JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 20, NO. 6, DECEMBER 2011 1225

JMEMS LettersNonlinear Springs for Bandwidth-Tolerant

Vibration Energy Harvesting

Son D. Nguyen and Einar Halvorsen

Abstract—We experimentally investigate the usefulness of softeningsprings in a microelectromechanical systems electrostatic energy harvesterunder colored noise vibrations. It is shown that the nonlinear harvesterhas performance benefits when the vibration’s center frequency variesin the frequency range of its softening response. With a vibration 3-dBbandwidth of 50 Hz, less than 3-dB variation in output power can beobtained over a 85-Hz wide range of vibration center frequencies. Com-pared to a simulated linear-spring device, the nonlinear device gives moreoutput power for a wide range of vibration bandwidths. The nonlineardevice shows less than 1-dB variation in output power when the vibrationbandwidth varies from 12 to 120 Hz and is centered on the resonantfrequency. [2011-0198]

Index Terms—Energy harvesting, microelectromechanical devices, non-linear systems, power generation, vibrations.

I. INTRODUCTION

Traditional vibration energy harvesters are designed as linear res-onant structures. They have a very narrow bandwidth and operateefficiently only when the excitation frequency is very close to theresonant frequency of the harvesters. However, ambient vibrationsmay have a wide spectrum of frequencies or varying vibration spectra.There have been several attempts to overcome this limitation by tuningthe resonant frequency or widening the bandwidth of the harvesters[1]. Methods have been developed to tune the resonant frequency sothat it can match the excitation frequency [2], [3]. Bandwidth wideningcan be achieved using a generator array, a mechanical stopper [4], orbistable structures [5], [6]. Several authors have exploited nonlinearsuspensions, often created with magnets, to extend the bandwidth ofthe harvesters by hardening [7]–[10] and/or softening nonlinearities[11], [12]. Nonlinear harvesters are usually characterized by sinusoidalexcitations [9]–[13] or white noise excitations [9], [11], but real-worldvibrations need not be close to these ideal cases. Vibration in a machineroom is one example [14]. Of the very few works that consider finitebandwidth random vibrations [15]–[17], i.e., colored noise, only oneis experimental [15]. Hence, very little is known about the potentialbenefits of nonlinear energy harvesting outside the idealized casesof infinite or zero bandwidth. Furthermore, the response with finitebandwidth vibrations cannot be inferred from the idealized casesbecause the superposition principle is not valid for a nonlinear device.

In this letter, we present experiments on a nonlinear-spring harvesterunder colored noise vibrations over a range of vibration bandwidthsand center frequencies. The device utilizes nonlinear beams in micro-machined silicon and does not rely on magnets. Except for a slightly

Manuscript received June 29, 2011; revised August 27, 2011; acceptedSeptember 17, 2011. Date of publication October 27, 2011; date of currentversion December 2, 2011. This work was supported in part by The ResearchCouncil of Norway under Grant 191282. Subject Editor A. Seshia.

The authors are with the Department of Micro and Nano SystemsTechnology, Faculty of Technology and Maritime Sciences, VestfoldUniversity College, 3103 Tønsberg, Norway (e-mail: duy.s.nguyen@hive.no;einar.halvorsen@hive.no).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JMEMS.2011.2170824

Fig. 1. (i) Schematic drawing of MEMS electrostatic energy harvester.(ii) Spring force versus displacement of the nonlinear spring calculated by FEMand compared to a linear spring. (iii) Photograph of a corner of the electrostaticenergy harvester with a nonlinear spring and part of transducer 2.

smaller proof mass, it is similar to the device characterized with whitenoise vibrations in our previous work [11].

II. EXPERIMENTAL RESULTS

Fig. 1 shows the device’s geometry and its spring force versusdisplacement as calculated by the finite-element method (FEM) [11].We note that the softening spring effect occurs only in one direction,while the other direction is still hardening. The overall effect issoftening, which can be seen clearly in the frequency sweeps in Fig. 2,where the broadening of the frequency response only occurs in downsweeps. This is due to the proof mass mean position shifting towardthe range of lower stiffness as the amplitude increases. At a very slowsweep rate and small acceleration (inset figure in Fig. 2), the resonantfrequency f0 is about 668 Hz, and the harvester’s 3-dB bandwidth Bis about 1.37 Hz. In all our experiments, the load resistors are fixedat 12.5 MΩ (transducer 1) and 17.5 MΩ (transducer 2), which arethe optimal loads for each transducer in the linear-spring regime. Thischoice ensures that the loading does not artificially favor the nonlinearoperation.

In order to characterize the harvester under colored noise vibrations,we generate random noise vibrations with a two-sided power spectraldensity (PSD) given by

Sa(f) =1

π

∆A2f2

(f2c − f2)2 + ∆2f2

(1)

where A is the rms acceleration, fc is the center frequency of thecolored noise acceleration, and ∆ is the full bandwidth (at 3 dB). A iskept fixed when comparing the average output power under differentbandwidths and center frequencies of the vibrations.

Fig. 3 shows the average output power versus the vibration’s centerfrequency fc for a vibration bandwidth of 50 Hz and at different rmsaccelerations. Due to the softening effect, the maximum output poweris found at lower frequencies than f0 for higher accelerations. The3-dB bandwidth ∆c of the responses in Fig. 3 is a measure of thetolerance toward center frequency variations of the vibrations. It isworth noting that ∆c increases with increasing rms acceleration A

1057-7157/$26.00 © 2011 IEEE

1226 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 20, NO. 6, DECEMBER 2011

Fig. 2. Linearly increasing and decreasing frequency sweeps for rms acceler-ations of 0.045, 0.064, 0.088, 0.119, and 0.150 g. (Dot-and-dash lines) Downsweeps. (Solid lines) Up sweeps. The bias is 36 V.

