An initial moving-boundary value problem associated with the spinning wave equation.

摘要

In this thesis we consider a mathematical model for a spinning string with specified initial-conditions and general Dirichlet-type boundary-conditions specified at the ends of a string, which are allowed to move in a prescribed manner. The resulting problem is known as an initial moving-boundary value problem and represents a model for the dynamics of a tether connecting two separating rocket payload components, which have been spin-stabilized prior to separation.;Two distinct versions of this problem are considered: (1) The "homogenous" problem occurs when the endpoints of the string are allowed to move only along the system's longitudinal axis (i.e. no transverse displacements are allowed). (2) The "non-homogeneous" problem allows the endpoints to experience transverse displacements from the longitudinal axis during the deployment of the tether.;The solution of each of these problems involves a number of transformations of variables, which reduce the system of coupled partial differential equations to the standard hypergeometric ordinary differential equation. In the homogeneous case, the resulting solution is obtained through the use of a (complex) Fourier exponential series expansion. Two illustrative examples are constructed. In the non-homogeneous case, the introduction of the Laplace transform gives rise to a "theoretical" solution whose practicality is questionable. An example illustrating this situation is completed using the method of d'Alembert.;Although much of this discussion allows for a general separation scheme of the two payloads, all of the examples assume that the end points separate at a constant velocity.