Approximate Solutions for Undamped Nonlinear Oscillations Using He’s Formulation

Stylianos Vasileios Kontomaris (), Georgios Chliveros and Anna Malamou
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Stylianos Vasileios Kontomaris: Faculty of Engineering and Architecture, Metropolitan College, 15125 Athens, Greece
Georgios Chliveros: Faculty of Engineering and Architecture, Metropolitan College, 15125 Athens, Greece
Anna Malamou: Radar Systems and Remote Sensing Lab, School of Electrical & Computer Engineering, National Technical University of Athens, 15780 Athens, Greece

J, 2023, vol. 6, issue 1, 1-12

Abstract: Solving nonlinear oscillations is a challenging task due to the mathematical complexity of the related differential equations. In many cases, determining the oscillation’s period requires the solution of complicated integrals using numerical methods. To avoid the complexity, there are many empirical equations in the literature that can be used instead of rigorous mathematical analysis to provide an acceptable approximation. In this paper, a recently developed method, He’s formulation, is applied to find the period in many different cases of nonlinear oscillators. The cases are those of the Duffing equation, the Helmholtz nonlinear oscillator, the simple pendulum and the case of a vertical oscillation under the influence of a nonlinear elastic force. The results of the method are accurate; thus, He’s formulation is a strong tool for solving nonlinear oscillations.

Keywords: He’s approximation; vibrations; restoring force; nonlinear differential equations (search for similar items in EconPapers)
JEL-codes: I1 I10 I12 I13 I14 I18 I19 (search for similar items in EconPapers)
Date: 2023
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