Titlebook: Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems; Vasile Marinca,Nicolae Herisanu,Bogdan Marinca Book 2021 The Editor(s)

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IntroductionA nonlinear system is a set of nonlinear equations—differential, integral, functional, algebraic, difference, or abstract operator equations, or a combination of some of these—used to describe a physical device or process that otherwise cannot be clearly defined by a set of linear equations of any kind.

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The Optimal Auxiliary Functions MethodTo apply the Optimal Auxiliary Functions Method (OAFM), we consider the following general nonlinear differential equation.

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The First Alternative of the Optimal Auxiliary Functions MethodIn this chapter, we will actually solve the Eq. (2.13) from which the first approximation . can be determined.

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Free Oscillations of Euler–Bernoulli Beams on Nonlinear Winkler-Pasternak FoundationThe use of beams of an elastic foundation has recently become widespread in engineering. Several research papers have appeared in literature on this topic. Horibe and Asano proposed a boundary integral equation method for calculating the large deflection of beams on an elastic foundation of the Pasternak type [1].

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Free Vibration of Tapered BeamsTapered beams can model engineering structures which require a variable stiffness along the length, such as moving arms and turbine blades [.,.,.], or can be modeled as a slender, flexible cantilever beam carrying a lumped mass with rotary inertia at an intermediate point along its span hence it exhibits large-amplitude vibrations [., .].