Vibrations of a Rotating Hanged String with a Heavy Tip Mass

The vibrations of a string-mass system are considered. A heavy mass is suspended from a string, and the system is subjected to axial rotation. The partial differential equation modeling the system’s dynamics is first derived. For uniform axial rotation, exact analytical solutions are given. For harmonically fluctuating rotation speeds, an approximate solution is found using the method of multiple scales to analyze the system’s principal parametric resonances. The stability of the system is examined both analytically and numerically. The analytically derived stability boundaries qualitatively match the behavior observed in numerical simulations. With some transformations, the system can also be reduced to a standard Mathieu equation. Keywords: string-mass; perturbation; rotation; principal parametric resonances; stability; Mathieu equation