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)ejw0t]. By setting Im [f(t)ejw0t] 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.
Complex Envelopes for Oscillations in Slightly Nonlinear Systems
W. H. Pierce
Electrical Engineering Department, University of Louisville, Louisville, Ky.
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Pierce, W. H. (December 1, 1971). "Complex Envelopes for Oscillations in Slightly Nonlinear Systems." ASME. J. Appl. Mech. December 1971; 38(4): 1012–1016. https://doi.org/10.1115/1.3408903
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