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

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