Effect of nonlinear stiffness on the motion of a flexible pendulum
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In this paper, we study the effect of a harmonic forcing function and the strength of a nonlinearity on a two-degrees-of-freedom system namely, an elastic pendulum, with internal resonance (for example nonlinearly elastic springs). The equations can also be used to model the coupling between a ship's pitch and roll. The system considered here is modeled by a mass hanging from a spring that is pinned at one end to the ground. The mass is free to move in the radial direction, is also free to rotate about the pin joint, and subject to a periodic forcing function. The forcing function used in this paper is in the tangential direction. The amplitude of the forcing function is used here as the control parameter and the system's dynamics are studied through the variation of this parameter. The first part of the paper is dedicated to establishing the route by which the motion of the system goes from a periodic attractor to a chaotic attractor. It was found that the route to chaos always begins with a secondary Hopf bifurcation followed by consecutive torus-doubling bifurcations, ending with torus breaking. A comparison was also made between the use of a linear spring, a weakly nonlinear spring, and a strongly nonlinear spring. This comparison showed that although the route to chaos was not altered, the bifurcations leading to chaos and the chaotic motion itself occurred at different frequency regimes. We observed that the nonlinearity could aid the stabilization of the periodic attractor beyond the previously seen threshold of instability. Yet, if the strength of the nonlinearity is sufficiently large, it can lead to chaos in frequency regimes where chaos was not observed previously. The strongly nonlinear system showed chaotic behavior for frequency regimes that displayed only periodic motion for both the linear system and the weakly nonlinear system. The route to chaos for these frequency ranges was also found to be different from that previously studied. This leads us to the hypothesis that chaos in this range was due to the nonlinearity of the spring and not the coupling effect.
