Fractional Differential Equations

https://hal.archives-ouvertes.fr/hal-01357931/document

Nonlinear normal modes, Part I: A useful framework for the structural dynamicist

Mathematical models of elastic structures have become very sophisticated: given the crucial material properties (mass density and the several elastic moduli), computer-based techniques can be used to construct exotic finite element models. By contrast, the modeling of damping is usually very primitive, often consisting of no more than mere guesses at “modal damping factors.” The aim of this paper is to raise the modeling of viscoelastic structures to a level consistent with the modeling of elastic structures. Appropriate material properties are identified which permit the standard finite element formulations used for undamped structures to be extended to viscoelastic structures. Through the use of “dissipation” coordinates, the canonical “M , K ” form of the undamped motion equations is expanded to encompass viscoelastic damping. With this formulation finite element analysis can be used to model viscoelastic damping accurately.

Dynamics of viscoelastic structures: a time-domain finite element formulation

Abstract : It is shown that Galerkin's procedure is always available for twice continuously differentiable periodic differential systems if a periodic solution is restricted to an isolated periodic solution. This gives an explicit condition for applying Galerkin's procedure and reveals the wide applicability of Galerkin's procedure to general nonlinear periodic differential systems. The proof of this result is based on existence theorems which are derived from idea contained in Newton's iterative method. The relationship of Galerkin's procedure to the method of averaging is also discussed. Finally the paper exemplifies Galerkin's procedure with a certain nonlinear equation. (Author)

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Galerkin's Procedure for Nonlinear Periodic Systems (関数方程式の近似解法研究会報告集)

The goal of this paper is to review current methods of energy harvesting, while focusing on piezoelectric energy harvesting The piezoelectric energy harvesting technique is based on the materials' property of generating an electric field when a mechanical force is applied This phenomenon is known as the direct piezoelectric effect Piezoelectric transducers can be of different shapes and materials, making them suitable for a multitude of applications To optimize the use of piezoelectric devices in applications, a model is needed to observe the behavior in the time and frequency domain In addition to different aspects of piezoelectric modeling, this paper also presents several circuits used to maximize the energy harvested

/pdf/piezoelectric-energy-harvesting-solutions-a-review-449lsu70t1.pdf

Piezoelectric Energy Harvesting Solutions: A Review.

http://www.ikb.poznan.pl/roman.lewandowski/Passive/jabartosz.pdf

Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers

http://www.ikb.poznan.pl/roman.lewandowski/Passive/YJSVI10314.pdf

Roman Lewandowski

Papers

Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers

Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractionalderivatives

Computational formulation for periodic vibration of geometrically nonlinear structures—part 1: Theoretical background

Computational formulation for periodic vibration of geometrically nonlinear structures—part 2: Numerical strategy and examples

Application of the Ritz method to the analysis of non-linear free vibrations of beams