On Some Peculiar Aspects of Axial Motions of Closed Loops of String in the Presence of a Singular Supply of Momentum

Abstract

We consider the dynamics of a closed loop of inextensible string which is undergoing an axial motion. At each instant, one material point of the string is in contact with a singular supply of linear momentum (also known as an external constraint). Several peculiar features of this problem which have not been previously discussed are presented. These include the possible presence of an arbitrary number of kinks, the vanishing nature of the singular supply of momentum, and the critical nature of the tension in the string. When the linear momentum is supplied by a mass-spring-dashpot system, we are also able to establish an exact expression for the frequency of the resulting vibrations, prove that dissipation cannot be present, show that these vibrations only occur for discrete speeds of axial motion, and establish that Coulomb friction is absent.