Nonlinear Energy Harvesting F. Cottone,* H. Vocca, and L. Gammaitoni��� NiPS Laboratory, Dipartimento di Fisica, Universita �� di Perugia, and Instituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06100 Perugia, Italy (Received 18 September 2008 published 23 February 2009) Ambient energy harvesting has been in recent years the recurring object of a number of research efforts aimed at providing an autonomous solution to the powering of small-scale electronic mobile devices. Among the different solutions, vibration energy harvesting has played a major role due to the almost universal presence of mechanical vibrations. Here we propose a new method based on the exploitation of the dynamical features of stochastic nonlinear oscillators. Such a method is shown to outperform standard linear oscillators and to overcome some of the most severe limitations of present approaches. We demonstrate the superior performances of this method by applying it to piezoelectric energy harvesting from ambient vibration. DOI: 10.1103/PhysRevLett.102.080601 PACS numbers: 05.40.Ca, 05.10.Ln, 05.45. a, 84.60. h The efficient powering of small-scale electronic mobile devices [1���3] is still an open problem. Old-style solutions, i.e., disposable batteries, cannot always be employed due to a number of reasons, chief among others the practical impossibility of replacement once exhausted. For such reasons, a new approach based on the exploitation of energy harvested where and when available has attracted considerable attention. Specifically, vibration energy har- vesting and ambient light exploitation are believed to con- stitute a potentially viable solution. Ambient vibrations come in a vast variety of forms from sources as diverse as wind induced movements, seismic noise, and car���s motion. Present working solutions for vibration-to- electricity [4���7] conversion are based on linear, i.e., reso- nant, mechanical oscillators that convert kinetic energy via capacitive, inductive, or piezoelectric methods [8���10] by tuning their resonant frequency in the spectral region where most of the energy is available. However, in the vast majority of cases, the ambient vibrations have their energy distributed over a wide spectrum of frequencies, with significant predominance of low frequency compo- nents, and frequency tuning is not always possible due to geometrical or dynamical constraints [10,11]. To overcome these difficulties, we propose a different approach based on the exploitation of the properties of nonlinear (i.e., nonresonant) oscillators. Specifically, we demonstrate that a bistable oscillator, under proper oper- ating conditions [12] can provide better performances compared to a linear oscillator in terms of the energy extracted from a generic wide spectrum vibration. In fact, a nonlinear oscillator, as the one that we discuss here, by default can present a wide spectral response (much wider than a linear or resonant one) and can be operated in such a way that its frequency response matches more closely what is available in the environment. In this regard, we point out that we are dealing here with open systems far from equilibrium, and the energy conversion mechanism is af- fected by both the spectral distribution and by the intensity of the vibrational energy available and is directly con- nected to the amplitude of the motion of the oscillating elements. Moreover, we note that the dynamical features discussed here are not limited to the sole piezoelectric energy conversion but can be applied also to other prin- ciples, e.g., capacitive and inductive. For the sake of demonstration, we realized a toy-model oscillator made by a piezoelectric inverted pendulum (Fig. 1) where on top of the pendulum mass a small magnet (tip magnet) has been added. The effect of ground vibration force is reproduced by applying a properly designed mag- netic excitation on two small magnets attached near the base of the pendulum. Under the action of the excitation, the pendulum oscillates, alternatively bending the piezo- electric beam and thus generating a measurable voltage signal. The dynamics of the inverted pendulum tip can be controlled with the introduction of an external magnet conveniently placed at a certain distance and with po- larities opposed to those of the tip magnet. The external magnet introduces a force dependent from that opposes the elastic restoring force of the bended beam. As a result, the inverted pendulum dynamics can