Donaldson's single-point second-order model [13] is used to close the Reynolds stress transport equations in cylindrical coordinates. A reduced set of equations are then solved for the decay of axisymmetric vortices and jets. A self-similar solution to the axisymmetric vortices is obtained numerically. The characteristics of the mean flow variables as well as the Reynolds stresses in this solution are discussed. Comparisons of the current results with Donaldson[13J and Donaldson and Sullivan[16] are also presented. The results show that the vortex core is free from turbulent shear stresses. The turbulent kinetic energy is also found to be relatively weak within the core region. The overshoot of the circulation is found to be 5% of the circulation at infinity over a wide range of Reynolds numbers. The effects of Reynolds number on the decay of the vortices are computed and discussed. Some of the quantities, such as mean flow circulation and turbulent kinetic energy, are found to be sensitive to the Reynolds number. However, the overshoot is found to be insensitive to the Reynolds number but its location does. A set of suitable model constants for the axisymmetric jets is also found and a self similar solution for the jet case is obtained. Comparisons of the computed results with some recent experimental data are presented. / Master of Science

A novel method of reducing the spin of a rotating symmetric body, similar to many earth orbit satellites, is by allowing small, despin weights to unwind from about the satellite so that they absorb some, or all, of the satellite angular momentum. This technique which has been used successfully on several U.S. satellites is commonly referred to as yo-yo despin. Several studies of the motion of a system such as this have been published where it was assumed that the motion was two-dimensional (i.e., without coning). This dissertation presents a comprehensive study of the yo-yo despin problem which includes a derivation of two-dimensional results as well as a three-dimensional or exact solution. The results presented are sufficient for rudimentary design computations and provide examples of the corrections necessary to apply to two-dimensional computations for their applications as estimates for the general motion. An approximate solution of the three-dimensional equations of motion is also presented along with an example of the accuracy obtained by the approximation. The equations of motion are derived in a straightforward manner using the vectorial methods of Newtonian mechanics. The Euler equations for a rigid body are used to describe the motion of the rigid body itself. The moment acting on the body through the tension in the yo-yo cables is unknown and it is necessary to apply the second law of Newton to a despin weight so that sufficient independent differential equations are available for the solution of the problem variables. These relations give three first-order differential equations and two second-order ones. An expression for the cable tension is also obtained. This system of equations is integrated numerically by a standard Runge-Kutta process. Two singularities require special attention: first, at the initial instant the fundamental inversion matrix for the Newton equations is singular; and second, special care must be taken at a point in the integration where a discontinuity is found to occur. Outside of these special points, the integration process is quite routine although some cases require precautions near the end of the despinning process in order that the integration is stopped before violent tumbling occurs. In order to discuss the motion relative to a fixed reference axis the Euler angles, and Euler angle rate equations are also integrated. A point of interest concerning the derivation of the equations of motion is that the Lagrange technique cannot be used without modification due to internal constraints which do work. After a numerical study of several typical examples, one concludes that for initial coning angles of less than 10° a two-dimensional analysis is sufficient for determining many important design variables such as maximum cable tension and despin time, although the cable length is somewhat overestimated and problems may occur in the release of the weights if only a two-dimensional analysis is considered. If one desires information on the angular trajectory of the body in inertial coordinates, a study of the problem must be made using the exact three-dimensional relations or the approximate three-dimensional relations. The approximate expressions save the investigator a great deal of effort and apparently provides excellent results. / Ph. D.

A vortex-panel method for potential flow is used as a basis for modeling surface effects and creating a finite-element interface so that an arbitrary body can be analyzed. The basic model consists of triangular panels of linearly varying vorticity which represent the body, vortex cores on the lifting edges of the body, and vortex filaments representing the wake. The interface modification is made by using a finite-element application's output as the basis for an input file for the model, executing the main program, and writing body and wake output readable by the finite-element application. The surface-effect modification is made by including an image of the body below the real body to create a surface boundary condition through symmetry. / Master of Science

When a mechanical system is subject to constraints its motion is in some way restricted. In accordance with Newton's second law, motion is a direct result of forces acting on a system; hence, constraint is inextricably linked to force. The presence of a constraint implies the application of particular forces needed to compel motion in accordance with the constraint; absence of a constraint implies the absence of such forces. The objective of this thesis is to formulate a comprehensive, consistent, and concise method for identifying a set of forces needed to constrain the behavior of a mechanical system modeled as a set of particles and rigid bodies. The goal is accomplished in large part by expressing constraint equations in vector form rather than entirely in terms of scalars. The method developed here can be applied whenever constraints can be described at the acceleration level by a set of independent equations that are linear in acceleration. Hence, the range of applicability extends to servo-constraints or program constraints described at the velocity level with relationships that are nonlinear in velocity. All configuration constraints, and an important class of classical motion constraints, can be expressed at the velocity level by using equations that are linear in velocity; therefore, the associated constraint equations are linear in acceleration when written at the acceleration level. Two new approaches are presented for deriving equations governing motion of a system subject to constraints expressed at the velocity level with equations that are nonlinear in velocity. By using partial accelerations instead of the partial velocities normally employed with Kane's method, it is possible to form dynamical equations that either do or do not contain evidence of the constraint forces, depending on the analyst's interests.

Inner ear hair cells have been identified as the sites of mechanoelectrical transduction from a mechanical event (e.g. hearing, motion) to an electrical event (e.g. neural response). Deflection of bundles of hair-like stereocilia extending from these cells has been associated with the transduction process. Stereocilia bundle structure and stiffness controls deflection and thus the fundamental sensitivity of the transduction process. The finite-element method was used along with analytical techniques to characterize individual stereocilium and stereocilia bundle stiffnesses. A three ‘stack’ bundle with a Young’s modulus of 3 GPa (F-actin protein) and Poisson’s ratio of 0.4 (nearly incompressible) resulted in a stiffness of K = 2.1 x 10⁻³ N/m. This value is within the range of experimentally determined stiffmesses. Tip-link and subapical band interconnecting structures each contribute significantly to bundle stiffness and each could act as the gating-spring in transduction models, which propose gating structures as a means of regulating ionic activity and therefore neural activity. Stiffness depends most strongly on individual stereocilium geometry and material description, tip-link orientation and material description, and stereocilia bundle width. Stiffness depends least on stereocilia height variations and subapical bands configuration. Linear analysis was reliable up to deflections of 3.5 um, the upper limit of physical response. Preliminary dynamic response indicates a natural frequency of 382 kHz for the vibration mode resembling physical deformation behavior. Future models should include hexagonal bundle arrangements, transversely isotropic stereocilia material descriptions, and viscoelastic tip-link behavior. / Master of Science

Thesis (Ph.D.)--Joint Program in Physical Oceanography (Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences and the Woods Hole Oceanographic Institution), 2000. / Includes bibliographical references (p. 284-290). / by Brian Kenneth Arbic. / Ph.D.