Robust Resonance due to Non-standard Frequency Modulation

Abstract

It is shown that harmonic signals incorporating a new type of weak non-standard frequency modulation (wNSFM) have unexpected spectral properties, namely, ever expanding broadband frequency spectra with progressing time. As such, they represent a new class of signals with spectra with strong frequency-time coupling. Applying this wNSFM signal to excite the classical single-degree-of-freedom linear, time-invariant damped/undamped oscillator yields new unique types of highly robust resonance phenomena. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient and usfficent conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are robust to changes in the parameters of the wNSFM. Our results reveal a new class of interesting resonance phenomena in linear resonators subject to non-standard weakly frequency-modulated excitations.