Planar string vibrations in the presence of smooth curved boundary obstacles at both ends

Abstract

This research investigates the planar vibrations of a string with smooth unilateral obstacles placed at both ends. The equation of motion for the system has been derived using the Lagrangian framework. The presence of obstacles at the ends introduces a moving boundary resulting from the wrapping and unwrapping of the string, hence increasing the complexity of the study. It is necessary to implement an appropriate scaling technique on the spatial domain to address the issue of moving boundaries. This scaling method effectively transforms the moving boundary problem into a fixed boundary problem. However, this transformation introduces nonlinearity into the governing equation. The linear vibration characteristics of the system are investigated through the process of linearizing the governing equation around the static configuration of the string. This analysis reveals the harmonic nature of the natural frequencies and their relationship to the geometry of the obstacle. The Galerkin projection method is employed to study the modal interactions, which exhibit significant properties such as amplitude and frequency modulations. A general expression for the reduced system of ordinary differential equations has been developed in such a way that the modal interactions can be studied for a large number of modes. Furthermore, there have been critical observations on the impact of the relative curvature and size of the two obstacles on the modulation frequency and modal interactions. We find that the amplitude modulation frequency due to modal interactions can be tuned depending on the relative size of the two obstacles.