{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/88099"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/88099","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Methodology for nonlinear quantification of a cantilever beam with local nonlinearities","abstract":"This study presents a methodology for the identification of linear and nonlinear regions of operation for a system that behaves almost linearly in the limit of extreme values of (a) certain parameter(s). An Euler-Bernoulli cantilever beam with two nonlinear configurations is used to develop and validate the methodology. One configuration consists of a cantilever beam with a cubic spring attached at a specific distance from the beam root to achieve a smooth nonlinear effect. The other configuration is a cantilever beam undergoing vibro-impact between symmetrically-spaced stops. Both systems have the property that, in the limit of small and large values of a parameter, the system is almost linear and can be modeled with negligible error as fixed-free or fixed-pinned, depending on the configuration. For the beam with a cubic spring attachment, the forcing amplitude is the varied parameter. For the vibro-impact beam, the parameter is the clearance between the stops and the beam at static equilibrium. Proper orthogonal decomposition is employed to obtain an optimal basis used to describe the systems with varying parameter values. The frequencies of the modes that comprise the basis are estimated using the Rayleigh quotient. The variations of these frequencies are studied to successfully identify parameter values for which the system is approximately linear and those for which it is highly nonlinear. A criterion based on the Betti-Maxwell reciprocity theorem is used to validate the existence of nonlinear behavior for the set of parameter values suggested by the described methodology. It was found that transition regions in which the dynamics of the system shifted from one linear system to the other exist and that these regions occur for different sets of parameter values for each mode. The effect of heavy damping on proper orthogonal decomposition is found to be important, particularly for the vibro-impact beam, due the method's dependence on the system response. An attempt is also made to isolate parameter values for which transient resonance capture occurs and prove its existence through use of empirical mode decomposition and the Hilbert transform.","abstract_html":"This study presents a methodology for the identification of linear and nonlinear regions of operation for a system that behaves almost linearly in the limit of extreme values of (a) certain parameter(s). An Euler-Bernoulli cantilever beam with two nonlinear configurations is used to develop and validate the methodology. One configuration consists of a cantilever beam with a cubic spring attached at a specific distance from the beam root to achieve a smooth nonlinear effect. The other configuration is a cantilever beam undergoing vibro-impact between symmetrically-spaced stops. Both systems have the property that, in the limit of small and large values of a parameter, the system is almost linear and can be modeled with negligible error as fixed-free or fixed-pinned, depending on the configuration. For the beam with a cubic spring attachment, the forcing amplitude is the varied parameter. For the vibro-impact beam, the parameter is the clearance between the stops and the beam at static equilibrium. Proper orthogonal decomposition is employed to obtain an optimal basis used to describe the systems with varying parameter values. The frequencies of the modes that comprise the basis are estimated using the Rayleigh quotient. The variations of these frequencies are studied to successfully identify parameter values for which the system is approximately linear and those for which it is highly nonlinear. A criterion based on the Betti-Maxwell reciprocity theorem is used to validate the existence of nonlinear behavior for the set of parameter values suggested by the described methodology. It was found that transition regions in which the dynamics of the system shifted from one linear system to the other exist and that these regions occur for different sets of parameter values for each mode. The effect of heavy damping on proper orthogonal decomposition is found to be important, particularly for the vibro-impact beam, due the method's dependence on the system response. An attempt is also made to isolate parameter values for which transient resonance capture occurs and prove its existence through use of empirical mode decomposition and the Hilbert transform.","abstract_has_math":false,"creators":["Herrera, Christopher Angelo"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Aerospace Engineering","degree_department":null,"school":null,"contributors":["Bergman, Lawrence A.","Vakakis, Alexander F.","McFarland, Donald M."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-29T20:38:47Z","date_published":"2015-09-29T20:38:47Z","updated_at":"2026-07-22T22:26:31Z","subjects":["Nonlinear Dynamics","Proper Orthogonal Decomposition (POD)","Empirical Mode Decomposition (EMD)","Vibro-impact","Nonlinear Quantification"],"languages":["en"],"rights":["Copyright 2015 Christopher Herrera"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/88099","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bergman, Lawrence A.","Vakakis, Alexander F.","McFarland, Donald M."]