Abstract
The present paper focuses on nonlinear oscillations of a horizontally supported Jeffcott rotor. An approximate solution to the system of governing equations having quadratic and cubic nonlinearities is obtained in two cases of practical interest: simultaneous and internal resonance. The Optimal Auxiliary Functions Method is employed in this study, and each governing differential equation is reduced to two linear differential equations using the so-called auxiliary functions involving a moderate number of convergence-control parameters. Explicit analytical solutions are obtained for the first time in the literature for the considered practical cases. Numerical validations proved the high accuracy of the proposed analytical solutions, which may be used further in the study of stability and in the design process of some highly performant devices.
Highlights
Approximate Analytical Solutions toThe nonlinear dynamics of rotors have long attracted attention, being an interesting subject with considerable technical depths and breadths
Karlberg and Aidanpää [3] considered the nonlinear vibrations of a rotor system with clearance, analyzing the two-degree-of-freedom unbalanced shaft in relation to a non-rotating massless housing
The Optimal Auxiliary Functions Method (OAFM) is used in the present study to obtain a first-order approximate analytical solution to governing nonlinear equations with quadratic and cubic nonlinearities in two cases: simultaneous and internal resonance
Summary
The nonlinear dynamics of rotors have long attracted attention, being an interesting subject with considerable technical depths and breadths. The objective of this article is to apply a new and accurate approach to nonlinear differential equations governing the oscillations of a horizontally supported Jeffcott rotor, namely the Optimal Auxiliary Functions Method (OAFM). The OAFM is used in the present study to obtain a first-order approximate analytical solution to governing nonlinear equations with quadratic and cubic nonlinearities in two cases: simultaneous and internal resonance. Our technique does not imply the presence of a small or large parameter the governing equations, or the boundary/initial conditions, and can be applied to a variety of engineering domains. The validity of this original method is proved by comparing the results with numerical results. Further in the study of stability, and in the design process of some highly performant devices
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