Autoparametric interaction in a double pendulum system

Abstract

This article investigates a four-degree-of-freedom mechanical model comprising a horizontal bar onto which two identical pendula are fitted. The bar is suspended from a pair of springs and the left-hand-side pendulum is excited by means of a harmonic torque. The article shows that autoparametric interaction is possible by means of typical external and internal resonance conditions involving the system natural frequencies and excitation frequency, yielding an interesting case when the right-hand-side pendulum does not oscillate, but stays at rest. It is demonstrated that applying the standard method of multiple scales to this system leads to slow-time and subsequently steady-state equations representative of periodic responses; however, in common with previous findings reported in the literature for systems of four or more interacting modes, global solutions are not obtainable. This article then concentrates on discussing a proposed new modification to the method of multiple scales in which the effect of detuning is accentuated within the zeroth-order perturbation equations and it is then demonstrated that the numerical solutions from this approach to multiple scales yield results that are virtually indistinguishable from those obtained from direct numerical integration of the equations of motion. It is also shown that the algebraic structure of the steady-state solutions for the modified multiple scales analysis is identical to that obtained from a harmonic balance analysis for the case when the right-hand-side pendulum is decoupled. This particular decoupling case is prominent from examination of both the original equations of motion and the steady-state solutions irrespective of the analysis undertaken. This article concludes by showing that the translation and rotation of the bar are, in this particular case, mutually coupled and opposite in sign.