Exceeding the classical time-bandwidth product in nonlinear time-invariant systems - A. Mojahed; K. L. Tsakmakidis; L. A. Bergamn; Vakakis, A.F. (Nonlinear Dynamics, 2022)
The classical “time-bandwidth” limit for linear time-invariant (LTI) devices in physics and engineering asserts that it is impossible to store broadband propagating waves (large Δω’s) for long times (large Δt’s). For standing (non-propagating) waves, i.e., vibrations, in particular, this limit takes on a simple form, <span id="MathJax-Element-2-Frame" class="MathJax" data-mathml="Δt<mspace width="thinmathspace" ></mspace>Δω=1">ΔtΔω=1, where Δω is the bandwidth over which localization (energy storage) occurs, and Δt is the storage time. This is related to a well-known result in dynamics, namely that one can achieve a high Q-factor (narrowband resonance) for low damping, or small Q-factor (broadband resonance) for high damping, but not simultaneously both. It thus remains a fundamental challenge in classical wave physics and vibration engineering to try to find ways to overcome this limit, not least because that would allow for storing broadband waves for long times, or achieving broadband resonance for low damping. Recent theoretical studies have suggested that such a feat might be possible in LTI terminated unidirectional waveguides or LTI topological “rainbow trapping” devices, although an experimental confirmation of either concept is still lacking. In this work, we consider a nonlinear but time-invariant mechanical system and demonstrate experimentally that its time-bandwidth product can exceed the classical time-bandwidth limit, thus achieving values both above and below unity, in an energy-tunable way. Our proposed structure consists of a single-degree-of-freedom nonlinear oscillator, rigidly coupled to a nondispersive waveguide. Upon developing the full theoretical framework for this class of nonlinear systems, we show how one may control the nonlinear flow of energy in the frequency domain, thereby managing to disproportionately decrease (increase) Δt, the storage time in the resonator, as compared with an increase (decrease) of the system’s bandwidth Δω. Our results pave the way toward conceiving and harnessing hitherto unattainable broadband and simultaneously low-loss wave-storage devices, both linear and nonlinear, for a host of key applications in wave physics and engineering.