Mode competition in a system of two parametrically driven pendulums with nonlinear coupling

Abstract

This paper is part three in a series on the dynamics of two coupled, parametrically driven pendulums. In the previous parts Banning and van der Weele (1995) and Banning et al. (t997) studied the case of linear coupling; the present paper deals with the changes brought on by the inclusion of a nonlinear (third-order) term in the coupling. Special attention will be given to the phenomenon of mode competition.The nonlinear coupling is seen to introduce a new kind of threshold into the system, namely a lower limit to the frequency at which certain motions can exist. Another consequence is that the mode interaction between lc~ and 2~q (two of the normal motions of the system) is less degenerate, causing the intermediary mixed motion known as MP to manifest itself more strongly.

Mode competition in a system of two parametrically driven pendulums; the dissipative case

“…This degeneracy can be lifted by the inclusion of additional nonlinearities in our system. In a forthcoming paper we shall see that an extra, non-linear term in the coupling between the pendulums already works wonders [8].…”

“…Here type A is being annihilated in a saddlenode bifurcation together with its counterpart bifurcated from the (unstable) lflmotion. The second change that leaps to the eye is the unfolding of the bifurcation line where the Hamiltonian type A gained stability by means of a symmetry breaking bifurcation and in the process gave birth to the so-called ML-motion; we will come back to this motion in forthcoming papers [5,8]. In this (Hamiltonian) symmetry breaking the elements t and RTt were lost.…”

“…One way to lift (part of) this degeneracy is to include extra non-linear terms in our equations of motion, for instance a third-order coupling term. A more detailed discussion is therefore postponed to a forthcoming paper dealing with non-linear coupling [8].…”

“…Indeed, A and MP together form a link between 1~ and 2ft. We will devote a short subsection to this mixed motion, but leave a detailed discussion for a later paper dealing with nonlinearities in the coupling [8]. The reason for this is that, for linear coupling, the birthlines R= and G run very close together (especially, near the intersection point of the two resonance tongues); this can be remedied by including a third-order term in the coupling between the pendulums.…”

Mode competition in a system of two parametrically driven pendulums; the role of symmetry

“…But even for a system in which the birthlines are associated with period doublings as in our two-pendulum model, and even if it has exactly the same symmetry structure, small differences in the nonlinear terms of the equations of motion may cause the interaction to look quite dissimilar. A good example of this is provided by the compound pendulum of Skeldon and Mullin [6] and Skeldon [7], or simply by our own model with a different choice of (nonlinear) coupling [3].…”