BROADBAND VIBRATION ENERGY HARVESTING ...

BROADBAND VIBRATION ENERGY HARVESTING FROM A VERTICAL CANTILEVER PIEZOCOMPOSITE BEAM WITH TIP MASS Onur Bilgen†, Michael I. Friswell*, S. Faruque Ali‡, Grzegorz Litak§ †

†

Corresponding author. Email: [email protected] Mechanical and Aerospace Engineering, Old Dominion University, 238 Kaufman Hall, Norfolk, VA 23529, USA * College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK ‡ Department of Applied Mechanics, Indian Institute of Technology, Madras, Chennai, 600 036, India § Department of Applied Mechanics, Technical University of Lublin, PL-20-618 Lublin, Poland

ABSTRACT

An inverted cantilevered beam vibration energy harvester with a tip mass is evaluated for its electromechanical efficiency and power output capacity in the presence of pure harmonic, pure random and various combinations of harmonic and random base excitation cases. The energy harvester employs a composite piezoelectric material device that is bonded near the root of the beam. The tip mass is used to introduce non-linearity to the system by inducing buckling in some configurations and avoiding it in others. The system dynamics include multiple solutions and jumps between the potential wells, and these are exploited in the harvesting device. This configuration exploits the non-linear properties of the system using base excitation in conjunction with the tip mass at the end of the beam. Such a non-linear device has the potential to work well when the input excitation does not have a dominant harmonic component at a fixed frequency. The paper presents an extensive experimental analysis, results and interesting conclusions derived directly from the experiments supported by numerical simulations. INTRODUCTION

Continuing improvements such as reduction of cost/size/weight enable the use of electronics in almost every synthetic or biological system. However, the energy limitation of portable power sources still is the fundamental bottleneck for virtually all applications. In this context, the ambient energy from solar loads, wind loads, thermal gradients or mechanical vibration in structures, vehicles, etc. can be harvested and used to provide the energy needed for necessary tasks. Reviews of the literature on energy harvesting are given in Sodano et al. [1], Anton and Sodano [2], Cook-Chennault et al. [3], and Priya and Inman [4]. Furthermore, focused reviews on the topic of nonlinear vibration energy harvesting are recently presented by Pellegrini et al. [5] and Harne and Wang [6]. There is a significant research and development effort for the purpose of enabling smart materials for energy harvesting in applications ranging from battery-free pace-makers (see Karami and Inman [7]) to structural health monitoring of civil or aerospace structures (see Beeby et al. [8], Priya [9], Lefeuvre et al. [10,11]). Some smart materials, such as piezoelectric materials, can sense and harvest from mechanical stimuli as well as producing mechanical actions. Compared to electro-magnetic, thermo-electric, photo-voltaic energy conversion methods, a complete electro-mechanical, also referred to as piezoelectric, conversion system can easily be implemented in micro-meter scale. Perhaps, the fundamental benefit of a piezoelectric energy harvester is its “solid-state” nature which makes it possible, and sometimes practical, to implement in small systems or sub-systems requiring relatively low levels of power. Devices in the form of “solid-state” cantilevered beams, or its derivatives, have been proposed and accepted as the modern alternative to the well-known electromagnetic generator to harvest energy from vibrations. The following sections present three fundamentally different approaches for maximizing the power output of a vibration energy harvester. 1

Linear Vibration Energy Harvesters

Applications of piezoelectric materials are very broad and they are widely used as actuators and sensors for active vibration control of beams and plates (see Hagood and Anderson [12], Ha et al. [13], Crawley [14], Ghiringhelli et al. [15], Chee et al. [16], Benjeddou [17], Chopra [18], Leo [19]). Cantilever type vibration energy harvesting devices, such as beams, use a passive substrate to support a piezoelectric transduction element. The maximum power output of such a device occurs when the fundamental frequency of the device is near the dominant frequency of ambient vibration insuring a resonance response maximizing the strain into piezoelectric material. Sodano et al. [20, 21] investigated the transduction performance of several monolithic and piezocomposite devices. Electromechanical models and experimental validation are presented by Erturk and Inman [22], and Erturk et al. [23]. Bilgen [24] presented the resistive shunt energy harvesting comparison of several cantilever beam vibration energy harvesters including: 1) thin unimorphs employing the Macro-Fiber Composite devices, 2) non-uniform cross-section unimorph beams with thick substrates employing monolithic and fiber-composite piezoelectric devices and 3) uniform cross-section beams with single crystal and polycrystalline piezoelectric materials. Guyomar et al. [25] and Wickenheiser et al. [26] shows further examples of analysis and maximization of mechanical and electronic parameters of vibration energy harvesters. Ng and Liao (27), duToit et al. (28), Roundy (29), Renno et al. (30) have proposed methods to optimize the parameters of the system to maximize the harvested energy. Shu and Lien (31, 32), and Shu et al. (33) conducted detailed analyses of power output response of piezoelectric energy harvesting systems. Karami and Inman [34] presented the electromechanical modeling of a low frequency zigzag micro energy harvester. Employment of Single-Crystal Piezoelectric Materials

The majority of the research in employing piezoelectric materials focuses on the widely available Lead-Ziconium-Titanate (PZT) piezoelectric material composition. Single crystal piezoelectric materials have been recently proposed to enhance the power conversion when compared to polycrystalline piezoelectric materials. The application of single crystal piezoelectric materials for energy harvesting was first reported by Hong and Moon [35]. The design was later improved by using Polydimethylsiloxane (PDMS) for substrate and tip mass by Mathers et al. [36]. In a different design, reported by Badel et al. [37], steel was used as the substrate of the single crystal energy harvester. In addition to using single crystal material, the authors used nonlinear voltage conditioning/conversion circuits to enhance the power production. The trend was continued by Rakbamrung et al. [38]. It was shown that if nonlinear voltage processing is implemented, the performance of PZT harvesters will be close to the leadmagnesium-niobate-lead-titanate (PMN-PT) energy harvesting devices. To overcome the size limit of single crystal piezoelectric materials, Ren at al. [39] proposed making a composite from several individual segments. The individual PMN-PT patches were bonded by a soft epoxy. The harvesting performance of the suggested stack device was investigated further by Ren et al. [40] in 2010. Erturk et al. [41] verified the performance of single crystal energy harvesting devices with the analytical models. Karami et al. [42] presented an extensive experimental and theoretical parametric characterization of beam-like, uniform cross-section, unimorph structures employing single crystal piezoelectric materials. Three types of piezoelectric materials (single crystal PMN-PZT, polycrystalline PZT-5A, and PZT-5H type monolithic ceramics), and two types of substrate materials were compared in a unimorph cantilevered beam configuration. To further optimize the harvested power, the relation between the thickness of the substrate and the power output for different substrate materials is examined. Geometrically Non-Linear Vibration Energy Harvesters

