A Note on a Simple Equation for Solving Nonlinear Undamped Oscillations

The objective of this paper is to simplify the procedure for determining the angular frequency of undamped nonlinear oscillations.

Firstly, an energy-based numerical method is employed to accurately determine the angular frequency of any undamped nonlinear oscillation. Subsequently, an oscillation with a polynomial-type restoring force, represented by an odd function, is introduced. The reason is that, in most cases, the restoring force can be approximated by an odd Maclaurin series function. Using a simple sinusoidal solution and the Fourier series expansion of trigonometric functions, while omitting terms higher than the first order, an approximate equation valid for small amplitudes is derived. For large oscillation amplitudes, the term raised to the highest order of power becomes dominant; therefore, the coefficients of the provided equation are slightly adjusted.

The newly derived equation was applied to the cubic-quintic Duffing oscillator, the simple pendulum, the capillary oscillator, the oscillator with cubic and harmonic restoring forces, the tangent oscillator, and the hyperbolic tangent oscillator with accuracy. The procedure involves two steps: the first one is to approximate the restoring force with a Maclaurin series function, and the second one is to apply the proposed equation.

In this paper, a new generic elementary equation is presented to determine the angular frequency of nonlinear undamped oscillations.