{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/41191"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/41191","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Comparison of approximate and exact methods for determining the frequencies of vibrating beams","abstract":"The classical method, required for its solution, the application of boundary conditions to the solution of the beam equation. Except for the case cf the beam with one concentrated load at the center, it was not considered a practical solution. The transcendental equation obtained in the solution of the unsymmetrical case, considered in part B, was found too cumbersome to handle. It was not attempted in parts C and D. The Rayleigh Method proved to be a simple, accurate and reasonably rapid method for all cases considered. The Dunkerley Equation gave very satisfactory results for parts A, B, and C. It was rapid to use, accurate and in most cases the data could be found in prepared tabulations. Results were inaccurate for the two span beam, indicating the necessity for caution in its application to multi-span beams. The Ritz Method, which is a refinement of the Rayleigh Method, proved to be exceedingly accurate when applied to the beam with the single concentrated load. However, it was found, that as the number of terms in the assumed deflection equation increased, the work became more time consuming. It was used only in parts A and B. The Influence Coefficient Method and the application of D'Alembert's Principle, which methods are quite similar, proved to be simple, accurate, and rapid. However, as the number of degrees of freedom increased, the degree of the algebraic equation increased, which complicated the solution. The Iteration Method is probably the method to be used if the number of degrees of freedom exceeds three. As the number of modes increases the number of iterations would increase, but the individual operations in themselves would remain simple. This method proved simple and accurate to use. For the cases considered, it was more time consuming to use than either the Influence Coefficient Method or the application of D'Alembert's Principle. However, for higher degree situations, it should prove to be a more practical method.","abstract_html":"The classical method, required for its solution, the application of boundary conditions to the solution of the beam equation. Except for the case cf the beam with one concentrated load at the center, it was not considered a practical solution. The transcendental equation obtained in the solution of the unsymmetrical case, considered in part B, was found too cumbersome to handle. It was not attempted in parts C and D. The Rayleigh Method proved to be a simple, accurate and reasonably rapid method for all cases considered. The Dunkerley Equation gave very satisfactory results for parts A, B, and C. It was rapid to use, accurate and in most cases the data could be found in prepared tabulations. Results were inaccurate for the two span beam, indicating the necessity for caution in its application to multi-span beams. The Ritz Method, which is a refinement of the Rayleigh Method, proved to be exceedingly accurate when applied to the beam with the single concentrated load. However, it was found, that as the number of terms in the assumed deflection equation increased, the work became more time consuming. It was used only in parts A and B. The Influence Coefficient Method and the application of D'Alembert's Principle, which methods are quite similar, proved to be simple, accurate, and rapid. However, as the number of degrees of freedom increased, the degree of the algebraic equation increased, which complicated the solution. The Iteration Method is probably the method to be used if the number of degrees of freedom exceeds three. As the number of modes increases the number of iterations would increase, but the individual operations in themselves would remain simple. This method proved simple and accurate to use. For the cases considered, it was more time consuming to use than either the Influence Coefficient Method or the application of D'Alembert's Principle. However, for higher degree situations, it should prove to be a more practical method.","abstract_has_math":false,"creators":["Stirling, Yates III"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Applied Mechanics","degree_department":"Applied Mechanics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1954,"date_issued":"1954","date_published":"1954","updated_at":"2026-07-22T22:19:10Z","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-02162010-020409"],"render_values":[{"text":"etd-02162010-020409","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/41191","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Applied Mechanics"]},{"key":"dc:creator","label":"Author","values":["Stirling, Yates III"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:29:54Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:29:54Z","2010-02-16"]},{"key":"dc:date.issued","label":"Date","values":["1954"]},{"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":["Applied 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"]}]},{"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-02162010-020409"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/41191"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The classical method, required for its solution, the application of boundary conditions to the solution of the beam equation. Except for the case cf the beam with one concentrated load at the center, it was not considered a practical solution. The transcendental equation obtained in the solution of the unsymmetrical case, considered in part B, was found too cumbersome to handle. It was not attempted in parts C and D. The Rayleigh Method proved to be a simple, accurate and reasonably rapid method for all cases considered. The Dunkerley Equation gave very satisfactory results for parts A, B, and C. It was rapid to use, accurate and in most cases the data could be found in prepared tabulations. Results were inaccurate for the two span beam, indicating the necessity for caution in its application to multi-span beams. The Ritz Method, which is a refinement of the Rayleigh Method, proved to be exceedingly accurate when applied to the beam with the single concentrated load. However, it was found, that as the number of terms in the assumed deflection equation increased, the work became more time consuming. It was used only in parts A and B. The Influence Coefficient Method and the application of D'Alembert's Principle, which methods are quite similar, proved to be simple, accurate, and rapid. However, as the number of degrees of freedom increased, the degree of the algebraic equation increased, which complicated the solution. The Iteration Method is probably the method to be used if the number of degrees of freedom exceeds three. As the number of modes increases the number of iterations would increase, but the individual operations in themselves would remain simple. This method proved simple and accurate to use. For the cases considered, it was more time consuming to use than either the Influence Coefficient Method or the application of D'Alembert's Principle. However, for higher degree situations, it should prove to be a more practical method."]},{"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":["Comparison of approximate and exact methods for determining the frequencies of vibrating beams"]}]}],"canonical_facts":{"dc:contributor.department":["Applied Mechanics"],"dc:creator":["Stirling, Yates III"],"dc:date.accessioned":["2014-03-14T21:29:54Z"],"dc:date.available":["2014-03-14T21:29:54Z","2010-02-16"],"dc:date.issued":["1954"],"dc:description.abstract":["The classical method, required for its solution, the application of boundary conditions to the solution of the beam equation. Except for the case cf the beam with one concentrated load at the center, it was not considered a practical solution. The transcendental equation obtained in the solution of the unsymmetrical case, considered in part B, was found too cumbersome to handle. It was not attempted in parts C and D. The Rayleigh Method proved to be a simple, accurate and reasonably rapid method for all cases considered. The Dunkerley Equation gave very satisfactory results for parts A, B, and C. It was rapid to use, accurate and in most cases the data could be found in prepared tabulations. Results were inaccurate for the two span beam, indicating the necessity for caution in its application to multi-span beams. The Ritz Method, which is a refinement of the Rayleigh Method, proved to be exceedingly accurate when applied to the beam with the single concentrated load. However, it was found, that as the number of terms in the assumed deflection equation increased, the work became more time consuming. It was used only in parts A and B. The Influence Coefficient Method and the application of D'Alembert's Principle, which methods are quite similar, proved to be simple, accurate, and rapid. However, as the number of degrees of freedom increased, the degree of the algebraic equation increased, which complicated the solution. The Iteration Method is probably the method to be used if the number of degrees of freedom exceeds three. As the number of modes increases the number of iterations would increase, but the individual operations in themselves would remain simple. This method proved simple and accurate to use. For the cases considered, it was more time consuming to use than either the Influence Coefficient Method or the application of D'Alembert's Principle. However, for higher degree situations, it should prove to be a more practical method."],"dc:description.degree":["Master of Science"],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-02162010-020409"],"dc:identifier.uri":["http://hdl.handle.net/10919/41191"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Comparison of approximate and exact methods for determining the frequencies of vibrating beams"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Applied Mechanics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:19:10Z"}