One of the drawbacks of linear energy harvesters, regardless of the type of its active material, is that generally they are efficient only when the excitation frequency is around the resonance 2

frequency (see Daqaq [43]). Therefore, most linear energy harvesting devices are designed on the assumption that the (base) excitation has some known form, typically harmonic excitation. However, there are many situations where energy harvesting devices are operating under unknown or random excitations, and in such situations harvesters with a broadband or adaptive response are likely to be beneficial. One approach is to adaptively change the parameters of the linear harvester so that its natural frequency becomes close to the excitation frequency as it changes (Wang et al. [44]). Such adaptive systems may be difficult to implement in general and also may not adapt well to a broadband excitation. Ferrari et al. [45] used an array of cantilever beam harvesters tuned to different frequencies. An alternative approach to maximize the harvested energy over a wide range of excitation frequency uses nonlinear structural systems and a range of devices have been proposed by Cottone et al. [46] and Gammaitoni et al. [47, 48]. The key aspect of the nonlinear harvesters is the use of a double well potential function, so that the device will have two equilibrium positions (Cottone et al. [46], Mann and Owens [49], Ramlan et al., [50], Ferrari et al. [51], Quinn et al. [52]). Gammaitoni et al. [47], and Masana and Daqaq [53] highlighted the advantages of a double well potential for energy harvesting, particularly when inter-well dynamics are excited. The simplest equation of motion with a double well potential is the well-known Duffing oscillator, which has been extensively studied, particularly for sinusoidal excitation. The dynamics is often complex, sometimes with coexisting periodic solutions, and sometimes exhibiting a chaotic response. The Duffing oscillator model has been used for many energy harvesting simulations, with the addition of electromechanical coupling for the harvesting circuit. One popular implementation of such a potential is a piezomagnetoelastic system based on the magnetoelastic structure that was first investigated by Moon and Holmes [54] as a mechanical structure that exhibits strange attractor motions. Erturk et al. [55] investigated the potential of this device for energy harvesting when the excitation is harmonic and demonstrated an order of magnitude larger power output over the linear system (without magnets) for nonresonant excitation. One problem with multiple solutions to harmonic excitation is that the response can respond in the low-amplitude solution; Sebald et al. [56] proposed a method to excite the system to jump to the high-amplitude solution at low energy cost. Stanton et al. [57], and Erturk and Inman [58] investigated the dynamic response, including the chaotic response, for such a system. Cottone et al. [46] used an inverted beam with magnets and also considered random excitation. Mann and Sims [59], and Barton et al. [60] used an electromagnetic harvester with a cubic force nonlinearity. Litak et al. [61] and Ali et al. [62] investigated nonlinear piezomagnetoelastic energy harvesting under random broadband excitation. McInnes et al. [63] investigated the stochastic resonance phenomena for a nonlinear system with a double potential well. Friswell et al. [64] presented an experimentally validated analysis of an inverted cantilever beam with tip mass subjected to harmonic excitations. The tip mass is adjusted such that the system is near buckling and therefore has a low effective resonance frequency. Borowiec et al. [65] analyzed the efficiency of the inverted cantilever beam, focusing on the region of stochastic resonance where the beam motion has a large amplitude. Numerical analysis showed that increasing the noise level allows the motion of the beam system to escape from single well oscillations and thus generate more power. Current Motivation

The sections above highlighted some of the recent works in the literature related to piezoelectric vibration energy harvesting subjected to harmonic and/or broadband ambient excitations. In most cases the geometric non-linearities are introduced, although increasing the mechanical complexity of the system, to achieve a reasonable amount of energy when the excitation frequency is low. One example of such ambient condition occurs in long span bridges and tall buildings. A low frequency, mechanically linear, piezo-elastic or piezo-magneto-elastic 3

harvester is difficult to realize due to small physical dimensions of the devices, although some solutions have been proposed (see Karami and Inman [34]). To this end, Friswell et al. [64] proposed a new mechanical configuration, consisting of a piezocomposite cantilever beam with a tip mass that is mounted vertically and excited in the transverse direction at its base. This device is highly nonlinear with two potential wells for large tip masses, when the beam is buckled. The system dynamics may include multiple solutions and jumps between the potential wells, and these were exploited in the harvesting device. An electromechanical model is first developed and the response to harmonic base excitation is investigated. The model is validated using an experimental prototype with three different tip masses, representing three interesting cases: a linear system; a low natural frequency, nonbuckled beam; and a buckled beam. The most practical configuration was identified as the prebuckled case, where the proposed system had a low natural frequency, a high level of harvested power, and an increased bandwidth over a linear harvester. Note that previous examination of the proposed harvester presented in Friswell et al. [64] focused on pure-tone excitation. In the current paper, the inverted cantilever beam with tip mass is examined for its power output when subjected to a combination of harmonic and random excitations motivated by the fact that almost all ambient sources of vibration will be multi-tone. An early example of the analytical and numerical analysis of a Duffing’s oscillator, without energy harvesting function, subjected to combined harmonic and random excitation has been presented by Iyengar [66]. Outline

This paper reports an extensive experimental investigation of the broadband response of the inverted cantilever beam vibration energy harvester. First, the device is presented and the results from the previous modeling and experimental effort are briefly discussed. Second, numerical simulations of response to are presented. Next, the experimental setup is presented highlighting the experimental procedure. Following, the experimental results are presented and discussed. The paper concludes with a brief discussion of conclusions and future direction. THE INVERTED CANTILEVER BEAM WITH TIP MASS

This section briefly describes the coupled electromechanical governing equation of motion based on the Euler-Bernoulli beam theory presented by Friswell et al. [64]. The beam is assumed to have a total length 𝐿, and carries a concentrated tip mass, 𝑀! , with moment of inertia 𝐼! , at a position 𝐿! from the base of the beam as illustrated in Figure 1. It is assumed that the tip mass is significantly larger than the beam mass and hence a single mode approximation of the beam deformation is sufficient. In addition, the beam is assumed to have uniform inertia and stiffness properties. The beam has cross-sectional area, 𝐴, mass density 𝜌, equivalent Young’s modulus 𝐸, and second moment of area 𝐼.

4

Figure 1: An illustration of the inverted cantilever beam with tip mass subjected to transverse base excitation. The piezoelectric patch is placed along the beam, near the clamped end. Adapted from [64].

The displacement-curvature relation of the beam is nonlinear due to the large transverse displacement of the beam. It is assumed that the thickness of the beam is small compared with the length so that the effects of shear deformation and rotatory inertia of the beam can be neglected. The beam is such that the first torsional resonance frequency is much higher than the excitation frequency and the lumped mass is kept symmetric with respect to the center line. Hence the vibration is purely planar and the torsional modes of the beam are neglected in the analysis. These assumptions are mostly consistent with observations in the experiments. The horizontal and vertical displacements at the tip mass are 𝑣 and 𝑢 respectively, and 𝑠 represents the distance along the neutral axis of the beam. Point 𝑃 denotes an arbitrary point on the beam whose position is described by 𝑠, 𝑣! , and 𝑢! . The displacement at any point in the beam is represented as a function of the tip mass displacement through a function for the beam deformation, 𝜓(𝑠), as 𝑣! 𝑠, 𝑡 =   𝑣! 𝐿, 𝑡 𝜓 𝑠 =  𝑣 𝑡 𝜓 𝑠

(1)

The displacement may be approximated by any function satisfying the boundary conditions at 𝑠 = 0, for example (Esmailzadeh and Nakhaie-Jazar [67]): 𝜓! 𝑠 = 𝜆! 1 − cos

𝜋𝑠 2𝐿

(2)

where 𝜆! is a constant such that 𝜓 𝐿! = 1, 𝑖. 𝑒.    𝜆! = 1/ 1 − cos 𝜋𝐿! /2𝐿 . Now, suppose that a piezoelectric layer is added to the beam in a unimorph configuration. The moment about the beam neutral axis produced by a voltage 𝑉 across the piezoelectric layers (Crawley and de Luis [68], Crawley and Anderson [69]) may be written as 𝑀! 𝑠, 𝑡 = 𝛾! 𝑉 𝑡

(3)

where the constant 𝛾! depends on the geometry, configuration and the piezoelectric device. Hence, for a unimorph with the piezoelectric layer in the 31 configuration, with thickness ℎ! , width 𝑏! , the constant is 𝛾! = 𝐸! 𝑑!" 𝑏! ℎ + ℎ! /2 − 𝑧 , where ℎ is the thickness of the beam, 𝐸! is the piezoelectric material modulus, 𝑑!" is the piezoelectric constant, and 𝑧 is the effective neutral axis (Park et al. [70]). These expressions assume a monolithic piezoceramic device perfectly bonded to the beam. In the current research, a piezocomposite device called the MacroFiber Composite (MFC) is employed due to its ability to conform to large strains. Bilgen et al. 5

