The linear oscillator under parametric excitation with fluctuating frequency

Abstract

The title problem is considered with the parametric excitation frequency varying sinusoidally about a mean value. The functional form assumed for the parametric variation of the oscillator's stiffness is e cos (2 Ωt + α cos β t) with a mean frequency 2 Ω and a frequency fluctuation of magnitude αβ . The structure of the principal instability zones for small e is examined by various approximate procedures and the influence of the parameters investigated. The parametric excitation is essentially periodic and is equivalent to a multi-frequency input with a frequency spacing which depends on the ratio of 2 Ω to β . The single instability zone of order e of the Mathieu equation ( α = 0) is shown to separate into a group of instability zones of the same order each associated with one of the input frequencies. Estimates of the size of these zones are given. The problem is relevant to the vibration of machine structures or mechanisms in which the excitation frequency, in the course of a cycle of operation for example, fluctuates about its average value.