Fig. 3. Average output power versus vibration center frequency for ∆ =50 Hz and rms accelerations of 0.097, 0.231, 0.360, and 0.488 g. Eachaverage output power point is obtained by averaging the rms values of threeexperiments. Each of these experiments lasts for 40 s.

beyond a certain value. For A = 0.097 g, ∆c ≈ 50 Hz, which is equalto the vibration’s bandwidth ∆. For larger A, ∆c > ∆, i.e., ∆c = 60,75, and 85 Hz for A = 0.231, 0.360, and 0.488 g, respectively. If theresponse bandwidth is taken instead at 1 dB below the peak power, itsvalues are 30, 31, 40, and 50 Hz, respectively, for the four values of A.The 1-dB bandwidth of the vibration is 25.4 Hz.

The output PSD as a function of frequency for some selectedvibration center frequencies fc at A = 0.488 g is shown in Fig. 4. Forfc = 660 Hz, which is close to f0, the harvester bandwidth is about48 Hz. When fc moves to lower frequencies, the harvester’s bandwidthis wider (about 63 Hz for fc = 640 and 72 Hz for fc = 620 Hz) butstill keeps the high cutoff frequency around the resonant frequency. Atfc = 600 Hz, the harvester’s bandwidth reduces because of the low-ered spectral weight of the vibration in the sensitive frequency rangeof the harvester. The harvester responds with a narrow bandwidtharound the resonant frequency when the frequency is further reducedto fc = 570 Hz because the frequency mismatch is too large to excitethe harvester into the strongly nonlinear regime of softening springbehavior. Although the peak value of the output PSD may appear high,the integrated PSD is not. It is proportional to the output power inFig. 3, where we obtained 26.9 nW for fc = 570 Hz, which is 7.2 dBlower than 140.3 nW obtained for fc = 660 Hz.

Next, we study the effect of the vibration bandwidth ∆ on the av-erage output power. The average output power against ∆ for different

Fig. 4. Output PSD versus frequency for different vibration center frequen-cies, vibration bandwidth ∆ = 50 Hz, and rms acceleration A = 0.488 g.

Fig. 5. Average output power versus vibration bandwidth ∆ for differentvibration center frequencies fc and an rms acceleration of 0.010 g.

fc’s and very small rms accelerations is shown in Fig. 5. At this level ofacceleration, the contribution of the spring nonlinearity is insignificant,and hence, the harvester operates in a linear-spring regime. It canbe seen that the harvester then works best when fc is close to f0 =668 Hz and ∆ < B [18]. The output power significantly reduces whenfc is far from f0 (i.e., fc = 630 and 610 Hz) or when ∆ > B because,then, much of the spectral content of the vibration is filtered out. Thatis the main limitation of energy harvesters with linear springs.

Fig. 6 shows the variations of the average output power with thevibration’s bandwidth ∆ for an rms acceleration of 0.488 g, whichis large enough to give the spring softening effect. We also plot thesimulated average output power for a harvester using linear springsunder the same excitation. When fc = f0 = 668 Hz and ∆ is small,which is the condition for the best performance of a linear harvester,the average output power with nonlinear springs is much lower thanthat with a linear spring. The linear spring would result in 1.5 µWat ∆ = 0.2 Hz with the proof mass reaching the maximum possibledisplacement, i.e., hits the mechanical end stops (not shown in Fig. 6).However, when ∆ is larger than about 80 Hz, the average power ofthe linear spring harvester is smaller than the average power of thenonlinear-spring harvester. When fc shifts to lower frequencies, i.e.,at 630 or 610 Hz, the performance of the harvester using nonlinearsprings is much better than the harvester using linear springs, alwaysgiving higher average power than the linear-spring harvester. In partic-ular, at fc = 630 Hz and ∆ = 30 Hz, the average output power withnonlinear springs is higher than that with linear springs by a factor of3.8. The output power for the nonlinear device varies less than 1 dBover a range of bandwidths from 12 to 120 Hz for fc = 668 Hz.

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Fig. 6. Measured average output power of nonlinear-spring device versusvibration bandwidth ∆ for different center frequencies, fc = f0 = 668 Hz,fc = 630 Hz and fc = 610 Hz, and an rms acceleration of 0.488 g. SPICEsimulations for a comparable linear-spring harvester.

III. CONCLUSION

We have experimentally investigated the usefulness of softeningsprings in a microelectromechanical systems (MEMS) vibration en-ergy harvester under colored noise excitations. Varying the vibrationcenter frequency, we found a response bandwidth of up to 85 Hz,exceeding the vibration bandwidth of 50 Hz. For vibration centerfrequencies 38 or 58 Hz below the harvester’s resonant frequency,the nonlinear-spring harvester always achieves more power than alinear-spring harvester regardless of the vibration’s bandwidth. Thenonlinear device showed less than 1-dB variation in output powerwhen the vibration bandwidth varied from 12 to 120 Hz around theresonant frequency. In conclusion, the nonlinear device is toleranttoward variations in vibration center frequency and bandwidth. It ispromising that the desired nonlinearity can be obtained purely bymechanical design.

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