Ninth Annual Graduate Student Miniconference in Computational Mathematics
Jalil Hasanyan
Virginia Tech http://www.math.vt.edu/people.php?type=Graduates&pid=jalil633 

Abstract 
Stability of a moving current carrying conductive string in a magnetic field
A wide range of physical systems may be modeled as axially moving strings such as belts, tapes, wires and fibers in an applied fields of electromagnetic origin. Such systems have broad applications in chemical and textile industries or in space applications. Regarding specific applications, the string system can be used for load transportation, like that of space elevators. For example, a mode of space elevators would include conductive carbon nanotubes which can be treated like a electrically conductive string system in an applied electromagnetic fields. The Stability of lateral vibrations of a moving currentcarrying string in a homogeneous magnetic field is investigated. It is assumed that the string is moving with a constant velocity between two rings which are at finite distance from each other. Directions of the magnetic field and the motion of the string coincide. The problem is first considered in a static setting. A critical value of the magnetic field is shown to exists when the uniform (along a string line) motion of a string is unstable. The critical value depends on the speed of motion, applied magnetic field, and current. In the dynamic setting, the condition for stability is also discussed. It is shown that there is a divergence between the critical values in the dynamic and static cases. Nonlinear resonance vibrations of a moving string are investigated when a periodic nonstationary current flows through the string. For an approximate solution, the method of multiple scales is applied. Domains of parameters are defined when the string falls into a prechaotic state (i.e. the frequency of vibrations is doubled. The stability of the system with a periodic magnetic field is also considered. 