[71] showed that the MFC can be modeled effectively as a monolithic structure considering the effect of the fibrous nature of the MFC device on the coupling coefficient, and also the effect of the bond and insulating Kapton layers. A detailed discussion of the MFC will be presented later. Using the single mode approximation, the kinetic and potential energies of the system in terms of the transverse displacement of the tip mass are written. Next, the equation of motion of the beam-mass-piezoelectric system is derived in terms of the displacement of the tip mass using Lagrange’s equations as [64] 𝑁!! 𝐼! + 𝑀! + 𝜌𝐴𝑁! + 𝜌𝐴𝑁! + 𝑀! 𝑁!! + 𝑁!! 𝐼! 𝑣 ! 𝑣 +   𝜌𝐴  𝑁! + 𝑀! 𝑁!! + 𝑁!! 𝐼! 𝑣𝑣 ! +   𝐸𝐼𝑁! − 𝑁! 𝜌𝐴𝑔 − 𝑁! 𝑀! 𝑔 + 2𝐸𝐼𝑁! 𝑣 ! 𝑣 −   Θ! 𝑉 − Θ! 𝑣 ! 𝑉 =   − 𝜌𝐴𝑁! + 𝑀! 𝑍

(4)

Using the displacement model, 𝜓 𝑠 , the constants 𝑁! to 𝑁! are given in Friswell et al. [64]. The mechanical stiffness and mass density of the piezoelectric layers are also included in these constants. On the electrical side the piezoelectric devices may be considered as a capacitor, and the charge they produce is given by Θ! 𝑣 + Θ! 𝑣 ! where Θ! = 𝛾! 𝜓 ! 𝐿! and Θ! = 1/ ! 2 𝛾! 𝜓 ! 𝐿! . The electrical circuit considered is represented by a resistive shunt connected across the piezoelectric material. The electrical governing equation is [64] 𝐶! 𝑉 + 1/𝑅! 𝑉 + Θ! 𝑣 + Θ! 𝑣 ! 𝑣 = 0

(5)

where 𝑅! is the load resistor and 𝐶! is the capacitance of the piezoelectric material. Here, it is assumed that the vertical cantilever beam is subjected to a base displacement excitation 𝑍 𝑡 = 𝑍! cos 𝜔𝑡 + 𝑍! 𝑛(𝑡) where 𝑍! is the peak amplitude of the harmonic signal, 𝜔 is its excitation frequency, 𝑍! is the peak amplitude of the noise, and 𝑛 𝑡 is the white noise with maximum amplitude of one. PREVIOUS ANALYSIS OF RESPONSE TO PURE HARMONIC EXCITATION

This section briefly presents the electrical response of the inverted cantilever beam to pure harmonic excitation. The response of the beam to harmonic excitation with three tip mass configurations (𝑀! = 0, 10.5, and  14.0  g) corresponding to the free-beam, pre-buckled and post-buckled conditions respectively has been presented previously in Ref. [64], therefore a very brief and qualitative analysis is presented here for the purpose of demonstrating the maximum power generation capability of the inverted cantilever beam and the mechanical state(s) that causes such power generation. First, the theoretical analysis of the mechanical response is presented showing the characteristics of the motions (e.g. the existence of multiple solutions) that result in a variety of electrical response. Next, the results from the previous experimental analysis are presented showing the power generation capability as a function of harmonic base excitation amplitude and frequency. Theoretical Phase Portraits

For the preliminary numerical analysis presented in this section the beam-mass system is excited at the base with harmonic excitation. As expected, the simulations show that the postbuckled response has two stable equilibrium positions. Linearization about both equilibrium positions provides the same natural frequencies as the system is assumed to be symmetric. In reality, the beam tends to have preference towards one of the two equilibrium positions. The natural frequency of the inverted cantilever beam decreases with increasing tip mass and is zero at the Euler buckling load corresponding to an estimated tip mass of 10.0 g. Further increase in the tip mass cause the beam to buckle, and the natural frequencies about the stable equilibrium positions increase. Thus, the inverted beam-mass system is able to resonate at low frequencies 6

a)

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Velocity (mm/s) Velocity (mm/s) Tip Velocity, mm/s

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close to the buckling condition. The post-buckled equilibrium positions are quite sensitive to the tip mass. The response dynamics is presented in Figure 2 through the detailed analysis of time responses for typical tip mass values. Figure 2(a) shows period one response at the theoretical pre-buckled resonance. Figure 2(b) also shows a period one response, but the beam is in the theoretical post-buckled regime where most of the responses are chaotic and produce lower power output. An example of the chaotic response is given in Figure 2(c). Figure 2(d) shows the response for the highest average output power, which shows some chaotic response superimposed on a period one response. As noted above, the theoretical buckling load corresponds to a tip mass of 10.0 g. 500

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-500 -500

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Figure 2: Theoretical phase portraits of the tip mass response for a base excitation of 𝒁𝒉 = 𝟏𝟔  𝐦𝐦 at frequency 0.50 Hz and for a load resistance 𝑹𝑳 = 𝟏𝟎𝟎  𝐤𝛀: (a) 𝑴𝒕 = 𝟖. 𝟕𝟔  𝐠, 𝑷𝒂𝒗𝒆 = 𝟕. 𝟔𝟓  𝛍𝐖, (b) 𝑴𝒕 = 𝟏𝟎. 𝟑𝟗  𝐠, 𝑷𝒂𝒗𝒆 = 𝟏𝟔. 𝟑  𝛍𝐖, (c) 𝑴𝒕 = 𝟏𝟎. 𝟓  𝐠, 𝑷𝒂𝒗𝒆 = 𝟐. 𝟒𝟎  𝛍𝐖, (d) 𝑴𝒕 = 𝟏𝟎. 𝟔𝟏  𝐠, 𝑷𝒂𝒗𝒆 = 𝟏𝟕. 𝟖  𝛍𝐖. Adapted from [64].

In Figure 2, the dots represent the Poincaré points. The results were obtained using zero initial conditions for the tip mass displacement and velocity. Other examples of low- and highamplitude solutions for large values of tip mass are presented in Ref. [64]. The low-amplitude response oscillates in one of the potential wells, whereas the high-amplitude response is a period seven response crossing back and forth between the potential wells. It should be emphasized that other solutions also exist for other values of tip mass, for example period three or period five solutions. Experimental Power Output

In the previous experimental analysis of the response to harmonic excitation, three tip mass conditions were evaluated. In the current discussion, the responses of the free-beam (𝑀! = 0  𝑔) and the pre-buckled beam (𝑀! = 10.5  g) configurations to purely harmonic excitation are briefly presented to form a basis for comparison with broadband excitation. The previous experiment can be described as several cascaded loops, where both the setup and the procedure are very similar to the current experiments. Once a tip mass configuration was selected, the load resistance, base displacement excitation amplitude and frequency were swept in a desired range. The frequency sweep was conducted in both increasing and decreasing directions. The beam was excited for 30 complete cycles of the base excitation in order to minimize the effect of transient motion. Only the response during the last 10 cycles of the excitation were recorded and analyzed. The frequency was incremented in very small steps and the continuity of waveform was ensured between each frequency step therefore unwanted disturbances (e.g. rapid accelerations) to the 7

beam were minimized. This is important to note since such disturbances will result in a “premature” transition to the alternative solution as the frequency is increased or decreased. Figure 3 presents the average power-output-per-harmonic-base-acceleration response of the inverted cantilever beam with no tip mass. As expected, the response is approximately linear, with a resonance frequency range of 2.43 – 2.47 Hz for the base displacement range of 5 – 20 mm. Z h :5 mm

ave ,

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7 W / (mm/ s2) 2

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-4

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2.2

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2.8

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! , Hz

Figure 3: Electrical response of the inverted cantilever beam with no tip mass to harmonic base excitation. 𝑹𝑳 ≅ 𝑹𝑶𝑷𝑻 = 𝟏. 𝟔𝟔  𝐌𝛀. The notation “:” represents “=” and is used to shorten labels.

