Interaction Between Periodic Elastic Waves and Two Contact Nonlinearities

Junca, Stéphane; Lombard, Bruno

doi:10.1142/s0218202511500229

Cited by 12 publications

(19 citation statements)

References 32 publications

(42 reference statements)

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“…Such systems are mathematically more complex than ordinary differential equations governing, for example, the behaviour of reed instruments. In principle, the harmonic balance method can be used to solve this kind of systems [21]. However, the current work does not aim to develop from scratch a new software package specifically dedicated to the flute model, but rather to take advantage of existing validated numerical tools.…”

Section: Introductionmentioning

confidence: 99%

Calculation of the Steady-State Oscillations of a Flute Model Using the Orthogonal Collocation Method

Terrien¹,

Vergez²,

Fabre³

et al. 2014

Acta Acustica united with Acustica

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In this paper we exploit the method of numerical continuation combined with orthogonal collocation to determine the steady-state oscillations of a model of flute-like instruments that is formulated as a nonlinear neutral delay differential equation. The delay term in the model causes additional complications in the analysis of the behaviour of the model, in contrast to models of other wind instruments which are formulated as ordinary differential equations. Fortunately, numerical continuation provides bifurcation diagrams that show branches of stable and unstable static and periodic solutions of the model and their connections at bifurcations, thus enabling an in depth analysis of the global dynamics. Furthermore, it allows us to predict the thresholds of the different registers of the flute-like instrument and thus to explain the classical phenomenon of register change and the associated hysteresis.

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Section: Introductionmentioning

confidence: 99%

Calculation of the Steady-State Oscillations of a Flute Model Using the Orthogonal Collocation Method

Terrien¹,

Vergez²,

Fabre³

et al. 2014

Acta Acustica united with Acustica

View full text Add to dashboard Cite

show abstract

“…The asymptotically stable region a > |b| in the plane of parameters (a, b) is exactly the region given by theorem 1, i.e. the energy method is optimal for the linear case to get the asymptotically stable region; 2. a = |b|: the stability follows from the same arguments as in the case a > |b|: energy method: theorem 1, and a study of the characteristic roots as in [14]. Moreover, crossing this edge, all real parts become positive in b < −|a|.…”

Section: On the Four Edges One Hasmentioning

confidence: 95%

“…Without assumptions (3) in corollary 2, one may encounter more complex situations, with 2 or more solutions. Note that these assumptions are satisfied in the physically-relevant situation examined in [14].…”

Section: Stability With Constant Source Termmentioning

confidence: 98%

“…Equation (1) is typically issued from hyperbolic partial differential equations with nonlinear boundary conditions [11,7]. A closely related system of neutral equations has been derived, for instance, in the case of elastic wave propagation across two nonlinear cracks [14]: y is then the dilatation of the crack, the shift 1 is the normalized travel time between the cracks, f and g denote the nonlinear contact law, and s is the T -periodic excitation. A natural issue when dealing with (1) is to prove existence, uniqueness and stability of periodic solutions.…”

Section: Introductionmentioning

confidence: 99%

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Stability of a critical nonlinear neutral delay differential equation

Junca

Lombard

2014

Journal of Differential Equations

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This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about the asymptotic stability of the constant and periodic solutions. A complete analysis of the stability diagram is given. Lastly, existence of periodic solutions is discussed, involving a Diophantine condition on the period.

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“…Using this model, Yan et al [49], [50] studied the second-harmonic generation at a kissing bond on one of the double interfaces between an adhesive layer and two aluminum blocks. The wave interaction with double nonlinear spring-type interfaces was also analyzed by Junca and Lombard [51].…”

Section: Introductionmentioning

confidence: 99%

Second-harmonic generation in a multilayered structure with nonlinear spring-type interfaces embedded between two semi-infinite media

2018

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The acoustic second-harmonic generation behavior in a multilayered structure with nonlinear spring-type interlayer interfaces is analyzed theoretically to investigate the frequency dependence of second-harmonic amplitudes in the reflected and transmitted fields when the structure is subjected to the normal incidence of a monochromatic longitudinal wave. The multilayered structure consists of identical linear elastic layers and is embedded between two identical linear elastic semi-infinite media. The layers are bonded to each other by spring-type interfaces possessing identical linear stiffness but different quadratic nonlinear parameters. By combining a perturbation analysis with the transfer-matrix method, analytical expressions are derived for the second-harmonic amplitudes of the reflected and transmitted waves. The second-harmonic amplitudes due to a single nonlinear interface are shown to vary remarkably with the fundamental frequency, reflecting the pass and stop band characteristics of the Bloch wave in the corresponding infinitely extended layered structure. By calculating the spatial distribution of second-harmonic amplitude inside the multilayered structure, the influence of the position of the nonlinear interface as well as the number of layers on the frequency dependence of second-harmonic amplitudes of the reflected and transmitted waves is elucidated. When all interlayer interfaces possess the identical nonlinearity, the second-harmonic amplitudes on both sides of the structure are shown to increase monotonically with the number of layers in the frequency ranges where both fundamental and double frequencies are within the pass bands of Bloch wave. The influence of two non-dimensional parameters, i.e., the relative linear compliance of the interlayer interfaces and the acoustic impedance ratio between the layer and the surrounding semi-infinite medium, on the second-harmonic amplitudes is elucidated.

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Interaction Between Periodic Elastic Waves and Two Contact Nonlinearities

Cited by 12 publications

References 32 publications

Calculation of the Steady-State Oscillations of a Flute Model Using the Orthogonal Collocation Method

Calculation of the Steady-State Oscillations of a Flute Model Using the Orthogonal Collocation Method

Stability of a critical nonlinear neutral delay differential equation

Second-harmonic generation in a multilayered structure with nonlinear spring-type interfaces embedded between two semi-infinite media

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