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dc.contributor.authorAlhazza, Khaleden_US
dc.date.accessioned2002-12-13en_US
dc.date.accessioned2014-03-14T20:20:09Z
dc.date.available2014-03-14T20:20:09Z
dc.date.issued2002-12-05en_US
dc.identifier.otheretd-12112002-125947en_US
dc.identifier.urihttp://hdl.handle.net/10919/30042
dc.description.abstractThe objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von K\'{a}rm\'{a}n-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented. We used two approaches to study the responses of shells to a primary resonance: a $direct$ approach and a $discretization$ approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency- and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared. For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating $2n+(2n)^2$ nonlinear first-order ordinary-differential equations simultaneously, where $n$ is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results. The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown. Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo symmetry-breaking, saddle node, and period-doubling bifurcations.en_US
dc.publisherVirginia Techen_US
dc.relation.haspartAlhazza_dissertation.pdfen_US
dc.rightsI hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Virginia Tech or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.en_US
dc.subjectShallow Shellsen_US
dc.subjectResonanceen_US
dc.subjectNonlinear Vibrationsen_US
dc.subjectGalerkin Discretizationen_US
dc.subjectModal Interactionen_US
dc.titleNonlinear Vibrations of Doubly Curved Cross-PLy Shallow Shellsen_US
dc.typeDissertationen_US
dc.contributor.departmentMechanical Engineeringen_US
thesis.degree.namePhDen_US
thesis.degree.leveldoctoralen_US
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen_US
dc.contributor.committeememberMasoud, Ziyad N.en_US
dc.contributor.committeememberInman, Daniel J.en_US
dc.contributor.committeememberLeo, Donald J.en_US
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-12112002-125947/en_US
dc.contributor.committeecochairNayfeh, Ali H.en_US
dc.contributor.committeecochairAhmadian, Mehdien_US
dc.date.sdate2002-12-11en_US
dc.date.rdate2003-12-13


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