In Figure 3, both the increasing and decreasing frequency sweeps are shown which approximately lay on top of each other. The electrical output/input ratio decreases near resonance as the base displacement amplitude is increased. This is expected since a higher level of energy is dissipated due to strain and aerodynamic damping at higher beam velocities. Figure 4 presents the average power-output-per-harmonic-base-acceleration response of the inverted cantilever beam with 10.5 g tip mass. At this value of tip mass the beam has not buckled, although the simulations show that the maximum power is generated near buckling condition. This prediction is consistent with the experimental response when pre-buckled and post-buckled cases are compared. The pre- and post-buckling resonance frequencies of the device can be easily tuned by moving the tip mass and hence the resonance may be tuned to the harmonic excitation frequency.

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ave ,

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-6

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0.5

0.6 ! , Hz

0.7

0.8

Figure 4: Electrical response of the inverted cantilever beam with 10.5 g tip mass to harmonic base excitation. Solid line indicates increasing frequency sweep (𝝎 ↑) and the dashed line indicates decreasing sweep. 𝑹𝑳 = 𝟑. 𝟑𝟒  𝐌𝛀 for 𝒁𝒉 = 𝟏𝟓 − 𝟐𝟎  𝐦𝐦 and 𝝎 ↑. 𝑹𝑳 = 𝟗. 𝟗𝟒  𝐌𝛀 for others.

The plotted responses correspond to the experimentally determined optimum resistances for each sweep. The existence of low- and high-amplitude responses is clearly seen from the electrical output. In order to aid the comparison of responses to pure harmonic and broadband excitations, the average power-output is plotted against the base-acceleration for the beam with 10.5 g tip mass in Figure 5. 10

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Figure 5: Electrical response of the inverted cantilever beam with 10.5 g tip mass to harmonic base excitation. Solid line indicates increasing frequency sweep (𝝎 ↑) and the dashed line indicates decreasing sweep. 𝑹𝑳 = 𝟑. 𝟑𝟒  𝐌𝛀 for 𝒁𝒉 = 𝟏𝟓 − 𝟐𝟎  𝐦𝐦 and 𝝎 ↑. 𝑹𝑳 = 𝟗. 𝟗𝟒  𝐌𝛀 for others.

When the beam with no tip mass is compared to the pre-buckled beam, a significant reduction of net power output is observed as expected simply owing to the fact that the excitation frequency is significantly lower. If the power output is normalized by the base-acceleration, as shown in Figure 3 and Figure 4, both the output/input ratio and the useful bandwidth of the energy harvester is increased for the pre-buckled beam while the center of the bandwidth is reduced as desired. NUMERICAL SIMULATIONS

The system is simulated to qualitatively understand the effects random excitations to the overall power generation. An exact prediction of the system response is difficult because of 9

unmodeled nonlinearities (e.g. piezoelectric hysteresis, out-of-plane motion, etc.) and because the single degree of freedom model for the beam is insufficient at large displacements. However the low-amplitude response is used to validate the underlying linear model and estimate material damping, and these parameters are used in the current simulations. A full set of theoretical analyses using the model are presented by Friswell et al. [72]. The frequency of excitation in the simulation is taken as ω = 0.46 Hz, which is in middle of the theoretical frequency region for multiple solutions (as previously shown in Ref. 64). This is slightly different to the harmonic frequency in the experiments, because the region of multiple solutions is a relatively small range of frequencies and is very sensitive to the non-linear model parameters that were previously estimated from the linear low-excitation-amplitude experiments. Time simulations are used to test the effect of random forcing on the steady state arising from the sinusoidal. The initial conditions are set to obtain the high-amplitude solution, and for the first 20 seconds of the simulation the random forcing is set to zero to enable the steady state response to develop. Then the amplitude of the random forcing component is linearly increased over 20 s. The random forcing is white noise base displacement with a bandwidth of 5 Hz. The displacement response, and the corresponding average power, is then computed to determine if the system is responding in the high- or low-amplitude solution. Figure 6 shows the resulting average power as the noise level is increased. For low noise levels the high-amplitude response often occurs, but for high levels of random forcing the lowamplitude response is preferred. For intermediate noise levels the response starts at the highamplitude solution and at some time instant will jump to the low-amplitude solution; this indicates that the low-amplitude and low-energy solution is preferred at this excitation frequency, although there is also a small probability that the response will jump back to the highamplitude solution. This is more clear at high noise levels in Figure 6, which shows that the response can jump to the high-amplitude solution. If initial conditions are chosen so that the lowamplitude solution is obtained, before adding the random excitation, the high-amplitude solution would not be obtained until the noise level is very high. 18 16

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4000

ZB(n oi se)ave , mm/ s2

Figure 6: Average power output for increasing white noise base excitation combined with a sinusoidal base displacement at 𝝎 = 𝟎. 𝟒𝟔  Hz and 𝒁𝒉 = 𝟏𝟎  mm. The load resistance is 𝑹𝑳 = 𝟓. 𝟎𝟎  𝐌𝛀.

Since the non-linear model parameters are estimated using results from low-harmonicexcitation-amplitude experiments, it is expected that the simulations should under predict the power output for broadband excitation. There are several reasons for this: First, in broadband excitation, the base displacement input is inevitably exciting higher modes of the beam which can be observed visually and can be seen in the measured beam displacement signal in the 10

previous experiments. Second, the location of the MFC is near the base of the cantilevered beam covering a very short distance. Since the MFC is not placed anywhere near the node of a loworder mode, one does not expect an electrical cancellation across the length of the MFC. Combining the two observations above, one can conclude that the measured power in experiment should be larger than the predicted values in the model since the model accounts for only a single mode of deformation. The relative impact of higher modes vs. the first bending mode is expected to be a function of base excitation signal strength, signal bandwidth, etc. For some broadband excitation cases, the second and possibly the third (and higher) bending modes are expected to be significantly excited in comparison to the first bending mode. CURRENT EXPERIMENTAL ANALYSIS

The current experimental work investigates the electrical response of the pre-buckled inverted cantilevered beam subjected to a combination of a broadband random signal and a harmonic pure-tone signal. A white noise is selected as the random signal where the signal has a uniform power spectral density. Such ideological power spectral density, of course, is not achievable with a power limited electromechanical system, therefore the white noise signal is low-pass (LP) filtered at pre-defined cut-off values creating a band-limited white noise. The objective is to understand the effect of various electromechanical parameters of the system which is subjected to a band-limited white noise. The electromechanical response of the beam is characterized by its power output through a resistive shunt due to a transverse base-displacement excitation. A load resistor is used to characterize the energy harvesting performance; therefore the analysis is focused on the fundamental nonlinear dynamic behavior of the inverted cantilever beam energy harvester. Motivation

The experimental analysis is conducted mainly for two purposes. The main purpose is to provide experimental data for validation of the model within a desired parameter space. The second purpose is to determine the effects of other nonlinearities (e.g. geometric, piezoelectric hysteresis, out-of-plane motion, etc.) which are not included in the model. The main variables of interest of an inverted cantilever beam vibration energy harvester are 1) tip mass amount, 2) electrical impedance (load resistance in this case), 3) base excitation strength (harmonic and noise) and 4) base excitation frequency content (pure-tone frequency and noise bandwidth). Previous analytical modeling and experimental validation showed significant dependence of power output on the tip mass in the inverted-cantilevered-beam vibration energy harvester concept. More specifically, whether the inverted beam was pre-buckled or post-buckled has been determined to be a key factor. A pre-buckled inverted cantilever beam with a tip mass close to the equivalent Euler buckling load showed the best broadband electrical power output, measured in terms of power-per-base-acceleration-squared, for the case where the base excitation was a pure tone harmonic signal (see Ref. 64). Test Procedure

