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John W. M. Bush Aristoff, Jeffrey Michael Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Department of Mathematics 2010-04-28T17:16:30Z 2010-04-28T17:16:30Z 2009 2009 http://hdl.handle.net/1721.1/54661 606901132 Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. Cataloged from PDF version of thesis. Includes bibliographical references (p. 117-123). This thesis has two parts. In Part I, we present the results of a combined experimental and theoretical investigation of the vertical impact of spheres on a water surface. Particular attention is given to characterizing the shape of the resulting air cavity in the limit where cavity collapse is strongly influenced by surface tension. A parameter study reveals the dependence of the cavity structure on the governing dimensionless groups. A theoretical description is developed to describe the evolution of the cavity shape and yields an analytical solution for the pinch-off time and depth. We also examine low-density spheres that decelerate substantially following impact, and characterize the deceleration rate and resulting change in behavior of the associated water-entry cavities. Theoretical predictions compare favorably with our experimental observations. Finally, we present a theoretical model for the evolution of the splash curtain formed at high speeds, and couple it to the underlying cavity dynamics. In Part II, we present the results of a combined experimental and theoretical investigation of the motion of a sphere on an inclined flexible beam. A theoretical model based on Euler-Bernoulli beam theory is developed to describe the dynamics, and in the limit where the beam reacts instantaneously to the loading, we obtain exact solutions for the sphere trajectory and descent time. For the case of an initially horizontal beam, we calculate the period of the resulting oscillations. Theoretical predictions compare favorably with our experimental observations in this quasi-static regime. Inertial effects are also addressed. (cont.) The time taken for descent along an elastic beam, the elastochrone, is shown to always exceed the classical brachistochrone, the shortest time between two points in a gravitational field. by Jeffrey Michael Aristoff. Ph.D. 123 p. eng Massachusetts Institute of Technology M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 Mathematics. On falling spheres : the dynamics of water entry and descent along a flexible beam Dynamics of water entry and descent along a flexible beam Thesis application/pdf Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="7482f4a8-cf08-49fd-895b-f0dfebdc3301"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Language>eng</Language> <Title>On falling spheres : the dynamics of water entry and descent along a flexible beam</Title> <Subtitle>Dynamics of water entry and descent along a flexible beam</Subtitle> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2009</PublicationDate> <Authors> <Author> <DisplayName>Aristoff, Jeffrey Michael</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://dspace.mit.edu/handle/1721.1/7582</License> <Keyword>Mathematics.</Keyword> <Abstract>This thesis has two parts. In Part I, we present the results of a combined experimental and theoretical investigation of the vertical impact of spheres on a water surface. Particular attention is given to characterizing the shape of the resulting air cavity in the limit where cavity collapse is strongly influenced by surface tension. A parameter study reveals the dependence of the cavity structure on the governing dimensionless groups. A theoretical description is developed to describe the evolution of the cavity shape and yields an analytical solution for the pinch-off time and depth. We also examine low-density spheres that decelerate substantially following impact, and characterize the deceleration rate and resulting change in behavior of the associated water-entry cavities. Theoretical predictions compare favorably with our experimental observations. Finally, we present a theoretical model for the evolution of the splash curtain formed at high speeds, and couple it to the underlying cavity dynamics. In Part II, we present the results of a combined experimental and theoretical investigation of the motion of a sphere on an inclined flexible beam. A theoretical model based on Euler-Bernoulli beam theory is developed to describe the dynamics, and in the limit where the beam reacts instantaneously to the loading, we obtain exact solutions for the sphere trajectory and descent time. For the case of an initially horizontal beam, we calculate the period of the resulting oscillations. Theoretical predictions compare favorably with our experimental observations in this quasi-static regime. Inertial effects are also addressed.</Abstract> <Abstract>(cont.) The time taken for descent along an elastic beam, the elastochrone, is shown to always exceed the classical brachistochrone, the shortest time between two points in a gravitational field.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>