The paradox of the hanging string: An explanation using singular perturbations

• Published in: Journal of sound and vibration Vol. 148; no. 2; pp. 343 - 351
• Main Authors: Triantafyllou, M.S., Triantafyllou, G.S.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 22.07.1991
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/0022-460X(91)90581-4

The problem of linear transverse oscillations of an elastic perfectly flexible string, hanging freely under its own weight, presents a paradox, in that a solution can be obtained only when the lower end is free. It is shown by using singular perturbations that inclusion of bending stiffness removes the paradox; the solution, however, develops a boundary layer near the lower end to accommodate the boundary conditions. As a result, the behaviour of the solution in the limit as the bending stiffness tends to zero, is singular. Thus, for a string with a simply supported lower end, the natural frequencies asymptotically approach those of the string with a free end, whereas for prescribed motion of the lower end, the thin boundary layer near the lower end effectively alters the amplitude of imposed motion of the string. For actual structures with small bending stiffness, such as risers and drilling strings, this singular behaviour implies the development of large bending stresses near the lower end.