The experiment can be described as several cascaded loops sweeping the parameters of interest at discrete values. Steps 1-5, described below, are repeated for all discrete values of parameters within the pre-selected domains. 1) The low-pass filter cut-off frequency, LP, is selected: 1.0, 2.5 and 5.0 Hz. 2) The white noise peak signal amplitude, 𝑍! , is selected: 0, 20, 40, 60, 80 and 100 mm. 3) The load resistance, 𝑅! , is selected. 4) The harmonic signal amplitude, 𝑍! , is chosen: 0, 10, 20 mm. 5) The harmonic signal frequency, 𝜔, is set: 0.25, 0.50, 0.75, 1.00 Hz. The noise and harmonic signal strengths noted above is not exceeded due to several reasons which are artificially introduced by the experimental setup. The main reason being the magnets, 11

used to form the tip-mass, attracting and attaching to the metallic slider track when high curvatures along the beam length are present. The beam is excited for approximately 100 complete cycles of the harmonic signal, where applicable, for each combination of parameters. This resulted in a total acquisition period of 100 s (for 1.00 Hz), 133 s (for 0.75 Hz), 200 s (for 0.50 Hz) and 400 s (for 0.25 Hz) for each set of parameters. It will be mentioned later that only a fraction of the total acquisition will be used for power calculations. Each test case is started from zero initial conditions where every test had at least 10 seconds of idle time in between; however a few cases still had small but measurable non-zero initial conditions due to the large inertia of the system. Note that certain combinations of noise and pure-tone signal strengths result in purely-harmonic and purely-random excitation cases – this is intentionally implemented in the test procedure to observe all possible excitation scenarios. Hardware

A National Instruments (NI) cDAQ 9172 data acquisition system, controlled with a code written in LabVIEW software, is employed to automatically examine the electromechanical response of the harvester. The control signal for the base excitation is produced by an NI 9263 cDAQ module with 16bit resolution (set to +/-10 V range) at a generation rate of 10 kHz. As noted above, the control signal from the module is low-pass filtered using a Kemo (Type VBF/24) elliptic filter at three different cut-off frequencies. This is done to 1) minimize the high frequency noise from the digital-to-analog (D/A) converter and 2) partially simulate the natural LP filtering effect of the host structure where the energy would be extracted from. Note that the filtered output of the D/A is not measured as the reference signal; therefore the lag effect of the filter is avoided. Instead, the actual base displacement is measured. The base excitation signal is connected to a Bytronic Pendulum Control System. This device is composed of a belt driven linear slider that moves in a track which is actuated by a DC motor. A multi-turn potentiometer is used to monitor the position of the belt, hence the position of the linear slider. A displacement feedback controller ensures that the displacement is proportional to the control signal. Here, it is worth noting that the linear slider has relatively low inertia and that there is no return spring (as in an electromagnetic shaker). The fact that slider has low inertia causes the base motion to be effected by small imperfections in the linear track. In addition, since the system is driven by the forcing of the DC motor only (e.g. no return spring), the actual excitation deviates from the desired smooth excitation near 𝑍 = 0 condition. Both of these imperfections in the excitation waveform are measured by the potentiometer; however their effects on the general motion and the power output of the harvester are initially assumed negligible. The measurement of input-dynamics confirms that the static response of the linear actuator is 25.4 mm/V for harmonic excitation; however, the bandwidth reduces as desired output displacement magnitude is increased. It will be discussed later that the actual basedisplacement is measured. The attenuation, however, does result in a narrower base excitation bandwidth than the LP cutoff. For the current tests, where the maximum LP filter cut-off frequency is 5 Hz, the attenuation due to power limitation of the actuation system will be small for almost all of the test cases. A clamping mechanism, which applies uniform pressure across the clamped surface of the beam, is attached to the linear slider. Note that all experiments presented in this paper are conducted without removing the beam from the clamp; therefore consistent boundary conditions are achieved across all cases. A beam made of spring steel of thickness 0.245 mm, width 15.9 mm and free length 293 mm is used as the inverted cantilever. The beam mounted on the linear slider is shown in Figure 7.

12

a)

b) Base

Beam

Potentiometer

Drive Motor

Linear Track

c)

Figure 7: Picture of the experimental setup: a) Base of the beam showing the base clamp and the MFC device near the root; b) Tip mass of 10.5 g shown in a nearly vertical position at a stable equilibrium; c) Linear actuation system with the inverted cantilever beam mounted.

The tip mass is implemented using several disk-like neodymium magnets with a diameter of 10 mm, height of 5 mm and approximate weight of 1.75 g each, whose positions could be adjusted easily. In the model the tip mass is assumed to be at the end of the beam; the portion of the beam above the magnets in the experiment is assumed to have negligible effect on the dynamic behavior of the system and could be included in the model if desired. The Macro-Fiber Composite Device

A single piezocomposite patch, the Macro-Fiber Composite model M2814-P2 manufactured by Smart Material Corp., of nominal active length 28 mm and active width 14 mm is bonded to the beam near the clamped end in a unimorph configuration. The Macro-Fiber Composite was developed at the NASA Langley Research Center (Wilkie et al. [73]). An MFC is a flexible, planar actuation device that employs rectangular cross-section, unidirectional piezoceramic fibers (PZT 5A) embedded in a thermosetting polymer matrix (High and Wilkie [74]). An electromechanical characterization of the mechanical and piezoelectric behavior of the MFC device can be found in Bilgen et al. [75]. Since the purpose here is to maximize power output, an MFC device type with through-the-thickness poling and electrodes (type P2) is chosen, which operates in the 31 electromechanical mode. From the manufacturer’s data sheets, the 31 mode device has approximately 40 times higher capacitance compared to the interdigitated 33 mode device (type P1) with similar piezoelectric material volume. The patch is aligned to the beam symmetrically in the widthwise direction and as close to the base as possible, but not touching the boundary. The MFC is adhered to the beam using a 3M DP460 type two-part epoxy and let for cure under vacuum. Due to the surface structure of the MFC devices, the bond layer is typically higher than those of monolithic ceramics. An average epoxy thickness of 24 µm is assumed which has been measured for a sample of 12 MFC unimorphs examined in Bilgen et al. [24]. As noted above, the active area of the patch is composed of periodic pairs of piezoceramic and epoxy fibers. The thickness and width of each piezoceramic fiber are approximately 180 µm and 350 µm respectively as reported by the manufacturer. From Bilgen [24] each epoxy layer between the fibers has a width of 51.25 µm. The total thickness of the active region of the MFC is 305 µm and each of the top and bottom Kapton layers in the active region is approximately 60 µm thick. Considering the thickness of the epoxy layer and the Kapton layer, the separation between the surface of the PZT fibers and the steel substrate is assumed to be 84 µm. The total active width of the M2814-P2 device is 14 mm and there are approximately 35 piezoceramic fibers, yielding a total piezoceramic width of 12.25 mm. The ceramic properties of the MFC M2814-P2 device are given in Table 1. 13

Table 1: Electromechanical properties of the active area of the MFC M2814-P2 device. “*” indicates calculated properties; others are nominal values supplied by the manufacturer. Packaging (insulation material) properties are not shown. Ceramic Properties

