A novel analytical approximation approach for strongly nonlinear oscillation systems based on the energy balance method and He's Frequency-Amplitude formulation

Document Type : Research Paper


Department of Civil and Environmental Engineering, Polytechnic University of Milan, Italy.


Nonlinear oscillations are an essential fact in physical science, mechanical structures, and other engineering problems. Some of the popular analytical solutions to analyze nonlinear differential equations governing the behavior of strongly nonlinear oscillators are the Energy Balance Method (EBM), and He's Amplitude Frequency Formulation (HAFF). The lack of precision and accuracy despite needing several computational steps to resolve the system frequency is the main demerit of these methods. This research creates a novel analytical approximation approach with a very efficient algorithm that can perform the calculation procedure much easier and with much higher accuracy than classic EBM and HAFF. The presented method's steps rely on Hamiltonian relations described in EBM and the de ned relationship between frequency and amplitude in HAFF. This paper demonstrates the substantial precision of the presented method compared to common EBM and HAFF applied in different and well-known engineering phenomena. For instance, the approximate solutions of the equations govern some strongly nonlinear oscillators, including the two-massspring systems, buckling of a column, and duffing relativistic oscillators are presented here. Subsequently, their results are compared with the Runge-Kutta method and exact solutions obtained from the previous research. The proposed novel approach resultant error percentages show an excellent agreement with the numerical solutions and illustrate a very quickly convergent and more precise than mentioned typical methods.


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