Energy-Based Criterion for Testing the Nonlinear Response Strength of Strong Nonlinear Oscillators

Abstract

This article proposes a simple energy-based criterion developed to characterize four commonly identified responses, namely: linear, weakly nonlinear, moderately nonlinear and strongly nonlinear regimes. The response of the nonlinear simple pendulum was used for benchmarking the boundary conditions for each of the four response regimes and the test criterion was demonstrated using relevant examples. The test presented in this article is important for clarifying the obscurity surrounding the accuracy and range of validity of recent approximate analytical schemes used to investigate strong nonlinear oscillators. Furthermore, it is meant to create awareness of the need to develop more robust testing criteria.
 Keywords: strong nonlinear oscillation; periodic oscillation; approximate analytical solution

Estimated the frequencies of a coupled damped nonlinear oscillator with the non-Perturbative method

“…[23][24][25][26][27] Originally, this theme was proposed by J.H.He and his co-authors as an essential possibility for obtaining the conservative frequency-amplitude relationship of the Duffing oscillator and its family. [28][29][30][31][32][33][34] He's frequency formula is used to derive the frequency-amplitude relationship of the nonlinear equation, and the approximate analytical solution expression is given. However, in many cases, it is possible to compute accurate approximate analytical solutions of the equations.…”

Generic numerical and analytical methods for solving nonlinear oscillators

“…Thus, for large oscillation amplitudes,  A 5 m (even if A  ¥), equation ( 49) is the best choice to calculate the oscillation's period. However, in case of equation (49) the constant c 0 differs compared to the case of equation (40) for big oscillation's amplitudes. The reason is that for large oscillation's amplitudes, the term x 5 dominates completely.…”

“…However, the case in which k 3 = was preferred because it is simple and provides a good approximation regardless of the oscillation's amplitude. Additionally, it is important to note that a simple energy-based criterion was employed to categorize the tested cases as linear, weakly nonlinear, moderately nonlinear, and strongly nonlinear [49]. The criterion is based on calculating the actual maximum potential energy using equation (8) for a given amplitude U max ( ) and the maximum potential energy of an equivalent linear oscillator, given by U linear…”

“…For a linear response,  e 0.25%; r for a weak nonlinear response 0.25% < e r 2.5%; for a moderate nonlinear response 2.5% < e r 25%; and for a strong nonlinear response e r > 25% [49]. The results obtained using the numerical method (black dots) and the fitted curve to equation (14).…”