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Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity

Olaf Klein (2004)

Applications of Mathematics

The asymptotic behaviour for t of the solutions to a one-dimensional model for thermo-visco-plastic behaviour is investigated in this paper. The model consists of a coupled system of nonlinear partial differential equations, representing the equation of motion, the balance of the internal energy, and a phase evolution equation, determining the evolution of a phase variable. The phase evolution equation can be used to deal with relaxation processes. Rate-independent hysteresis effects in the strain-stress...

Conditions for periodic vibrations in a symmetric n-string

Claude Gauthier (2008)

Open Mathematics

A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.

Eliciting harmonics on strings

Steven J. Cox, Antoine Henrot (2008)

ESAIM: Control, Optimisation and Calculus of Variations

One may produce the qth harmonic of a string of length π by applying the 'correct touch' at the node π / q during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their 'touch' is a damper of magnitude b concentrated at π / q . The 'correct touch' is that b for which the modes, that do not vanish at π / q , are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree q - 1 ....

Energy of the harmonics in a vibrating string after the impact of a hammer

Franco Rampazzo (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questa Nota vengono usati alcuni risultati precedentemente ottenuti - v. [4] e [5] - riguardanti l'urto di un martelletto rigido e di una corda elastica. Da essi possono dedursi le condizioni della corda - deformazione e atto di moto - all'istante in cui essa rimane libera dall'influenza del martelletto. È dunque possibile determinare mediante l'analisi di Fourier, i valori delle energie delle varie armoniche, i quali, com'è ben noto, determinano il timbro del suono emesso dalla corda (timbro...

Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping

Abderrahmane Zaraï, Nasser-eddine Tatar (2010)

Archivum Mathematicum

A viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping is considered. Using integral inequalities and multiplier techniques we establish polynomial decay estimates for the energy of the problem. The results obtained in this paper extend previous results by Tatar and Zaraï [25].

Ingham-type inequalities and Riesz bases of divided differences

Sergei Avdonin, William Moran (2001)

International Journal of Applied Mathematics and Computer Science

We study linear combinations of exponentials e^{iλ_nt} , λ_n ∈ Λ in the case where the distance between some points λ_n tends to zero. We suppose that the sequence Λ is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L^2 (0,T). Here we prove that if the upper uniform density of Λ is less than T/(2π), the family of divided differences can...

On a nonlinear equation of the vibrating string

Angela Iannelli, Giovanni Prouse, Alessandro Veneziani (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A nonlinear model of the vibrating string is studied and global existence and uniqueness theorems for the solution of the Cauchy-Dirichlet problem are given. The model is then compared to the classical D'Alembert model and to a nonlinear model due to Kirchhoff.

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