procedure resembles the one used in Padé approximations. The accuracy of the approximation is improved iteratively, with each iteration providing estimates for the initial conditions of the homoclinic orbit, the value of ρH, and three undetermined constants in the local expansions. Within three iterations the error in ρH falls to the order of 0.1%. Comparisons with numerical integrations are made, and a discussion on ways to extend the technique to other types of homoclinic or heteroclinic orbits, and to improve its accuracy, is given.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1573-269X
    Keywords: Nonlinear subharmonic orbits ; sensitive dependence on initial conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We employ nonsmooth transformations of the independent coordinate to analytically construct families of strongly nonlinear periodic solutions of the harmonically forced nonlinear pendulum. Each family is parametrized by the period of oscillation, and the solutions are based on piecewise constant generating solutions. By examining the behavior of the constructed solutions for large periods, we find that the periodic orbits develop sensitive dependence on initial conditions. As a result, for small perturbations of the initial conditions the response of the system can ‘jump’ from one periodic orbit to another and the dynamics become unpredictable. An analytical procedure is described which permits the study of the generation of periodic orbits as the period increases. The periodic solutions constructed in this work provide insight into the sensitive dependence on initial conditions of chaotic trajectories close to transverse intersections of invariant manifolds of saddle orbits of forced nonlinear oscillators.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Acta mechanica 128 (1998), S. 59-80 
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary We analyze standing wave oscillations of a periodic system of infinite spatial extent composed of layers with cubic material nonlinearities that are coupled by weak linear stiffnesses. We show that nonlinear modal interactions in this system are unavoidable due to the existence of an infinite degenerate set of internal resonances. Considering only the dominant 1∶3 resonances, the standing wave problem is formulated in terms of a bi-infinite set of coupled nonlinear difference equations that govern the layer modal amplitudes. In the limit of weak coupling between layers this set of difference equations is analyzed by (i) direct perturbation analysis, and (ii) matched asymptotic expansions of the differential equations resulting from a continuum approximation. Strongly and weakly localized, as well as spatially extended (non-localized) solutions are computed. In addition, composite solutions derived by matching in-phase and out of phase segments of the aforementioned solutions can also be constructed. In the limit of weak coupling between layers and/or strong material nonlinearities, the solutions develop sensitive dependence on initial conditions, and the possibility of spatial chaos in the system exists.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 40 (1997), S. 1639-1665 
    ISSN: 0029-5981
    Keywords: layered media ; stress wave propagation ; integral transform ; propagation zone ; attenuation zone ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Axisymmetric stress wave transmission through the leading layers of layered structures of infinite radial but finite axial extent is numerically studied by employing two different computational approaches: a technique based on the numerical inversion of Double Integral Transformations (DIT), and a Finite Element (FE) analysis. Considering the first approach, careful selections of the limits of the numerical inversions and the sampling rates are required in order to overcome inherent numerical instabilities associated with exponential dichotomy. This type of numerical instability is more evident in layered media with weak coupling. In such systems, direct multiplications of layer transfer matrices are avoided by employing a global scheme to assemble well-conditioned global transfer matrices. Moreover, the specific structure of the propagation and attenuation zones of the structure are taken into account for increasing the efficiency and effectiveness of the transfer matrix manipulations. Satisfactory agreement between the DIT and FE numerical results is observed, at least for early times. Close to the region of application of the external pressure, the FE simulations suffer from the discretization of the applied load, node-to-node oscillations and reflections from ‘infinite’ elements (‘silent boundaries’). Using the aforementioned numerical techniques, transient wave transmission in two-layered systems (one with weak and one with strong interlayer coupling) is considered, and the effects of weak coupling on the wave transmission is studied. We show that at early times, weak coupling results in stress localization in the region close to the applied pressure, a result which can have potential application in the use of layered media as shock isolators. © 1997 by John Wiley & Sons, Ltd.
    Additional Material: 15 Ill.
    Type of Medium: Electronic Resource
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  • 9
    Publication Date: 1995-11-01
    Print ISSN: 0938-8974
    Electronic ISSN: 1432-1467
    Topics: Mathematics , Physics
    Published by Springer
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  • 10
    Publication Date: 2017-06-23
    Description: A century-old tenet in physics and engineering asserts that any type of system, having bandwidth , can interact with a wave over only a constrained time period t inversely proportional to the bandwidth ( t · ~ 2). This law severely limits the generic capabilities of all types of resonant and wave-guiding systems in photonics, cavity quantum electrodynamics and optomechanics, acoustics, continuum mechanics, and atomic and optical physics but is thought to be completely fundamental, arising from basic Fourier reciprocity. We propose that this "fundamental" limit can be overcome in systems where Lorentz reciprocity is broken. As a system becomes more asymmetric in its transport properties, the degree to which the limit can be surpassed becomes greater. By way of example, we theoretically demonstrate how, in an astutely designed magnetized semiconductor heterostructure, the above limit can be exceeded by orders of magnitude by using realistic material parameters. Our findings revise prevailing paradigms for linear, time-invariant resonant systems, challenging the doctrine that high-quality resonances must invariably be narrowband and providing the possibility of developing devices with unprecedentedly high time-bandwidth performance.
    Keywords: Physics, Applied
    Print ISSN: 0036-8075
    Electronic ISSN: 1095-9203
    Topics: Biology , Chemistry and Pharmacology , Geosciences , Computer Science , Medicine , Natural Sciences in General , Physics
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