Nonlinear oscillations for beginners. Calculating period using an elementary computational approach

Abstract

Determining the period of a nonlinear oscillation is a challenging task that requires a strong mathematical background in solving nonlinear differential equations. However, the procedure can be significantly simplified using the area under the 1/|υ|=f(x) graph, where υ is the velocity of the oscillating object and x is its displacement from its equilibrium position. The proposed method requires elementary computational tools and is appropriate for determining the period of any nonlinear undamped oscillation. Characteristic examples are presented, such as the simple pendulum, the oscillation with a power-law restoring force, and the cubic-quintic Duffing oscillator. The proposed approach provides accurate results and is appropriate for introductory physics and mechanics courses.