},{"key":"dc:creator","label":"Author","values":["Herrera, Christopher Angelo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-29T20:38:47Z","2015-08","2015-07-21","2015-8"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Aerospace Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"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":["Nonlinear Dynamics","Proper Orthogonal Decomposition (POD)","Empirical Mode Decomposition (EMD)","Vibro-impact","Nonlinear Quantification"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2015 Christopher Herrera"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/88099"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This study presents a methodology for the identification of linear and nonlinear regions of operation for a system that behaves almost linearly in the limit of extreme values of (a) certain parameter(s). An Euler-Bernoulli cantilever beam with two nonlinear configurations is used to develop and validate the methodology. One configuration consists of a cantilever beam with a cubic spring attached at a specific distance from the beam root to achieve a smooth nonlinear effect. The other configuration is a cantilever beam undergoing vibro-impact between symmetrically-spaced stops. Both systems have the property that, in the limit of small and large values of a parameter, the system is almost linear and can be modeled with negligible error as fixed-free or fixed-pinned, depending on the configuration. For the beam with a cubic spring attachment, the forcing amplitude is the varied parameter. For the vibro-impact beam, the parameter is the clearance between the stops and the beam at static equilibrium. Proper orthogonal decomposition is employed to obtain an optimal basis used to describe the systems with varying parameter values. The frequencies of the modes that comprise the basis are estimated using the Rayleigh quotient. The variations of these frequencies are studied to successfully identify parameter values for which the system is approximately linear and those for which it is highly nonlinear. A criterion based on the Betti-Maxwell reciprocity theorem is used to validate the existence of nonlinear behavior for the set of parameter values suggested by the described methodology. It was found that transition regions in which the dynamics of the system shifted from one linear system to the other exist and that these regions occur for different sets of parameter values for each mode. The effect of heavy damping on proper orthogonal decomposition is found to be important, particularly for the vibro-impact beam, due the method's dependence on the system response. An attempt is also made to isolate parameter values for which transient resonance capture occurs and prove its existence through use of empirical mode decomposition and the Hilbert transform.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2015-09-29 without embargo terms","The student, Christopher Herrera, accepted the attached license on 2015-07-21 at 02:39.","The student, Christopher Herrera, submitted this Thesis for approval on 2015-07-21 at 02:53.","This Thesis was approved for publication on 2015-07-21 at 13:24.","DSpace SAF Submission Ingestion Package generated from Vireo submission #8580 on 2015-09-29 at 13:23:25","Made available in DSpace on 2015-09-29T20:38:47Z (GMT). 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An Euler-Bernoulli cantilever beam with two nonlinear configurations is used to develop and validate the methodology. One configuration consists of a cantilever beam with a cubic spring attached at a specific distance from the beam root to achieve a smooth nonlinear effect. The other configuration is a cantilever beam undergoing vibro-impact between symmetrically-spaced stops. Both systems have the property that, in the limit of small and large values of a parameter, the system is almost linear and can be modeled with negligible error as fixed-free or fixed-pinned, depending on the configuration. For the beam with a cubic spring attachment, the forcing amplitude is the varied parameter. For the vibro-impact beam, the parameter is the clearance between the stops and the beam at static equilibrium. Proper orthogonal decomposition is employed to obtain an optimal basis used to describe the systems with varying parameter values. The frequencies of the modes that comprise the basis are estimated using the Rayleigh quotient. The variations of these frequencies are studied to successfully identify parameter values for which the system is approximately linear and those for which it is highly nonlinear. A criterion based on the Betti-Maxwell reciprocity theorem is used to validate the existence of nonlinear behavior for the set of parameter values suggested by the described methodology. It was found that transition regions in which the dynamics of the system shifted from one linear system to the other exist and that these regions occur for different sets of parameter values for each mode. The effect of heavy damping on proper orthogonal decomposition is found to be important, particularly for the vibro-impact beam, due the method's dependence on the system response. An attempt is also made to isolate parameter values for which transient resonance capture occurs and prove its existence through use of empirical mode decomposition and the Hilbert transform.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2015-09-29 without embargo terms","The student, Christopher Herrera, accepted the attached license on 2015-07-21 at 02:39.","The student, Christopher Herrera, submitted this Thesis for approval on 2015-07-21 at 02:53.","This Thesis was approved for publication on 2015-07-21 at 13:24.","DSpace SAF Submission Ingestion Package generated from Vireo submission #8580 on 2015-09-29 at 13:23:25","Made available in DSpace on 2015-09-29T20:38:47Z (GMT). No. of bitstreams: 3 HERRERA-THESIS-2015.pdf: 47691539 bytes, checksum: 55a9008de1b0271560a34c195eb954b4 (MD5) MastersThesis_caherre2_1.tex: 186445 bytes, checksum: 2a23ebdb8e853481585da40727da6153 (MD5) LICENSE.txt: 4216 bytes, checksum: d512fa2766807c565cb7dd9e26522277 (MD5) Previous issue date: 2015-07-21"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/88099"],"dc:language":["en"],"dc:rights":["Copyright 2015 Christopher Herrera"],"dc:subject":["Nonlinear Dynamics","Proper Orthogonal Decomposition (POD)","Empirical Mode Decomposition (EMD)","Vibro-impact","Nonlinear Quantification"],"dc:title":["Methodology for nonlinear quantification of a cantilever beam with local nonlinearities"],"dc:type":["text"],"thesis:degree_discipline":["Aerospace Engineering"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:31Z"}