A simple method for solving damped Duffing oscillators

Abstract

This paper investigates the accuracy of extending He’s frequency–amplitude equation to the damped Duffing oscillator. Exact numerical solutions are compared with analytical results obtained by extending He’s equation to damped oscillators for various values of amplitude (A) and damping coefficient (c). A simulation of the damped Duffing oscillator using a novel combination of two electrical circuits is also conducted, providing a real-world example of nonlinear oscillation. The oscillation period, calculated using He’s equation, is accurate for Ao < 1, where Ao represents the initial amplitude, regardless of c. The analytic solution is accurate only for Ao < 1 and c > 0.1. For smaller damping coefficients, discrepancies arise due to slow amplitude reduction and cumulative period calculation errors over time, necessitating a correction factor. For larger damping coefficients, the system quickly approaches the harmonic damping range, resulting in minor errors. In conclusion, the limits of applicability of He’s frequency–amplitude equation for damped Duffing oscillators were determined, and appropriate modifications were introduced.