Generic numerical and analytical methods for solving nonlinear oscillators

Abstract

Nonlinear oscillations are a challenging problem due to the high complexity of the underlying differential equations. This paper focuses on the nonlinear oscillator with cubic and harmonic restoring force as it represents a very general case. In particular, many well-known oscillators (e.g., the Duffing equation, the capillary oscillator, the cubic-quintic Duffing oscillator, the simple pendulum, etc.) are subcases of it. Accurate solutions in terms of Fourier series functions for different oscillation amplitudes were derived using an energy-based numerical method and fitting processes. In addition, an analytical approach was developed by finding the derivative of the restoring force function at the point x=c0A, where A is the amplitude of the oscillation, and c0 is a constant. Using this approach, the error was negligible (~0.02−0.22%) even for A → ∞. The methods and solutions for the nonlinear oscillator with cubic and harmonic restoring force can be equally applied to all its subcases.