{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71685"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71685","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The First Passage Problem in Random Vibration for a Simple Hysteretic Oscillator","abstract":"A method to determine the ordinary statistical moments of time to first passage and to determine the probability of first passage failure for a simple oscillator, incorporating the modified Bouc hysteresis model, has been developed. Two boundary value problems are formulated from Markov process theory and solved by a Petrov-Galerkin finite element method. The first is known as a generalized Pontriagin-Vitt equation, which when solved yields the ordinary moments of time of first passage as a function of the oscillator's initial displacement, initial velocity and initial hysteretic force. In the second formulation, an initial-boundary value problem related to the backward Kolmogorov equation is derived and solved directly for the cumulative function of oscillator reliability, also in terms of the initial state of the oscillator. A comparison of the finite element results with those obtained by Monte Carlo simulation is then given to demonstrate the accuracy of the finite element method. Finally, a method to estimate the reliability of the hysteretic oscillator having prescribed the first few moments of first passage time is considered.","abstract_html":"A method to determine the ordinary statistical moments of time to first passage and to determine the probability of first passage failure for a simple oscillator, incorporating the modified Bouc hysteresis model, has been developed. Two boundary value problems are formulated from Markov process theory and solved by a Petrov-Galerkin finite element method. The first is known as a generalized Pontriagin-Vitt equation, which when solved yields the ordinary moments of time of first passage as a function of the oscillator's initial displacement, initial velocity and initial hysteretic force. In the second formulation, an initial-boundary value problem related to the backward Kolmogorov equation is derived and solved directly for the cumulative function of oscillator reliability, also in terms of the initial state of the oscillator. A comparison of the finite element results with those obtained by Monte Carlo simulation is then given to demonstrate the accuracy of the finite element method. Finally, a method to estimate the reliability of the hysteretic oscillator having prescribed the first few moments of first passage time is considered.","abstract_has_math":false,"creators":["Spencer, Billie Floyd, Jr."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Theoretical and Applied Mechanics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T19:12:36Z","date_published":"2014-12-16T19:12:36Z","updated_at":"2026-07-22T22:26:05Z","subjects":["Applied Mechanics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8600321"],"render_values":[{"text":"(UMI)AAI8600321","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71685","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Spencer, Billie Floyd, Jr."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T19:12:36Z","10000-01-01","1985"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Theoretical and Applied Mechanics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Applied Mechanics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71685","(UMI)AAI8600321"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A method to determine the ordinary statistical moments of time to first passage and to determine the probability of first passage failure for a simple oscillator, incorporating the modified Bouc hysteresis model, has been developed. Two boundary value problems are formulated from Markov process theory and solved by a Petrov-Galerkin finite element method. The first is known as a generalized Pontriagin-Vitt equation, which when solved yields the ordinary moments of time of first passage as a function of the oscillator's initial displacement, initial velocity and initial hysteretic force. In the second formulation, an initial-boundary value problem related to the backward Kolmogorov equation is derived and solved directly for the cumulative function of oscillator reliability, also in terms of the initial state of the oscillator. A comparison of the finite element results with those obtained by Monte Carlo simulation is then given to demonstrate the accuracy of the finite element method. Finally, a method to estimate the reliability of the hysteretic oscillator having prescribed the first few moments of first passage time is considered.","Made available in DSpace on 2014-12-16T19:12:36Z (GMT). No. of bitstreams: 1 8600321.pdf: 4244495 bytes, checksum: 8330f7292f6e3079a0aaef0ee6f343c6 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71851 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","152 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."]},{"key":"dc:title","label":"Title","values":["The First Passage Problem in Random Vibration for a Simple Hysteretic Oscillator"]}]}],"canonical_facts":{"dc:creator":["Spencer, Billie Floyd, Jr."],"dc:date":["2014-12-16T19:12:36Z","10000-01-01","1985"],"dc:description":["A method to determine the ordinary statistical moments of time to first passage and to determine the probability of first passage failure for a simple oscillator, incorporating the modified Bouc hysteresis model, has been developed. Two boundary value problems are formulated from Markov process theory and solved by a Petrov-Galerkin finite element method. The first is known as a generalized Pontriagin-Vitt equation, which when solved yields the ordinary moments of time of first passage as a function of the oscillator's initial displacement, initial velocity and initial hysteretic force. In the second formulation, an initial-boundary value problem related to the backward Kolmogorov equation is derived and solved directly for the cumulative function of oscillator reliability, also in terms of the initial state of the oscillator. A comparison of the finite element results with those obtained by Monte Carlo simulation is then given to demonstrate the accuracy of the finite element method. Finally, a method to estimate the reliability of the hysteretic oscillator having prescribed the first few moments of first passage time is considered.","Made available in DSpace on 2014-12-16T19:12:36Z (GMT). No. of bitstreams: 1 8600321.pdf: 4244495 bytes, checksum: 8330f7292f6e3079a0aaef0ee6f343c6 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71851 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","152 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."],"dc:identifier":["http://hdl.handle.net/2142/71685","(UMI)AAI8600321"],"dc:subject":["Applied Mechanics"],"dc:title":["The First Passage Problem in Random Vibration for a Simple Hysteretic Oscillator"],"dc:type":["text"],"thesis:degree_discipline":["Theoretical and Applied Mechanics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:05Z"}