Dynamic pull-down theory for the Toda oscillator

The pull-down instability is a newly found phenomenon for oscillators with even nonlinearities. It is a seemingly periodic motion, which will suddenly pull down after some cycles. The phenomenon is similar to the pull-in instability in the micro-electromechanical systems. This paper restudies the well-known Toda oscillator, which is considered to be in periodic motion. Its pseudo-periodic property is studied and the pull-down instability is found for the first time ever. To this end, this paper establishes a variational principle for the Toda oscillator, and the criterion for prediction of the pull-down motion is obtained. A simple formulation to calculate the pull-down instability is proved and verified through examples. Furthermore, an approximate frequency formula is obtained by Li-He modified homotopy perturbation method.