Convergence Analysis of the Straightforward Expansion Perturbation Method for Weakly Nonlinear Vibrations

Soledad Moreno-Pulido, Francisco Javier García-Pacheco, Alberto Sánchez-Alzola and Alejandro Rincón-Casado
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Soledad Moreno-Pulido: Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, CA, Spain
Francisco Javier García-Pacheco: Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, CA, Spain
Alberto Sánchez-Alzola: Department of Statistics and Operation Research, College of Engineering, University of Cadiz, 11519 Puerto Real, CA, Spain
Alejandro Rincón-Casado: Department of Mechanics, College of Engineering, University of Cadiz, 11519 Puerto Real, CA, Spain

Mathematics, 2021, vol. 9, issue 9, 1-16

Abstract: There are typically several perturbation methods for approaching the solution of weakly nonlinear vibrations (where the nonlinear terms are “small” compared to the linear ones): the Method of Strained Parameters , the Naive Singular Perturbation Method , the Method of Multiple Scales , the Method of Harmonic Balance and the Method of Averaging . The Straightforward Expansion Perturbation Method (SEPM) applied to weakly nonlinear vibrations does not usually yield to correct solutions. In this manuscript, we provide mathematical proof of the inaccuracy of the SEPM in general cases. Nevertheless, we also provide a sufficient condition for the SEPM to be successfully applied to weakly nonlinear vibrations. This mathematical formalism is written in the syntax of the first-order formal language of Set Theory under the methodology framework provided by the Category Theory.

Keywords: numerical analysis; approximation theory; nonlinear vibration; perturbation method; Banach space; unitary algebra; sup norm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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