Bifurcations of whirling motions of an elastic string with fixed ends

This thesis is an investigation of the steady rotational motions of an elastic string model. In our approach we regard the steady rotational motions as relative equilibria, i.e. dynamic orbits coinciding with one-parameter orbits of the group of symmetries. Following the theory of relative equilibria, developed in its modern form by Smale [Sma70], we determine the configurations of relative equilibria as the critical points of an augmented potential energy. For the whirling string, the potential energy is augmented by the centrifugal potential energy, which is parameterized by the square of the rotational speed. We use local bifurcation theory to determine the existence of nontrivial rotational motions bifurcating from the rest state. More precisely, we apply Liapunov-Schmidt reduction and the equivariant branching lemma to determine the rotational speeds at which nontrivial rotational motions branch from the rest state. This approach towards the investigation of the bifurcations from the rest state is more systematic than the ad hoc method used in Caughey [Cau70].; We investigate the nonlinear stability of the rest state of the string. We first demonstrate that standard techniques fail to show that the energy function is a Liapunov function at this equilibrium. We define a notion of stability that incorporates the conditions under which Caughey determines “orbital stability” of the steady rotational motions of the string. Then we use this notion of stability to sharpen Caughey's conditional stability result for the rest state of the string.; We also analyze the steady rotational motions for a discrete analogue of the elastic string; a point mass attached to two massless springs. After identifying the relative equilibria of this system, we use the reduced energy-momentum method of Simo et al [SLM91] and Lewis [Lew92] to conclude that these steady rotational states are orbitally stable.