This dissertation addresses the generalization of rigid-body attitude kinematics, dynamics, and control to higher dimensions. A new result is developed that demonstrates the kinematic relationship between the angular velocity in N-dimensions and the derivative of the principal-rotation parameters. A new minimum-parameter description of N-dimensional orientation is directly related to the principal-rotation parameters. The mapping of arbitrary dynamical systems into N-dimensional rotations and the merits of new quasi velocities associated with the rotational motion are studied. A Lagrangian viewpoint is used to investigate the rotational dynamics of N-dimensional rigid bodies through Poincar??e??s equations. The N-dimensional, orthogonal angularvelocity components are considered as quasi velocities, creating the Hamel coefficients. Introducing a new numerical relative tensor provides a new expression for these coefficients. This allows the development of a new vector form of the generalized Euler rotational equations. An N-dimensional rigid body is defined as a system whose configuration can be completely described by an N??N proper orthogonal matrix. This matrix can be related to an N??N skew-symmetric orientation matrix. These Cayley orientation variables and the angular-velocity matrix in N-dimensions provide a new connectionbetween general mechanical-system motion and abstract higher-dimensional rigidbody rotation. The resulting representation is named the Cayley form. Several applications of this form are presented, including relating the combined attitude and orbital motion of a spacecraft to a four-dimensional rotational motion. A second example involves the attitude motion of a satellite containing three momentum wheels, which is also related to the rotation of a four-dimensional body. The control of systems using the Cayley form is also covered. The wealth of work on three-dimensional attitude control and the ability to apply the Cayley form motivates the idea of generalizing some of the three-dimensional results to Ndimensions. Some investigations for extending Lyapunov and optimal control results to N-dimensional rotations are presented, and the application of these results to dynamical systems is discussed. Finally, the nonlinearity of the Cayley form is investigated through computing the nonlinearity index for an elastic spherical pendulum. It is shown that whereas the Cayley form is mildly nonlinear, it is much less nonlinear than traditional spherical coordinates.

Thesis (Ph. D.)--University of Chicago, Dept. of Physics, 2001. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--Worcester Polytechnic Institute. / Keywords: fluid-structure interaction; low dimensional models; coupled map lattices; vortex shedding; cylinder wake patterns; flow control; multi-variable least squares algorithm; neural networks; adaptive estimation. Includes bibliographical references (p. 147-149).

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The cross-flow past a pair of equal-diameter circular cylinders, arranged in a staggered configuration, was investigated experimentally in a closed-circuit water tunnel at Reynolds numbers, based on the mean-flow velocity and the cylinder diameter, within the lower subcritical range. The wake formation process was studied employing dye-injection flow visualization and hot-film measurements. The main emphasis was placed on acquiring a physical understanding of the mechanisms leading to vortex shedding, and particularly on the effect of a forced oscillation transverse to the flow direction of either of the two cylinders. For comparison purposes, investigations were also carried out with both cylinders stationary. / Experimental results showed that, for a reasonably large angle of incidence, the flow in the wake of a stationary cylinder pair could be characterized by two distinct periodicities, each of which was dominant on one side of the wake. Furthermore, for lower Reynolds numbers (Re < 1.0x10 4), there was an integral relationship between the two Strouhal numbers, but this integral relationship was no longer maintained for Re > 1.0x10 4. On the other hand, the flow around stationary cylinders for a small angle of incidence was characterized by a single Strouhal number, which remained approximately constant over the entire Reynolds number range. / For all the cylinder configurations investigated the wake flow patterns remained essentially the same as those of the corresponding static cases, when either of the two cylinders was forced to oscillate with a nondimensional forcing frequency less than approximately 0.10. However, beyond this value, the wake underwent considerable modification vis-a-vis when the cylinders were stationary, and the flow pattern within the wake was strongly dependent on the value of the forcing frequency. In particular, there were distinct regions of synchronization between the dominant wake periodicities and the cylinder oscillation; these synchronization regions involved sub- and superharmonics as well as fundamental synchronizations. With either upstream or downstream cylinder oscillation, the wake on the mean-flow side of the downstream cylinder synchronized with the shear layers separated from its outer surface, whereas synchronizations on the mean-flow side of the upstream cylinder were caused by the periodicities formed from the interaction of the other three shear layers. / The flow phenomena associated with the synchronizations were described in detail via flow visualization. The organization of the wake was strongly dependent on whether it was the upstream or downstream cylinder which was oscillating. The synchronized wake on the mean-flow side of the downstream cylinder at both lower and higher oscillation frequencies for upstream cylinder oscillation was observed to form either by the shedding of independent vortices or by the coalescence of two or more vortices. However, for downstream cylinder oscillation, although the synchronizations on this side of the wake at lower oscillation frequencies were caused by the shedding of independent vortices or by the coalescence of vortices, those at higher oscillation frequencies were the consequence of the coalescence of vortices only. For large incidence angles, the number of shear layers separated from the downstream cylinder which interacted with those separated from the upstream cylinder was critical in causing the synchronizations on the mean-flow side of the upstream cylinder. / In most cases, the flow for all the cylinder configurations traversed between the same patterns as those obtained when the cylinders were placed stationary at their minimum and maximum transverse spacings; but there were also some situations where the oscillation of either cylinder pushed the flow outside the regimes associated with the stationary configurations. The synchronization ranges obtained when the upstream or downstream cylinder was oscillating were different from each other, and these ranges were much wider than the corresponding synchronization ranges for a single oscillating cylinder. For two cylinders, an analysis of the fundamental synchronization showed that the frequency range over which this occurred was much broader for upstream cylinder oscillation than for downstream cylinder oscillation. Also, the fundamental synchronization ranges for downstream cylinder oscillation were closer to those for single cylinder oscillation in comparison to those for upstream cylinder oscillation.

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