Value

Electromechanical Mode Ceramic Navy / Industry Type Length, Lc (mm) Width, bc (mm) Thickness, hc (mm) Thickness Ratio, hc / h Ceramic Mass, mc (g) 3 Density, ρc (gr/cm ) Strain / applied field, d33 (pm/V) Strain / applied field, d31 (pm/V) Electrode Spacing (mm) E -12 2 Compliance, s11 (10 m /N) -12 2 E Compliance, s13 (10 m /N) E -12 2 Compliance, s33 (10 m /N)

31 II / 5A 28 14 (12.25*) 0.180* 0.73* 0.48* 7.8 400 -170 0.180* 16.4 -7.22 18.8

Physical Measurements

The signals of interest are measured using an NI 9215 analog-to-digital (A/D) module with 16 bit resolution (set to +/- 10 V range) at a fixed sampling rate of 100 Hz. Two signals are measured: 1) The potentiometer output which is proportional to the base displacement. 2) Voltage output of the MFC piezocomposite device. A 30:1 voltage divider circuit with an equivalent 30.7 MΩ input impedance is used to monitor the voltage output of the piezocomposite device. Figure 8 shows the complete schematic of the experimental setup. NI cDAQ 9172

Output (D/A)

PC w/ NI LabView

LP Filter

Linear Slider w/ Disp. Feedback Control

Beam w/ MFC

Resistor Control Base Disp. Monitor (Potentiometer) Input (A/D) 30:1 Probe

Load Resistance

Figure 8: The complete experimental setup schematic.

The tip mass value is chosen as 10.5 g (six magnets, 𝐿! = 287  mm) representing the prebuckling load. A total of 10 different load resistance values are applied. Table 2 shows the effective load resistances seen by the MFC device. Note that the probe and A/D input impedances are included in the effective load resistance as well as the selected resistor. Table 2: The load resistance, 𝑹𝑳 , values used in the experiment. Nominal (MΩ) 5 6 7 8 9 10 12

Measured (MΩ) 5.05 6.09 7.56 8.13 9.16 9.92 12.1

14

14 15 31 (Open)

13.7 15.3 30.7

The nominal values shown in the table are used for labeling purposes. The average electrical power output is defined as: 𝑃!"# =

1 (𝑇! − 𝑇! )

!! !!

𝑉 ! 𝑡 /𝑅!  𝑑𝑡

(6)

where 𝑇! and 𝑇! are the end and beginning times of the acquisition, 𝑉(𝑡) is the measured voltage output. Similarly, the average base acceleration is calculated by: 𝑍!"# =

1 (𝑇! − 𝑇! )

!!

𝑍 𝑡 𝑑𝑡

(7)

!!

where 𝑍 is the base acceleration derived from the measured displacement of the base. The derivation will be discussed next. The average power-per-base-acceleration-square is simply ! defined as 𝑃/𝑍 ! !"# = 𝑃!"# /𝑍!"# . COMBINED RANDOM AND HARMONIC EXCITATION

This section presents the experimental results. First, the input excitation behavior is analyzed. The relationship between the nominal input values and the measured output of the base actuation system is characterized. Second, the power output of all tests are qualitatively discussed, highlighting fundamentally interesting phenomenon and the time history of base displacement and power output. Third, certain test cases are investigated further showing characteristics that result in maximum power generation. Power output as a function of load resistance is also discussed. Base Excitation Characteristics

As noted previously, the base acceleration is derived from the measured base displacement signal. This is done by down-sampling the measured displacement signal, splining, up-sampling, and finally numerically differentiating twice to obtain the acceleration. The down-sampling is conducted such that the highest displacement values are captured hence the derived acceleration is considered an approximation. Figure 9 presents the average base acceleration as a function of nominal values of test parameters such as peak noise amplitude, LP filter cut-off frequency, and the frequency and amplitude of the harmonic component of the excitation signal. In this paper, the average values of the absolute of the signals are reported consistently. In Figure 9, the average value of base acceleration is calculated as shown in Eq. (7) for the last 24% of the total acquisition period.

15

a)

b)

3

3

ave,

ZB

ZB

ave,

mm/ s2

10

mm/ s2

10

2

10

0

20

40

60

80

100

120

0

Z n , mm

c)

20

40

60

80

100

120

80

100

120

Z n , mm

d)

3

3

10

ZB

ave,

ave,

mm/ s2

mm/ s2

10

ZB

2

10

2

10

0

20

40

60

80

100

2

10

0

120

20

40

60 Z n , mm

Z n , mm

Figure 9: Average base-acceleration as a function of nominal base-displacement noise amplitude for all tested cases. a) 𝝎 = 𝟎. 𝟐𝟓  𝑯𝒛, b) 𝝎 = 𝟎. 𝟓𝟎  𝑯𝒛, c) 𝝎 = 𝟎. 𝟕𝟓  𝑯𝒛, d) 𝝎 = 𝟏. 𝟎𝟎  𝑯𝒛. Note that test points corresponding to the 𝒁𝒏 = 𝒁𝒉 = 𝟎  𝐦𝐦 conditions are removed. The tests corresponding to the 𝒁𝒏 > 𝟔𝟎  𝐦𝐦 and LP=5.0 Hz conditions are removed due to MFC failure as a result of large curvatures.

In Figure 9, it is clear when pure harmonic excitation values (𝑍! = 0  𝑚𝑚) are considered, the acceleration amplitudes are calculated accurately. Note that the average value of the absolute of a harmonic signal is 2/𝜋 (0.64) times the peak value. As an example, the measured average base displacement at 1 Hz signal with no noise is 12.8 mm which theoretically results in an acceleration of 505 mm/s2 – the derived acceleration for this case is 494 mm/s2. The Effect of Signal Bandwidth

The effect of LP filter cut-off value to noise strength is significant and as expected as seen in Figure 9. As the “broadband” excitation is band limited, the input energy is reduced which may limit the beams ability to hop between stable equilibriums. An important observation in Figure 9 is that the base acceleration as a function of “noise-to-signal- ratio”. As expected, for filter values of 2.5 and 5.0 Hz, the effect of noise strength, 𝑍! , is the dominant factor in base acceleration when compared to the harmonic component amplitude, 𝑍! . When noise is suppressed with the LP filter (e.g. as in the LP=1.0 Hz case), the harmonic component is dominant as frequency is increased up to the LP filter value. These basic and expected results are important to note for the power analyses presented in the following sections. Repeatability and Time-averaging

Observing the average base acceleration calculated for the final 24% acquisition period, as shown in Figure 9, the tests are very repeatable for a similar set of inputs showing that the time 16

averaging works as intended. In particular, zero-harmonic (𝑍! = 0  mm) tests were repeated four times as an artifact of the fully populated test matrix. As the acquisition time is a function of the harmonic frequency, four different acquisition lengths were recorded for the zero-harmonic tests. When the average base-acceleration values are observed, different runs result in consistent values. Visual Observations on the Mechanical Response

Although beam displacement is not measured, the first visual observation made during the tests is the significant low-frequency tip-displacement response of beam as a result of broadband excitation. In some cases, the beam achieved very large (and non-uniform) curvatures, and the MFC device was damaged during a test. This test corresponded to the base excitation with 5 Hz LP filtered noise with 60 mm peak harmonic amplitude, and the third resistor value of 7.56 MΩ. It is worth noting that the damage caused an irreparable short-circuit between the electrodes of the MFC, therefore stopping all power generation. All test results after the noted case above are discarded. Power Output for All Tested Excitation Conditions

The fundamental interest here is to examine the net power output of the inverted cantilever beam subjected to broadband excitation. Figure 10 presents the average power output as a function of measured values of base acceleration, and as a function of the nominal values of LP filter cut-off frequency, and the Zfrequency and amplitude of the harmonic signal. The results hp :20 ! :0.50 L P :1.0 correspond to averaging based on the last 24% acquisition period. Note that test points Z hp :20 of ! :0.50 L P :1.0 corresponding to the 𝑍! = 𝑍! = 0  mm conditions are removed. Only two resistance values are presented: 𝑅! = 5.05  MΩ and 𝑅! = 6.09  MΩ. 10

2

L P :5.0

Z h :20

! :0.50

10

1

L P :5.0 R :6 L P :5.0 R :5

Z h :10

! :0.50

L P :2.5 ! 6= 0.50

P

ave ,

7W

L P :2.5

L P :1.0

Z h :20

! :1.00

10

0

LP:5.0 ω≠0.50 0

1000

2000

3000

4000

ZBave , mm/ s2

Figure 10: Average power output as a function of measured base-acceleration based on the final 24% acquisition period for all tested cases. The dashed boxes indicate the cases with LP=1.0 Hz condition.

