Nonlinear oscillations of a single-degree-of-freedom, parametrically-excited system are considered. The stiffness involves quadratic and cubic nonlinearities and models a shallow arch or buckled mechanism. The excitation frequency is assumed to be close to twice the natural frequency of the system. Numerical integration is used to obtain phase plane portraits, power spectra, and Poincare´ maps for large-time motions. Period-doubling bifurcations and several types of limit cycles and chaotic behavior are observed. Approximate analytical techniques are applied to analyze some of the limit cycles and transitions of behavior. The results are used to estimate the parameter region in which chaos may occur.
Period Doubling and Chaos in Unsymmetric Structures Under Parametric Excitation
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Szemplin´ska-Stupnicka, W., Plaut, R. H., and Hsieh, J. (December 1, 1989). "Period Doubling and Chaos in Unsymmetric Structures Under Parametric Excitation." ASME. J. Appl. Mech. December 1989; 56(4): 947–952. https://doi.org/10.1115/1.3176195
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