Chaos Displayed by Solutions of a Forced Pendulum Equation with a Fifth-OrderPolynomial for the Damping Terms

摘要

Solutions of a forced pendulum equation of one degree-of-freedom with nonlineardamping terms have been studied for chaotic and dissipative behavior. This equation can simulate phenomena in which vortex shedding from oscillating and rotating bodies is involved. The characteristic properties of this equation, that is, its singular points, attractors, separatrices, etc. are investigated and related to their physical meaning. A novel attractor is found that represents simultaneously autorotation and self-sustained oscillation. Chaotic behavior is studied with Poincare maps and power spectra. Strange attractors exist which are insensitive to various pendulum coefficients. Of Navy interest are pendulum-type equations that model the rolling motion of ships. This motion can exhibit chaotic behavior. Another example of Navy interest is the usefulness of an extended pendulum equation to describe rotating and oscillating objects in a fluid, a problem whose solution otherwise would require the numerical integration of a set of nonlinear partial differential equations. This case will be studied in this report in more detail. Keywords: Power spectrum, Phase plane. (KR)