PHYSICAL REVIEW E 94, 062205 (2016) Harvesting wind energy to detect weak signals using mechanical stochastic resonance Barbara J. Breen, * Jillian G. Rix, Samuel J. Ross, and Yue Yu () Physics Department, Grinnell College, Grinnell, Iowa 50112, USA John F. Lindner, Nathan Mathewson, Elliot R. Wainwright, and Ian Wilson Physics Department, The College of Wooster, Wooster, Ohio 44691 USA (Received 6 September 2016; published 7 December 2016) Wind is free and ubiquitous and can be harnessed in multiple ways. We demonstrate mechanical stochastic resonance in a tabletop experiment in which wind energy is harvested to amplify weak periodic signals detected via the movement of an inverted pendulum. Unlike earlier mechanical stochastic resonance experiments, where noise was added via electrically driven vibrations, our broad-spectrum noise source is a single flapping flag. The regime of the experiment is readily accessible, with wind speeds ∼20 m/s and signal frequencies ∼1 Hz. We readily obtain signal-to-noise ratios on the order of 10 dB. DOI: 10.1103/PhysRevE.94.062205 I. INTRODUCTION Energy harvesting is an exciting research area in which attempts are made to extract clean energy from ambient sources to power small electronic devices [1–3]. Energy harvester energy sources are free. A classic example is a crystal radio receiver powered by the received radio waves. Especially in- teresting is the potential for nonlinear energy harvesting [4,5]. While linear energy harvesters typically tune their resonant frequencies to narrow spectral regions, nonlinear nonresonant oscillators can have much wider spectral responses. Stochastic resonance is a well-studied phenomenon where ambient noise amplifies weak signals in nonlinear sys- tems [6,7]. In bistable or threshold systems, broadband noise can boost faint signals, too weak for a sensor to detect otherwise, from subthreshold to superthreshold. Stochastic resonance has modeled a wide range of phenomena, from ice ages to hair cells [8–11]. Stochastic resonance has even been used for energy harvesting in bistable vibrating systems: Zheng et al. recently demonstrated stochastic resonance in a mechanical system of a cantilevered beam with an electrical vibrator as a noise source [12,13]. Here we describe stochastic resonance in a simple me- chanical system with an aeromechanical noise source. We achieved stochastic resonance by harvesting the noisy energy of a flapping flag to amplify weak periodic signals delivered to a bistable inverted pendulum. We find that the flapping flag can be an excellent broadband noise source and realize signal-to-noise ratios from 10 to 20 dB for moderate wind speeds of 20–25 m/s. In this article, Sec. II reviews theories of stochastic resonance and our bistable system. Section III describes our apparatus construction. Section IV details our experimental protocol. Section V analyzes our results. Section VI offers a summary, applications, and future work. II. THEORY A. Stochastic resonance Model a damped inverted pendulum at an angle θ by I ¨ θ =−γ ˙ θ − V ′ [θ ] + τ D sin[2πf D t ] + τ N ξ [t ], (1) * Corresponding author: breenbar@grinnell.edu where I is the rotational inertia, γ is the viscosity, τ D is the drive torque, f D is the drive frequency, τ N is the root-mean- square noise torque, and ξ [t ] is a random process with zero mean and unit variance. Overdots indicate time derivatives, and the prime indicates derivative with respect to the argument. The bistable potential V [θ ] has two wells separated by a barrier. If the barrier height between the wells is E and the vibrational noise has a variance σ 2 and correlation time τ , the probability of transition between the wells is proportional to the Kramers rate [14], whose leading behavior is given by the Arrhenius or Boltzmann factor P ∼ f K ∼ e −E/kT , (2) where the effective temperature kT = σ 2 τ/γ . In mechanical resonance, the periodic drive frequency equals a system’s natural frequency f D = f 0 . In stochastic resonance, the drive frequency is half the Kramers rate, f D = f K 2 , (3) so that on average the transitions between wells occur twice each drive period and T D = 2T K [15]. In practice, Fourier techniques reveal the statistical cor- relation between a weak periodic signal and noise in a bistable system. The power spectral density, or spectrum S , is proportional to the absolute square of the Fourier transform of the time series. The weak periodic drive embedded in the noise produces a narrow spectral peak at the drive frequency f D against a broad noise background. The signal-to-noise ratio R = S ∧ − N N = S ∧ N − 1  0, (4) where S ∧ = S [f D ] is the spectrum at the drive frequency and N is the background noise about the drive frequency. This definition ensures R = 0 in the absence of the drive. The signature of stochastic resonance is a local maximum of R at a nonzero value of noise. B. Inverted pendulum spring Our bistable element is an inverted pendulum of length ℓ rotating back and forth between two stops. We attach a spring of stiffness k and equilibrium length ℓ 0 to the pendulum bob 2470-0045/2016/94(6)/062205(4) 062205-1 ©2016 American Physical Society