{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/44112"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/44112","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"The effect of cracks on the dynamic behavior of bars and shafts","abstract":"Nondestructive methods of detecting cracks in structural components and machinery are important, both in preventing failures and in establishing maintenance procedures. This thesis considers how the vibration behavior of cracked members can be modelled mathematically and how these mathematical models may lead to advancements in crack detection procedures. Two separate cases are considered: the longitudinal vibration of a cracked bar and the coupled vibrations of a cracked rotating shaft. In the longitudinal vibration study, the equation of motion is developed for a cantilevered bar with a symmetric surface crack. Next, Galerkin's Method is used to obtain one- and two-term approximate solutions. Both forced and free vibrations of the bar are analyzed. Graphical results showing the relationships between displacement and crack size, crack position, and forcing frequency are presented and discussed. Spectral analysis is used to compare uncracked and cracked bar behavior. Finally, a sensitivity analysis of the forced vibration case is conducted to observe how the forcing frequency affects the rate of change of steady-state response at the onset of cracking. In the second part of the thesis, a similar analysis is conducted for a cracked, simply-supported Timoshenko shaft rotating at a constant angular speed. The equations of motion derived by Wauer (b) are used as the basis of the study. Again, Galerkin's Method is applied to obtain approximate solutions. Time histories and spectra are used to observe how changes in various parameters influence the vibration behavior. The effects of mass eccentricity and gravity are studied. Finally, the effect of a periodic axial impact load is considered.","abstract_html":"Nondestructive methods of detecting cracks in structural components and machinery are important, both in preventing failures and in establishing maintenance procedures. This thesis considers how the vibration behavior of cracked members can be modelled mathematically and how these mathematical models may lead to advancements in crack detection procedures. Two separate cases are considered: the longitudinal vibration of a cracked bar and the coupled vibrations of a cracked rotating shaft. In the longitudinal vibration study, the equation of motion is developed for a cantilevered bar with a symmetric surface crack. Next, Galerkin's Method is used to obtain one- and two-term approximate solutions. Both forced and free vibrations of the bar are analyzed. Graphical results showing the relationships between displacement and crack size, crack position, and forcing frequency are presented and discussed. Spectral analysis is used to compare uncracked and cracked bar behavior. Finally, a sensitivity analysis of the forced vibration case is conducted to observe how the forcing frequency affects the rate of change of steady-state response at the onset of cracking. In the second part of the thesis, a similar analysis is conducted for a cracked, simply-supported Timoshenko shaft rotating at a constant angular speed. The equations of motion derived by Wauer (b) are used as the basis of the study. Again, Galerkin's Method is applied to obtain approximate solutions. Time histories and spectra are used to observe how changes in various parameters influence the vibration behavior. The effects of mass eccentricity and gravity are studied. Finally, the effect of a periodic axial impact load is considered.","abstract_has_math":false,"creators":["Collins, Kevin Ralph"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Civil Engineering","degree_department":"Civil Engineering","school":null,"contributors":[],"advisors":[],"committee_chairs":["Plaut, Raymond H."],"committee_members":["Rojiani, Kamal B.","Holzer, Siegfried M."],"year":1989,"date_issued":"1989-03-29","date_published":"1989-03-29","updated_at":"2026-07-22T22:20:11Z","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-08012012-040538"],"render_values":[{"text":"etd-08012012-040538","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/44112","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Plaut, Raymond H."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Rojiani, Kamal B.","Holzer, Siegfried M."]},{"key":"dc:contributor.department","label":"Department","values":["Civil Engineering"]},{"key":"dc:creator","label":"Author","values":["Collins, Kevin Ralph"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:42:02Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:42:02Z","2012-08-01"]},{"key":"dc:date.issued","label":"Date","values":["1989-03-29"]},{"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":["Civil Engineering"]},{"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-08012012-040538"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/44112"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Nondestructive methods of detecting cracks in structural components and machinery are important, both in preventing failures and in establishing maintenance procedures. This thesis considers how the vibration behavior of cracked members can be modelled mathematically and how these mathematical models may lead to advancements in crack detection procedures. Two separate cases are considered: the longitudinal vibration of a cracked bar and the coupled vibrations of a cracked rotating shaft. In the longitudinal vibration study, the equation of motion is developed for a cantilevered bar with a symmetric surface crack. Next, Galerkin's Method is used to obtain one- and two-term approximate solutions. Both forced and free vibrations of the bar are analyzed. Graphical results showing the relationships between displacement and crack size, crack position, and forcing frequency are presented and discussed. Spectral analysis is used to compare uncracked and cracked bar behavior. Finally, a sensitivity analysis of the forced vibration case is conducted to observe how the forcing frequency affects the rate of change of steady-state response at the onset of cracking. In the second part of the thesis, a similar analysis is conducted for a cracked, simply-supported Timoshenko shaft rotating at a constant angular speed. The equations of motion derived by Wauer (b) are used as the basis of the study. Again, Galerkin's Method is applied to obtain approximate solutions. Time histories and spectra are used to observe how changes in various parameters influence the vibration behavior. The effects of mass eccentricity and gravity are studied. Finally, the effect of a periodic axial impact load is considered."]},{"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":["The effect of cracks on the dynamic behavior of bars and shafts"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Plaut, Raymond H."],"dc:contributor.committeemember":["Rojiani, Kamal B.","Holzer, Siegfried M."],"dc:contributor.department":["Civil Engineering"],"dc:creator":["Collins, Kevin Ralph"],"dc:date.accessioned":["2014-03-14T21:42:02Z"],"dc:date.available":["2014-03-14T21:42:02Z","2012-08-01"],"dc:date.issued":["1989-03-29"],"dc:description.abstract":["Nondestructive methods of detecting cracks in structural components and machinery are important, both in preventing failures and in establishing maintenance procedures. This thesis considers how the vibration behavior of cracked members can be modelled mathematically and how these mathematical models may lead to advancements in crack detection procedures. Two separate cases are considered: the longitudinal vibration of a cracked bar and the coupled vibrations of a cracked rotating shaft. In the longitudinal vibration study, the equation of motion is developed for a cantilevered bar with a symmetric surface crack. Next, Galerkin's Method is used to obtain one- and two-term approximate solutions. Both forced and free vibrations of the bar are analyzed. Graphical results showing the relationships between displacement and crack size, crack position, and forcing frequency are presented and discussed. Spectral analysis is used to compare uncracked and cracked bar behavior. Finally, a sensitivity analysis of the forced vibration case is conducted to observe how the forcing frequency affects the rate of change of steady-state response at the onset of cracking. In the second part of the thesis, a similar analysis is conducted for a cracked, simply-supported Timoshenko shaft rotating at a constant angular speed. The equations of motion derived by Wauer (b) are used as the basis of the study. Again, Galerkin's Method is applied to obtain approximate solutions. Time histories and spectra are used to observe how changes in various parameters influence the vibration behavior. The effects of mass eccentricity and gravity are studied. Finally, the effect of a periodic axial impact load is considered."],"dc:description.degree":["Master of Science"],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-08012012-040538"],"dc:identifier.uri":["http://hdl.handle.net/10919/44112"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["The effect of cracks on the dynamic behavior of bars and shafts"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Civil Engineering"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:11Z"}