Effect of Nonlinear Stiffness on the Motion of a Flexible Pendulum

In this paper, we study the effect of a harmonicforcing function and the strength of a nonlinearityon a two-degrees-of-freedom system namely, an elasticpendulum, with internal resonance (for examplenonlinearly elastic springs). The equations can alsobe used to model the coupling between a ship's pitchand roll. The system considered here is modeled by amass hanging from a spring that is pinned at one endto the ground. The mass is free to move in the radialdirection, is also free to rotate about the pin joint, and subject to a periodic forcing function. Theforcing function used in this paper is in thetangential direction. The amplitude of the forcingfunction is used here as the control parameter and thesystem's dynamics are studied through the variation ofthis parameter.The first part of the paper is dedicatedto establishing the route by which the motion of thesystem goes from a periodic attractor to a chaoticattractor. It was found that the route to chaos alwaysbegins with a secondary Hopf bifurcation followed byconsecutive torus-doubling bifurcations, ending withtorus breaking.A comparison was also made between the use of a linear spring, a weakly nonlinear spring, and astrongly nonlinear spring.This comparison showed that althoughthe route to chaos was not altered, the bifurcationsleading to chaos and the chaotic motion itselfoccurred at different frequency regimes. We observedthat the nonlinearity could aid the stabilizationof the periodicattractor beyond the previously seenthreshold of instability. Yet, if the strength of thenonlinearity is sufficiently large, it can lead tochaos in frequency regimes where chaos was notobserved previously. The strongly nonlinear systemshowed chaotic behavior for frequency regimes thatdisplayed only periodic motion for both the linearsystem and the weakly nonlinear system. The route tochaos for these frequency ranges was also found to bedifferent from that previously studied. This leads usto the hypothesis that chaos in this range was due tothe nonlinearity of the spring and not the coupling effect.