Investigation of a Nonlinear Vibrational Energy Harvester in the Stochastic Resonance Regime ^†

Abstract

In this paper the possibility to exploit advantageously the Stochastic Resonance phenomenon in a Nonlinear Energy Harvester to scavenge energy from wide band mechanical vibrations is experimentally demonstrated. The device is demonstrated to be capable of scavenging energy in case of a subthreshold sinusoidal vibration and a wideband noise (limited at 100 Hz) superimposed. The existence of an optimal value of the noise intensity maximizing the switching ratio of the bistable beam, then the performances, is experimentally demonstrated. The harvester is observed to generate power up to about 60 µW and 150 µW in case of a subthreshold sinusoidal input at 1 Hz and 3 Hz with a superimposed noise limited at 100 Hz.

1. Introduction

Mechanical vibration sources are ubiquitously available in the environment with different characteristics. Examples of such sources, available to be exploited for energy scavenging, are: seismic noise, vehicles motion, acoustic noise, multi-tone vibrating systems, human activity, domestic appliances, vibrating machines etc. [1]. Typically, energy coming from these sources is distributed over a wide frequency spectrum.

Since traditional linear solutions, based on resonant structures, are efficient when the energy is concentrated very close to their resonance frequency and are stable in time, nonlinear architectures have been proposed. In general, nonlinear architectures are characterized by a lower efficiency compared to linear solutions operated in the resonant regime, but, on average, constant (or slightly dependent on the frequency) on a wider range of frequencies. Such characteristic make nonlinear mechanical structures interesting to implement solutions for energy harvesting from vibrations.

Among different solutions available in state of the art, Snap Through Buckling (STB) structures emerge [2,3]. Main advantages of the STB configuration are related to its intrinsic nonlinear bistable dynamics. In particular the large displacement of the beam and the fast switching between the two stable states are the two main characteristics which can be advantageously exploited for the sake of power conversion process [4]. As a main drawback the STB configuration requires a minimum acceleration to activate the beam switching between its two stable states. Such threshold acceleration depends on the system characteristics (beam length, pre-compression, proof mass) [3,5].

Solutions exploiting a STB beam configuration have been recently presented by the authors [3,5,6,7,8] and demonstrated to be capable of power a wireless transmitter [9]. In [10,11] a magnetic repulsion mechanism to compensate for the asymmetric behavior of the device, in the vertical direction, due to the effect of the proof mass load, has been presented.

The frequency spectrum of many environmental vibrations sources is characterized by the presence of a periodic low frequency and low amplitude tone, which can change in time, with superimposed a wideband noise. Examples could be rotating machines, wheels of vehicles, people walking and running. In a typical real scenario such a periodic tone, as well as the noise, is weak. Actually, their amplitude, considered separately, are lower than the STB beam threshold, thus not sufficient to activate the beam switching process.

The presence of the superimposed noise, can be advantageously exploited to activate the beam switching. Actually, in the right conditions (amplitude of the periodic tone sufficiently close to the threshold and sufficiently large noise variance), the superimposed (to the periodic tone) noise can overpass the beam threshold thus making the beam switching. In particular, there exists an optimal value of the noise amplitude, σ_opt, which starts the beam to switch with the same frequency of the subthreshold periodic solicitation. In other terms the beam switching is synchronized with the input periodic solicitation. Such a condition is known as the Stochastic Resonance (SR) regime [12].

Actually, the existence of such a condition has been predicted by the behavioral model presented in [7,8]. In this paper, preliminary experimental results providing an evidence of the possibility to exploit the SR phenomenon in a nonlinear bistable vibrational energy harvester are presented.

2. The Nonlinear Harvester and the Experimental Setup

The nonlinear bistable harvester consists of a flexible STB beam, a proof mass placed in the middle of the beam and two lateral piezoelectric transducers. A schematization of a top view of the device is shown in Figure 1a.

The beam is a strip of PolyEthylene Terephthalate (PET) of dimensions 6 cm (pre-compressed) by 1 cm and thickness 140 µm. The pre-compression Δy = 1 mm applied along the Y axis and the proof mass of 5.2 g assured a good compromise between the frequency bandwidth and the switching threshold (minimum acceleration), compatible with the applications addressed [3]. The distance between the two stable states is Δx = 9 mm.

