A numerical-perturbation approach is used to study modal interactions in the dynamic response of infinitely long circular cylindrical shells to an external harmonic excitation. The excitation frequency is near the linear natural frequency of the breathing mode (i.e., primary resonance of the breathing mode) and the linear natural frequency of the breathing mode is approximately twice that of a flexural mode (i.e., two-to-one internal or autoparametric resonance). The method of multiple-time scales is used to derive a set of autonomous first-order nonlinear differential equations that describe the modulation of the amplitudes and phases of the interacting modes. The same approach is used to study the axisymmetric dynamic response of spherical shells to a radial harmonic excitation having a frequency near one of the linear natural frequencies of a flexural mode (i.e., primary resonance of a flexural mode) and in the presence of a two-to-one internal resonance between the excited mode and a lower flexural mode. The modulation equations derived for infinitely long circular cylindrical shells and for axisymmetric spherical shells are scaled to the same form and their fixed points, periodic solutions, and chaotic solutions are studied as the amplitude or the frequency of excitation varies. As the excitation amplitude varies, the fixed-point solutions of the modulation equations exhibit the jump and saturation phenomena. They also undergo supercritical and subcritical Hopf bifurcations as the frequency or the amplitude of excitation varies. Between the two Hopf-bifurcation frequencies, the fixed-point solutions are unstable and limit cycles exist. Some limit cycles experience symmetry-breaking (pitchfork) bifurcation followed by an infinite cascade of period-doubling bifurcations culminating in chaos. Other limit cycles lose stability through cyclic-fold bifurcations causing a transition to chaos.
The same procedure is used to study the nonlinear dynamic response of infinitely long circular cylindrical shells to a subharmonic excitation of order one-haIf of the breathing mode in the presence of a two-to-one internal resonance. The force-response curves exhibit saturation, jumps, and Hopf bifurcations. They also show that the shell does not respond until a certain threshold level of excitation is exceeded. The frequency-response curves exhibit jumps and pitchfork and Hopf bifurcations. For certain parameters and excitation frequencies between the Hopf-bifurcation values, limit-cycle solutions of the modulation equations are found. As the excitation frequency changes, the limit cycles deform and lose their stability through either pitchfork or cyclic-foId (saddle-node) bifurcations. Some of these saddIe-node bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles. Period-three motions are observed over a narrow range of excitation frequencies.
Lastly, a computerized symbolic manipulator is used to analyze the dynamic response of an infinitely long circular cylindrical shell to radial harmonic excitations. The excitation frequency is near the linear natural frequency of a flexural mode (i.e., primary resonance of a flexural mode). Due to the complete circular symmetry of the shell, each natural frequency corresponds to two orthogonal mode shapes. The mode with the same spatial variation as the excitation is called the driven mode and the other mode is called the companion mode. Modal interactions between the driven mode and the companion mode are studied. The steady-state response of the shell can involve either the driven mode alone (single-mode response) or both the driven and companion modes (two-mode response). The frequency-response curve exhibits jumps and Hopf bifurcations. Between the Hopf·bifurcation frequencies, the modulation equations exhibit multiple limit-cycle solutions. As the excitation frequency varies, these limit cycles go through either saddle-node collisions or incomplete sequences of period-doubling bifurcations. Some of the saddle-node bifurcations result in the birth of limit cycles and some result in transition to chaos. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/54270 |
| Date | January 1989 |
| Creators | Raouf, Raouf A. |
| Contributors | Engineering Mechanics, Nayfeh, Ali, Mook, Dean T., Hendricks, Scott L., Johnson, Eric R., Ragab, Saad A. |
| Publisher | Virginia Polytechnic Institute and State University |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | xi, 157 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20103348 |
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