Expansion Perturbation Method for Nonlinear Vibrations

This paper originates from and is motivated by a recently published manuscript, where the influence of various support types within the nonlinear vibrations of beams is analyzed. During the event of this latter paper, the authors looked for a proof of the simple Expansion Perturbation Method (SEPM), not reaching any manuscript containing a correct and rigorous definition and proof of the tactic. The most objective of this paper is to supply a mathematical formalism of SEPM. Currently, an outsized number of nonlinear vibration problems in Engineering are solved by the Nonlinear Finite Element Method. However, in many cases, it's necessary to seek out an analytical solution so as to raised understand the contribution of forces, masses or geometries. within the process of checking out an analytical solution, hypotheses, simplifications and linearizations are raised, which usually cause approximations of the precise analytical solutions. Traditionally, nonlinear problems are solved by perturbations methods so as to eliminate the generated secular terms. consistent with these techniques, the answer is represented by a couple of terms of an expansion, usually no quite two or three terms. Therefore, the deviation between the approximate analytical solution and therefore the refore the exact analytical solution depends on the amount of selected expansion terms and the amplitude of the vibration.