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

Soledad Moreno-Pulido,Alberto Sánchez-Alzola,Francisco Javier García-Pacheco,Alejandro Rincón-Casado

doi:10.3390/math9091036

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.

Highlights

This paper originates from and is motivated by a recently published manuscript [1], where the influence of different support types in the nonlinear vibrations of beams is analyzed
It is worth mentioning that we provide a mathematical proof for the validity of Straightforward Expansion Perturbation Method (SEPM) in the previously mentioned weakly nonlinear vibrations, whereas no mathematical proof is given for a general validation of HAM
After the application of the proposed methodology in this work, we obtain as a result the mathematical formalism which yields to the validity of the SEPM

Summary

Introduction

This paper originates from and is motivated by a recently published manuscript [1], where the influence of different support types in the nonlinear vibrations of beams is analyzed. During the development of this latter paper, the authors searched for a mathematical proof of the Straightforward Expansion Perturbation Method (SEPM), not reaching any manuscript containing a proper and rigorous definition and proof of the method. Nonlinear problems have been solved by perturbations methods in order to eliminate the generated secular terms. According to these techniques, the solution is represented by a few terms of an expansion, usually no more than two or three terms. The deviation between the approximate analytical solution and the exact analytical solution depends on the number of selected expansion terms and the amplitude of the vibration [2,3,4,5,6]

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