{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/45246"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/45246","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Dynamics of three-degree-of-freedom systems with quadratic nonlinearities","abstract":"The dynamics of two three-degree-of-freedom systems with quadratic nonlinearities are studied. The first system has two simultaneous two-to-one internal resonances. The second has a combination internal resonance. In both cases the response to a primary resonant excitation of the third mode is studied. The method of multiple time scales is used to obtain the equations that govern the amplitudes and phases of the first system. Then the fixed points of these equations are obtained and their stability is determined. The fixed points undergo Hopf bifurcations, and the overall system response can be periodic or periodically, quasiperiodically, or chaotically modulated. The method of the time-averaged Lagrangian is used to obtain the equations that govern the amplitudes and phases of the second system. The fixed points of these equations are obtained and their stability is determined. These fixed points undergo Hopf bifurcations, and the overall system response can be periodic or a two- or three-torus.","abstract_html":"The dynamics of two three-degree-of-freedom systems with quadratic nonlinearities are studied. The first system has two simultaneous two-to-one internal resonances. The second has a combination internal resonance. In both cases the response to a primary resonant excitation of the third mode is studied. The method of multiple time scales is used to obtain the equations that govern the amplitudes and phases of the first system. Then the fixed points of these equations are obtained and their stability is determined. The fixed points undergo Hopf bifurcations, and the overall system response can be periodic or periodically, quasiperiodically, or chaotically modulated. The method of the time-averaged Lagrangian is used to obtain the equations that govern the amplitudes and phases of the second system. The fixed points of these equations are obtained and their stability is determined. These fixed points undergo Hopf bifurcations, and the overall system response can be periodic or a two- or three-torus.","abstract_has_math":false,"creators":["Nayfeh, Tariq Ali"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Engineering Mechanics","degree_department":"Engineering Mechanics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1991,"date_issued":"1991","date_published":"1991","updated_at":"2026-07-22T22:20:34Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-10222009-125017"],"render_values":[{"text":"etd-10222009-125017","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/45246","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Engineering Mechanics"]},{"key":"dc:creator","label":"Author","values":["Nayfeh, Tariq Ali"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:47:59Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:47:59Z","2009-10-22"]},{"key":"dc:date.issued","label":"Date","values":["1991"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Engineering Mechanics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-10222009-125017"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/45246"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The dynamics of two three-degree-of-freedom systems with quadratic nonlinearities are studied. The first system has two simultaneous two-to-one internal resonances. The second has a combination internal resonance. In both cases the response to a primary resonant excitation of the third mode is studied. The method of multiple time scales is used to obtain the equations that govern the amplitudes and phases of the first system. Then the fixed points of these equations are obtained and their stability is determined. The fixed points undergo Hopf bifurcations, and the overall system response can be periodic or periodically, quasiperiodically, or chaotically modulated. The method of the time-averaged Lagrangian is used to obtain the equations that govern the amplitudes and phases of the second system. The fixed points of these equations are obtained and their stability is determined. These fixed points undergo Hopf bifurcations, and the overall system response can be periodic or a two- or three-torus."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Dynamics of three-degree-of-freedom systems with quadratic nonlinearities"]}]}],"canonical_facts":{"dc:contributor.department":["Engineering Mechanics"],"dc:creator":["Nayfeh, Tariq Ali"],"dc:date.accessioned":["2014-03-14T21:47:59Z"],"dc:date.available":["2014-03-14T21:47:59Z","2009-10-22"],"dc:date.issued":["1991"],"dc:description.abstract":["The dynamics of two three-degree-of-freedom systems with quadratic nonlinearities are studied. The first system has two simultaneous two-to-one internal resonances. The second has a combination internal resonance. In both cases the response to a primary resonant excitation of the third mode is studied. The method of multiple time scales is used to obtain the equations that govern the amplitudes and phases of the first system. Then the fixed points of these equations are obtained and their stability is determined. The fixed points undergo Hopf bifurcations, and the overall system response can be periodic or periodically, quasiperiodically, or chaotically modulated. The method of the time-averaged Lagrangian is used to obtain the equations that govern the amplitudes and phases of the second system. The fixed points of these equations are obtained and their stability is determined. These fixed points undergo Hopf bifurcations, and the overall system response can be periodic or a two- or three-torus."],"dc:description.degree":["Master of Science"],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-10222009-125017"],"dc:identifier.uri":["http://hdl.handle.net/10919/45246"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Dynamics of three-degree-of-freedom systems with quadratic nonlinearities"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Engineering Mechanics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:34Z"}