show two different types of behaviors as a function of the distance . Specifically, when the external magnet is far away, the inverted pendulum behaves like a linear oscillator whose dynamics is resonant with a resonance frequency deter- mined by the system parameters. This situation accounts well for the usual operating condition of traditional piezo- electric vibration-to-electric energy converters [6]. On the other hand, when is small enough, two new equilibrium positions appear. The random vibration makes the pendu- lum swing in a more complex way with small oscillations around each of the two equilibrium positions and large excursions from one to the other. In order to quantify the energy produced by the piezoelectric oscillator, we com- puted the power dissipated in a purely resistive load, by PRL 102, 080601 (2009) P H Y S I C A L R E V I E W L E T T E R S week ending 27 FEBRUARY 2009 0031-9007=09=102(8)=080601(4) 080601-1 �� 2009 The American Physical Society

measuring the voltage drop V over a resistive load RL [7], under the influence of a random vibration with Gaussian distribution (with zero mean and standard deviation ') and exponential autocorrelation function (with correlation time (). In Fig. 2 (upper panel), we show the average electrical power hV2i=RL as a function of for three different values of the noise standard deviation '. In all the cases, the power increases rapidly from the linear case (large ) up to a maximum value and then decreases when the magnets become closer and closer. A qualitatively similar behavior is observed (Fig. 2, lower panel) if we plot the pendulum rms position xrms, as a function of . In order to quantitatively account for the experiments, we developed a dynamical description of the inverted pendulum based on the following equation of motion: m��� x �� dU��x�� dx _ x KvV��t�� �� '$��t��: (1) The first term on the right-hand side accounts for the conservative force, where U��x�� is the potential energy of the pendulum [13] shown in Fig. 3. U��x�� �� Kx2 �� ��ax2 �� b 2�� 3=2 �� c 2 (2) with K, a, b, and c representing constants related to the physical parameters of the pendulum [14,15] (see Fig. 1): K �� Keff=2 with Keff effective elastic constant a �� d2��"0M2=2%d�� 2=3 with "0 the permeability constant, M �� 0:051 A m2, the effective magnetic moment, and d �� 2:97 a geometrical parameter related to the distance be- tween the measurement point and the pendulum length b �� a=d2 and c �� K=d2. The second term on the right-hand side of (1) _ x accounts for the energy dissipation due to the bending, and KvV��t�� accounts for energy transferred to the electric FIG. 2 (color online). Piezoelectric oscillator mean electric power (upper panel) and position xrms (lower panel) as a function of for three different values of the noise standard deviation '. The symbols correspond to experimental values measured from the apparatus in Fig. 1. The continuous curves have been obtained from the numerical solution of the stochastic differen- tial equation (1). Both in the experiment and in the numerical solution, the stochastic force has the same statistical properties with correlation time ( �� 0:1 s. Every data point is obtained from averaging the rms values of ten time series sampled at a frequency of 1 kHz for 200 s. The rms is computed after zero averaging the time series. The expected relative error in the numerical solution is within 10%. FIG. 1 (color online). Schematic of the experimental appara- tus. The inverted pendulum is a four-layer piezoelectric beam made by lead zirconate titanate (PSI-5A4E) 60 mm of free length, clamped at one end. The piezoelectric beam has a width of 5 mm and a thickness of 0.86 mm. The pendulum mass is a steel cylinder 140.0 mm long and with diameter of 4.0 mm, with three magnets attached (each magnetic dipole moment is 0:051 A m2). The inverted pendulum resonance frequency (in the linear regime) is 6.67 Hz. The displacement x is measured via an optical readout. The voltage signal from the piezo is measured through a load resistor RL placed in parallel. The actual magni- tude of the standard deviation of the vibrational force applied in the three cases is ' �� 3 10 4 N, 6 10 4 N, 12 10 4 N. The load resistance is R �� RL �� 100 M and the piezo capaci- tance C �� 112nF. The effective mass is m �� 0:0155 kg. The damping constant is �� 0:016 Hz. PRL 102, 080601 (2009) P H Y S I C A L R E V I E W L E T T E R S week ending 27 FEBRUARY 2009 080601-2