Dynamics of the pendulum with periodically varying length

Abstract

a b s t r a c tDynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of a child's swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with a numerical study. Chaotic motions of the pendulum depending on problem parameters are investigated numerically.

Comparisons between the pendulum with varying length and the pendulum with oscillating support

“…Here, a first difference between the two systems appears. Whilst it is well known [61,40,1,2,6,13,30] that for α sufficiently small and δ chosen accordingly, it is possible to stabilise the upward fixed point (θ,θ) = (π, 0) for the system (2), on the contrary we show that, as observed numerically in [18], stability of the upward fixed point cannot be achieved for any values of the parameters in the system (3). In Section 3 we calculate sufficient conditions on the damping coefficient in each system for the origin to achieve global attraction (up to a zero measure set).…”

“…We now consider the two pendulum systems (2) and (3), and estimate numerically the relative areas of the basins of attraction; previous study has be completed in [12,64,66] for the pendulum with oscillating support and in [16,18] for the pendulum with varying length. Throughout all this section, we fix the parameter values at α = 0.5 and δ = ε = 0.1.…”

“…One of the most commonly studied perturbations is the pendulum with vertically oscillating support [6,12,13,30,64], often also referred to as the parametrically forced pendulum, see for example [21,29,32,44,66]. Another much studied perturbation of the simple pendulum is the pendulum with periodically varying length [8,16,58,18,56]. In [22,53] it was noted that the pendulum with varying length is an example of a periodically forced system whose linearisation is different from Mathieu's equation, but no further analysis was completed.…”

“…In [22,53] it was noted that the pendulum with varying length is an example of a periodically forced system whose linearisation is different from Mathieu's equation, but no further analysis was completed. A brief comparison between the pendulum with oscillating support and the pendulum with varying length was made in [18], drawing the conclusions that the two systems are qualitatively different and further comparison of the two systems deserved a separate study. However there has not been further work focused on exploring the differences of the two systems.…”

“…Therefore to avoid further confusion and misguidance it is important not only to highlight the differences between the two systems but also to note where they are similar to one another. In the present paper we expand on the study made in [18] and directly compare some aspects of the two systems.…”