Although a definite conclusion cannot be made at this point in the discussion, the response of the harvester at the end of a long period can be viewed as a “pseudo-steady-state” response in very generic terms. In Figure 10, there are two fundamentally different clusters of response. First, it is immediately apparent that the existence of a harmonic excitation at a frequency of 0.50 Hz results in the highest power output which is in the 30 to 70 µμW range. The 𝑍! = 10  or  20  mm at 𝜔 = 0.50  Hz excitation conditions are capable of a high-amplitude response and mostly “immune” to the noise in the excitation. The noise immunity is shown both with respect to the increasing values of LP and 𝑍! . 17

An interesting feature captured is the reduction of power as the noise amplitude, 𝑍! is increased from 40 to 60 mm for the response that corresponds to LP=5.0 Hz, 𝑍! = 10  mm at 𝜔 = 0.50  Hz. This simply illustrates that there is a critical “noise-to-signal” ratio where the highamplitude response cannot be sustained after a certain threshold value. The same has been observed qualitatively in the simulations (presented in Figure 6). The second cluster is considered as the responses that do not result in a sustained highamplitude response. This cluster is within the 0.5 to 15 µμW range. Within this cluster, the large amplitude, 5.0 Hz filtered, 𝑍! = 20  mm, and high frequency, 𝜔 = 1.00  Hz, excitation condition result in very small power output (𝑃!"# = 3 − 5  µμW) when compared to the high-amplitude cluster (𝑃!"# = 30 − 70  µμW). Within the same “low-amplitude” cluster, the 1.0 Hz filtered excitation with 𝑍! = 20  mm, and high frequency, 𝜔 = 1.00  Hz, excitation condition result in an appreciable power output of 15  µμW. Comparing the two conditions mentioned above, with the difference being the bandwidth, it is further demonstrated that the power generation is adversely effected by the bandwidth of the noise when a harmonic excitation component is present. Before making further observations on Figure 10, the average response based on the final 72% of the total acquisition period is also examined (not shown here), and compared to the values based on the final 24% of the acquisition period. The first observation is that the average power output values based on the two different acquisition periods are nearly identical for all almost all cases. The noise susceptibility of the high-amplitude response is observed again; however revealing a different feature where the average power appears to be significantly lower for only two test cases for the 72% acquisition period. To identify the cause of this difference in the average power output, the time history of the measured base-displacement and voltage output response of the cases subjected to the following conditions are investigated: LP=5.0 Hz, 𝑍! = 60  mm, 𝑍! = 10  mm at 𝜔 = 0.50  Hz. Figure 11 shows the complete measured response of the harvester subjected to a load resistance of 𝑅! = 5.05  MΩ. Z, mm

50

0

-50 40 0

50

100 Time, sec

150

200

50

100 Time, sec

150

200

V, V

20 0 -20 -40

0

Figure 11: Full time history of the measured base-displacement, 𝒁 𝒕 , and the voltage output, 𝑽 𝒕 , response for the case subjected to the conditions: LP=5.0 Hz, 𝒁𝒏 = 𝟔𝟎  𝐦𝐦, 𝒁𝒉 = 𝟏𝟎  𝐦𝐦 at 𝝎 = 𝟎. 𝟓𝟎  𝐇𝐳 where the load resistance is 𝑹𝑳 = 𝟓. 𝟎𝟓  𝐌𝛀.

Similarly, Figure 12 presents the measured response of the harvester subjected to a load resistance of 𝑅! = 6.09  MΩ.

18

Z, mm

50

0

-50 40 0

50

100 Time, sec

150

200

50

100 Time, sec

150

200

V, V

20 0 -20 -40

0

Figure 12: Full time history of the measured base-displacement, 𝒁 𝒕 , and the voltage output, 𝑽 𝒕 , response for the case subjected to the conditions: LP=5.0 Hz, 𝒁𝒏 = 𝟔𝟎  𝐦𝐦, 𝒁𝒉 = 𝟏𝟎  𝐦𝐦 at 𝝎 = 𝟎. 𝟓𝟎  𝐇𝐳 where the load resistance is 𝑹𝑳 = 𝟔. 𝟎𝟗  𝐌𝛀.

In both Figure 11 and Figure 12, the voltage time history shows a finite probability of the existence of the high-amplitude solution within a finite amount of observation period. The predicted effect of jumping from the high- to the low-amplitude solution is also demonstrated in the experiments. As the two electrical loads are nearly equal from the harvesters’ perspective, it can be observed that the emergence of the high-amplitude response for a system under similar excitation conditions is time dependent. Considering the inherent uncertainty in the initial conditions, such observation is consistent with the characteristics of a bistable system subjected to broadband excitation. As the desire in this research is the sustained generation of high-level power, and for the sake of brevity, the average electrical output calculated from the final 24% of the acquisition period is accepted as the “pseudo-steady-state” response of the system. The results in the following sections are consistently derived from the final 24% acquisition period. Comparison of High- and Low-Amplitude Responses

From the full data set presented in Figure 10 it is clear that the harmonic excitation at 𝜔 = 0.50  Hz results in the largest power output. When plotted in logarithmic scale, the poweroutput-per-base-acceleration-square shows a perfectly linear decrease as a function of baseacceleration. From the annotations in Figure 13, one can easily tell the high-energy response produced by harmonic signal with 0.5 Hz frequency both at 𝑍! = 10  and  20  mm amplitudes. In terms of output-input ratio, low noise inputs result consistently in high power output. For reference purposes, the zero-harmonic-excitation cases are also shown in the figure. As discussed previously, the high-amplitude response due to harmonic excitation is significantly higher when compared to response to pure random excitation. In addition, the high-amplitude response is tolerant to noise up to a certain level. In the cases tested here, this threshold is measured as 𝑍! = 40 − 60  mm only for the 5.0 Hz filtered base excitation signal. Highamplitude response to lower bandwidth signals (e.g. 1.0 and 2.5 Hz) is consistently tolerant up to the highest tested nominal noise amplitude of 𝑍! = 100  mm.

19

P =ZB2

ave ,

7 W / (mm/ s2) 2

10 10 10 10 10 10

-2

Z h :00

L P :1.0

-3

-4

! :0.50 R L :5,6

L P :2.5 L P :1.0

L P :5.0

-5

L P :2.5

-6

Z h :20 Z h :10

L P :5.0

-7

10

2

10

3

10

4

ZBave , mm/ s2

Figure 13: Average power-output-per-base-acceleration-square response to harmonic excitation at 𝝎 = 𝟎. 𝟓𝟎  𝐇𝐳 as a function of measured base-acceleration. Only two resistance values are presented: 𝑹𝑳 = 𝟓. 𝟎𝟓  𝐌𝛀 and 𝑹𝑳 = 𝟔. 𝟎𝟗  𝐌𝛀.

