Complex Envelopes for Oscillations in Slightly Nonlinear Systems

Abstract

A complex envelope f (t) is used to replace the traditional sine and cosine envelopes of the response of a sinusoidally forced, slightly nonlinear system, so that the response is y(t) = Re [f (t)ejw0 t ]. By setting Im [f (t)ejw0 t ] equal to the Hilbert transform of y(t), simple equations for the derivative of f (t) are easily obtained by the method of averaging. The approximations implicit in the method of averaging are shown to introduce broad-band filtering, whose effects can be eliminated by use of all harmonics. An example of an off-center rotary load shaken by a general elliptical motion shows how resonant modes of the device’s mount can be selectively energized at different frequencies, provided the modes have high Q and adequate separation of resonant frequency.