Exceeding the classical time-bandwidth product in nonlinear time-invariant systems

Abstract

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 ${\Delta }\omega$’s) for long times (large Δt’s). For standing (non-propagating) waves, i.e., vibrations, in particular, this limit takes on a simple form, ${\Delta }t \,{\Delta }\omega = 1$, where ${\Delta }\omega$ is the bandwidth over which localization (energy storage) occurs, and ${\Delta }t$ 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) ${\Delta }t$, the storage time in the resonator, as compared with an increase (decrease) of the system’s bandwidth ${\Delta }\omega$. 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.

Appendix: Data acquisition and postprocessing

Due to the very low frequency of oscillation of the pendulum, i.e., $\sim 0.5\, {\text{Hz}}$, typical accelerometers and vibrometers cannot be used to accurately measure its response. For this reason, we recorded (video-captured) the motion of the pendulum at 240 frames-per-second with full HD quality until it settled to its fixed point. For the purpose of tracking the motion of the pendulum, we covered the steel ball with white tape, placed a small black tracking point on it (cf. Fig. 3b and 3c), and filmed its motion at a 14-inch offset from the pendulum. After a video is recorded, we converted it to a series of gray-scale frames to speed up the data acquisition process. To decrease the noise from the capture, we placed a 60 × 60 pixel moving window on the tracking point to which we then applied a 2D smoothing Gaussian filter with standard deviation of 2. Once the motion of the pendulum is quantified, the coordinate system was placed at its fixed point and, considering the length of the pendulum, its pixel location was converted to an angle. To calculate the angular velocity, we differentiated the time series of the angle time series and suppressed the noise caused by numerical differentiation with a 3rd order lowpass Butterworth filter with cutoff frequency of 10 Hz. Finally, the angular acceleration was obtained by differentiating the filtered angular velocity.

Furthermore, to obtain the envelope of the angular velocity signal, we computed its absolute value and extracted its local maxima, which were at least 0.75 s apart, and assumed values of at least 0.01 rad/s. Then, to obtain the envelope signal, the local maxima were interpolated by Akima Spline curves [25] to avoid extreme fluctuations and ensure C¹ continuity and then were extrapolated to time $t=0$ at the left and to the right until the envelope became zero. Once the envelope of the angular velocity signal was obtained, one could compute the bandwidth, decay-time constant, and time-bandwidth product of the system associated with the specific input that produced the angular velocity signal.