Nonlinear effects in vibrating strings observed through numerical analysis*

This study explores the nonlinear effects in vibrating strings through numerical methods, specifically the Runge-Kutta (RK45) method. Nonlinear phenomena in vibrating strings are of significant interest due to their diverse applications in various fields such as engineering, physics, and music. By employing the RK45 numerical method, using MATLAB programing language, this research aims to accurately model and analyze the behavior of freely vibrating strings as well as vibrations under the influence of external driving force. The results are then compared to approximate analytical models. The study investigates the influence of frequency of external force, on the nonlinear behavior of vibrating strings. Transverse waves on a string exhibit an elliptical pattern. That pattern is precessing with time.  By further investigation, a nonlinear resonance curve for amplitude vs. external angular frequency is formed. Through comprehensive simulations and analysis, insights into the complex dynamics of nonlinear vibrating strings are gained, contributing to a deeper understanding of their behavior and potential applications in the scientific and musical realm.