In static conditions, the beam rests in one of the two stable states. When a suprathreshold external force (acceleration) is applied along the transverse direction (X axis), the beam switches from its stable state to the other one and hits the piezoelectric transducers, which, vibrating to their natural frequency produce electric charges.

An optimal resistive load of 15 kΩ [6] assuring the maxim power transfer was connected to the output terminals of the piezoelectric transducers.

The proof mass on the beam consists of two identical neodymium (NdFeB) permanent magnets SN-10-04-N [13] having diameter of 10 mm and height of 4 mm (each one).

In order to continuously monitor the displacement of the proof mass (beam’s switching), an infrared distance measurement module, QTR-1A by Pololu, has been used.

An electromagnetic shaker consisting of two identical electromagnets WF-P34/18 driven by counter phase current signals through by a dedicated electronic has been employed for the sake of the device characterization.

A real view of the experimental setup (shaker and the harvester in between) is shown in Figure 1b.

Test voltage signals have been generated by the Agilent 33120A function generator and converted in current signals by the driving electronic. The driving currents, the piezoelectric transducers’ output voltages and the infrared distance sensor’s output voltage, were acquired by the MSO9064A scope by Agilent Technologies with a sampling frequency fs = 2 kHz, stored and processed, offline, by dedicated algorithms developed in Matlab^®.

Figure 1. The nonlinear STB energy harvester. (a) A schematization of a top view of the device; (b) A real view of the experimental setup with the electromagnetic shaker, the conditioning electronics and the harvester in between the two electromagnets.

Figure 1. The nonlinear STB energy harvester. (a) A schematization of a top view of the device; (b) A real view of the experimental setup with the electromagnetic shaker, the conditioning electronics and the harvester in between the two electromagnets.

3. Experimental Results

The experiments were aimed at estimating the response of the STB device in case of a low (subthreshold) deterministic periodic input with a superimposed noise limited at 100 Hz. In particular, the main target was to demonstrate experimentally the existence of an optimal value of the noise amplitude, σ_opt, which permits to operate the device in the Stochastic Resonance regime. In such an operative condition, the beam starts to switch with the same frequency of the subthreshold periodic solicitation (synchronization).

To this aim, the Signal to Noise Ratio (SNR) estimated by computing the ratio between the values of the Power Spectral Density (PSD) of the beam displacement, estimated at the periodic signal frequency with and without the periodic driving signal, has been observed while changing the variance of the noise.

Figure 2a,b show the (SNR) as a function of the variance of the superimposed noise limited at 100 Hz, in case of a subthreshold deterministic input at (a) 1 Hz and (b) 3 Hz. As expected an optimal value, σ_opt, of the noise level maximizes the SNR.

Table 1. Information about the threshold acceleration, a_{th_max}, the subthreshold acceleration of the adopted periodic input, a_{sub_max}, the STD of the optimal noise, σ_opt, and the electrical power, P_e, measured on a resistive load of 15 kΩ.

Frequency of the Subthreshold Input, (Hz)	STB Beam Threshold, ath_max (m/s2)	Subthreshold Input, asub_max (m/s2)	STD of the Noise, σopt, (m/s2)	Electrical Power, Pe, (μW)
1	19.71	12.81	7.65	60
3	22.49	14.01	9.23	150

provides information about the threshold acceleration, a_{th_max}, the subthreshold acceleration of the adopted periodic input, a_{sub_max}, the standard deviation, STD, of the optimal noise, σ_opt, and the electrical power, P_e, measured on the optimal resistive load of 15 kΩ.

Figure 2. The Signal to Noise Ratio (SNR) vs the variance of the superimposed noise limited at 100 Hz, in case of a subthreshold deterministic input at (a) 1 Hz and (b) 3 Hz. As expected an optimal value of the noise level, σ_opt, maximizes the SNR.

Figure 2. The Signal to Noise Ratio (SNR) vs the variance of the superimposed noise limited at 100 Hz, in case of a subthreshold deterministic input at (a) 1 Hz and (b) 3 Hz. As expected an optimal value of the noise level, σ_opt, maximizes the SNR.