Summary
In this paper, the transverse vibrations of a one-dimensional string attached to a parabolic-shaped obstacle at one end are studied. Gravity and axial vibrations are taken into account. The changing position of the attachment point leads to a moving boundary problem. A boundary fixing transformation is applied to transform the problem to a fixed domain, leading to additional nonlinearities in the problem formulation. Different cases are studied. If the tension in the string and the gravity balance in a certain sense, then the effects of the nonlinearities become apparent, and lead to nonstandard eigenvalue problems. Otherwise, a standard multiple-time scales perturbation approach and a characteristic coordinate transformation can be applied to analyze the problem and its solution.