Another observation can be made for the voltage output response of the harvester to the open-circuit load resistance as shown in Figure 14. The tests with 30.7 MΩ load represents the approximate open-circuit condition. This condition is also referred to as the sensor mode which results in approximately the maximum voltage output and represents the minimum electrical power dissipation. In addition, the open-circuit voltage partially allows the direct measurement of the strain at the base of the beam. The open circuit condition is useful to check the calibration of model for parameters. Z h :20 ! :0.50 L P :1.0

10

V =ZB2

-4

Z h :20 ! :0.50 L P :2.5 Z h :20 ! :0.50 L P :5.0

Z h :00 L P :1.0

Z h :10 ! :1.00 L P :1.0 Z hp:00 L P :2.5

ave ,

V / (mm/ s2) 2

10

-3

10

10

Z h :20 ! :1.00 L P :1.0 Z h :20 ! :1.00 L P :5.0

-5

Z h :10 ! :1.00 L P :5.0 -6

Z h :00 L P :5.0

10

2

10

3

ZBave , mm/ s2

Figure 14: Average voltage-output-per-base-acceleration-square response to harmonic excitation at 𝝎 = 𝟎. 𝟓𝟎 − 𝟏. 𝟎𝟎  𝐇𝐳 as a function of measured base-acceleration. Some of the 𝒁𝒉 = 𝟏𝟎  𝐦𝐦 cases are removed to aid clarity. Only the open-circuit load resistance values are presented (𝑹𝑳 = 𝟑𝟎. 𝟕  𝐌𝛀).

Observations similar to the power plots can be made from Figure 14 regarding the state of the mechanical response (e.g. whether the high-amplitude response is sustained). As observed in the power response, the logarithmic voltage-per-base-acceleration shows a perfectly linear decrease as a function of logarithmic base-acceleration for the 𝑍! = 20  mm excitation at 𝜔 = 0.50  Hz. Although not shown, an exactly same linear slope is observed for the 𝑍! = 10  mm harmonic excitation at 𝜔 = 0.50  Hz. 20

Finally, another fundamentally important response occurs at the short-circuit condition which results in the maximum current output. In the experiments, the lowest load resistance was 5.05 MΩ therefore the short-circuit condition is not observed. Effect of Electrical Load

In terms of electrical load, the large set of results can be analyzed from several points of view. As the purpose of this research is to maximize power generation, the results can be observed for “time-averaged-optimum-resistance” values. Analyzing the complete set of results, the 5.05 MΩ, 6.09 MΩ and 7.56 MΩ load resistance values are observed as the global and approximate “optimum” resistances for the base-excitation bandwidth (LP) values of 5.0, 2.5, and 1.0 Hz respectively. Figure 15 presents the average power-output-per-base-acceleration response to pure harmonic excitation at 𝜔 = 0.5  Hz as a function of load resistance. 10

-2

L P :2.5 Z h :10 10

L P :1.0 Z h :20

-3

ave ,

7 W / (mm/ s2) 2

L P :1.0 Z h :10

P =ZB2

L P :2.5 Z h :20

10

-4

L P :5.0 Z h :10,20

! :0.50 Z n :00 5

7

9

11 13 15

30

R L , M+

Figure 15: Average power-output-per-base-acceleration response to pure harmonic excitation at 𝝎 = 𝟎. 𝟓𝟎  𝐇𝐳 as a function of load resistance. The nominal noise amplitude, for all cases in this plot, is zero (𝒁𝒏 = 𝟎  𝐦𝐦).

From Figure 15, it is clear that the “optimum” load resistance may be slightly lower than the lowest load resistance of 5.05 MΩ for the cases with 5.0 Hz bandwidth; however the value is expected to be very close to 5.05 MΩ. It is important to note that the notion of a single optimum resistance is partially stretched here as this resistance value is not expected to be a fixed value. In reality, the load resistance, or the impedance, should be dynamically changed in response to the mechanical state and in return truly optimizing the amount of energy extracted from the system. Such dynamic (active or passive) impedance can be implemented through the use of analog/digital electronics; however such load characteristics are beyond the scope of this paper. Power Output for Pure Random Excitation

Another detailed look at the data involves the power output in response to pure band-limited noise. Figure 16 presents the average power output as a function of measured base-acceleration for three different bandwidth values. Only two resistances are presented: 𝑅! = 5.05  MΩ and 𝑅! = 6.09  MΩ.

21

10

1

L P :1.0

Z h :00 R L :5, 6

L P :2.5

P

ave ,

7W

L P :5.0

10

0

0

1000

2000

3000

4000

ZBave , mm/ s2

Figure 16: Average power output as a function of measured base-acceleration based on the final 24% acquisition period. The harmonic excitation is zero (𝒁𝒉 = 𝟎  𝐦𝐦).

Figure 17 shows the average power-output-per-base-acceleration response to pure bandlimited random excitation. 10

-2

10

10

-3

L P :1.0 -4

L P :2.5

P =ZB2

ave ,

7 W / (mm/ s2) 2

Z h :00

10

-5

L P :5.0 10

-6

5

7

11 13 15

30

R L , M+

Figure 17: Average power-output-per-base-acceleration response to pure random excitation as a function of load resistance. The nominal harmonic amplitude, for all cases in this plot, is zero (𝒁𝒉 = 𝟎  𝐦𝐦).

As an artifact of the test matrix, each test case for the zero-harmonic condition is repeated four times with the difference being the total acquisition period. In the figure, three of the four tests for each excitation case are shown which have 100, 200 and 400 second periods. As before, the average values are calculated from the final 24% of the total acquisition period. From Figure 16, it is clear that the maximum power output (~7.5 µμW) due to pure random excitation corresponding to the LP value of 2.5 Hz is very small when compared to the power output (~60 µμW) due to harmonic excitation at the same LP filter condition (see Figure 10). Based on the observations from Figure 17, the notion of an “optimum” resistance is less clear and appears to have been reduced when compared to the pure harmonic excitation. In the pure random excitation, there appears to be two relatively distinct load resistances that result in locally maximum power output. The multi-local maximum power output versus load resistance has been reported in multi-frequency excitations by Abdelkefi et al. [76]. 22

CONCLUSIONS

An inverted cantilevered beam vibration energy harvester, with a tip mass of equivalent load slightly lower than the Euler buckling load, is evaluated for its electromechanical efficiency and power output capacity in the presence of pure harmonic, pure random and various combinations of harmonic and random base excitation cases. The configuration exploits the non-linear properties of the system using low-frequency base excitation in conjunction with the tip mass at the end of the beam. The system response is experimentally shown to include multiple solutions and jumps between the potential wells, and these are exploited in the harvesting device. In the presence of harmonic and random vibrations it is shown both theoretically and experimentally that the level of noise has a threshold which causes the preference of the low-amplitude response; however in some cases may have a desirable effect. The generation of relatively high levels power-output-per-base-acceleration by the nonlinear system within a broad range of excitation frequencies when compared to frequency-sensitive linear system is also demonstrated in the presence of some level of noise. The pre-buckled system under high-amplitude harmonic and low-amplitude random excitation produces higher acceleration-normalized-power-output when compared to a linear system operating at resonance. A basic lesson can be deduced for the vibration energy harvester under investigation, and potentially for other bistable structures, which are subjected to combined random and harmonic vibrations. A simple device such as a passive mechanical-filter at the base of the inverted cantilever beam could potentially filter the ambient noise which otherwise will cause inefficient electromechanical conversion. This is assuming that the structure, weight, size, etc. of the harvester cannot be optimized due to other constraints. ACKNOWLEDGMENTS

Dr. Bilgen acknowledges the support of the Old Dominion University Equipment Trust Fund and the Frank Batten College of Engineering and Technology. Prof. Friswell acknowledges the funding of the EPSRC through the Engineering Nonlinearity programme grant, EP/K003836. Dr. Ali acknowledges funding from the Royal Society through a Newton Fellowship. Prof. Litak was supported by the Polish National Science Center under grant agreement 2012/05/B/ST8/00080. REFERENCES

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