Three types of problems in nonlinear dynamics are studied. First, we use a complex-variable invariant-manifold approach to determine the nonlinear normal modes of weakly nonlinear discrete systems with one-to-one and three-to-one internal resonances. Cubic geometric nonlinearities are considered. The system under investigation possesses similar nonlinear normal modes for the case of One-to-one internal resonance and nonsimilar nonlinear normal modes for the case of three-to-one internal resonance. In contrast with the case of no internal resonance, the number of nonlinear normal modes may be more than the number of linear normal modes. Bifurcations of the calculated nonlinear normal modes are investigated. For continuous systems without internal resonances, we consider a cantilever beam and compare two approaches for the determination of its nonlinear planar modes. In the first approach, the governing partial-differential system is discretized using the linear mode shapes and then the nonlinear mode shapes are determined from the discretized system. In the second approach, the boundary-value problem is treated directly by using the method of multiple scales. The results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes.
Second, we study the nonlinear response of multi-degree-of-freedom systems with repeated natural frequencies to various parametric resonances. The linear part of the system has a nonsemisimple one-to-one resonance. The character of the stability and various types of bifurcation are analyzed. The results are applied to the flutter of a simply supported panel in a supersonic airstream. In which case, the nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the excited modes are derived and used to calculate the equilibrium solutions and their stability and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions. Moreover, bifurcation analyses in the case of fundamental parametric resonance reveal that the addition of quadratic nonlinearities change qualitatively as well as quantitatively the response of systems with cubic nonlinearities. The quadratic nonlinearities change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold bifurcations, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.
Third, we investigate the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced modal frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/38337 |
| Date | 06 June 2008 |
| Creators | Chin, Char-Ming |
| Contributors | Engineering Mechanics, Nayfeh, Ali H., Mook, Dean T., Smith, Charles W., Renardy, Michael J., Hendricks, Scott L. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | viii, 200 leaves, BTD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 30505774, LD5655.V856_1993.C556.pdf |
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