**Non-Linear Energy Harvesting with Random Noise and Multiple Harmonics**

Ji-Tzuoh Lin, Barclay Lee and Bruce William Alphenaar

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50727

## **1. Introduction**

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Technology 2006;12 1079–1083.

[36] Miao P, Mitcheson P D, Holmes A S, Yeatman E M, Green T C, Stark B H. MEMS inertial power generators for biomedical applications. Journal of Microsystem

> Harvesting energy from background mechanical vibrations in the environment has been proposed as a possible method to provide power in situations where battery usage is impractical or inconvenient. The most commonly used method for energy harvesting is to generate power from the vibrations of a piezoelectric material [1-3]; other methods include electromagnetic inductive coupling [4-6] and charge pumping across vibrating capacitive plates [7-10]. It has been shown that a piezoelectric cantilever attached to a vibrating structure can be used to power wireless transmission nodes for sensing applications [9]. In order to generate sufficient power, the frequency of the vibration source must match the resonant frequency of the piezoelectric cantilever. If the source vibrates at a fixed, known frequency, the dimensions of the cantilever, and the proof mass can be adjusted to ensure frequency matching. Many naturally occurring vibration sources do not have a fixed frequency spectrum, however, and vibrate over a broad range of frequencies. Lack of coupling of the piezoelectric cantilever to the off-resonance vibrations means that only a small amount of the available power can be harvested.

> Recent reports have shown that the resonant frequency of a simply supported beam [11] or a piezoelectric cantilever [12] can be tuned by applying an axial force. Research also show that the resonant frequency of a cantilever can also be manipulated by applying a transverse force on the cantilever [13,14]. (In all these cases, the cantilevers response remained within the linear regime.) In principle, this effect could be developed into an active tuning scheme which matches the cantilever resonance to the maximum vibrational output of the environment at any particular time. Calculations indicate, however, that the power consumed by active tuning completely offsets any improvement obtained in the scavenging efficiency [15]. More promising are passive tuning schemes in which a fixed force modifies the frequency response of the cantilever beam, without requiring additional power input.

© 2012 Lin et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Lin et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For example, an attractive magnetic force acting above the cantilever beam reduces the spring constant of the cantilever and lowers the resonance frequency [13,14], while an attractive force acting along the axis of the cantilever applies axial tension, and increases the resonance frequency [12]. While this can be used to tune the resonant frequency, there is no increase in output power, and the cantilever motion can even be dampened by the magnetic force and the resulting power output reduced [12,13].

The use of a magnetic force to introduce non-linear oscillation in cantilever motion has recently been reported [16-18]. A pendulum made with piezoelectric material [16] was used to study the energy output under different strengths of random Gaussian noise. An improvement of between 400% and 600% was observed compared to a standard linear oscillator. A piezomagnetoelastic structure [17] with two external magnets was studied, in which chaotic motion was observed outside the resonance frequency. It was further reported [18] that the softening response of a cantilever due to a magnetic attractor expands the response bandwidth and also increases the off resonant amplitude significantly.

Stochastic motions have been long observed with a pendulum in a repulsive magnetic field [19-20] In a generalization effort, the optimal relationship among the physical parameters for a coupling enhancement was provided in [16] [Cottone et al., 2009] using Duffing oscillator. Improvements for the non-linear system have been attributed to an advantage in the amplification of the vibration response from energy harvesters in the stochastic regime [17- 18].

Here, we will first demonstrate how this capability can be used to improve power output from a broadband vibration source, having a 1/f frequency dependence (pink noise) [21]. Note that a 1/f vibration spectrum describes a vibration source in which the power spectral density of the vibration is inversely proportional to frequency. Since many naturally occurring vibration sources display a 1/f dependence, this provides evidence that the magnetic coupling could be used for more efficient energy harvesting in practical settings.

The second part of this chapter provides an in-depth study of the response of a magnetically coupled cantilever at different frequencies [22-23]. It is our observation that amplification of the cantilever output occurs not only under stochastic motion but also due to subharmonic and ultraharmonic resonance in the vicinity of the main resonant frequency. The partial solutions of subharmonic and ultraharmonic are intrinsically embedded in the magnetic coupled equation as derived in forced oscillations of weakly nonlinear systems [24]. For a particular weakly coupled cantilever experimented in this paper, maximum output is maintained at the resonant frequency through combination of ultra-harmonic components. In a singly parametric excited scan of voltage production with non-linear piezoelectric cantilever, four distinct types of efficiency improvements are observed, in which the signal is amplified above the linear cantilever operation: (1) ultraharmonic amplification below resonance; (2) stochastic amplifications in multi-frequency and multi-amplitude oscillations; (3) ultra-sub-harmonic amplification at multiple quarter frequencies; (4) sub-harmonic amplification at one-third frequencies. For data analysis, a 1-D non-linear system coupled with piezoelectric charge production is modeled to illustrate the dynamic functions.

### **2. Non-linear dynamics in Pink noise background**

### **2.1. Experimental setup and vibration backgroud**

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18].

force and the resulting power output reduced [12,13].

For example, an attractive magnetic force acting above the cantilever beam reduces the spring constant of the cantilever and lowers the resonance frequency [13,14], while an attractive force acting along the axis of the cantilever applies axial tension, and increases the resonance frequency [12]. While this can be used to tune the resonant frequency, there is no increase in output power, and the cantilever motion can even be dampened by the magnetic

The use of a magnetic force to introduce non-linear oscillation in cantilever motion has recently been reported [16-18]. A pendulum made with piezoelectric material [16] was used to study the energy output under different strengths of random Gaussian noise. An improvement of between 400% and 600% was observed compared to a standard linear oscillator. A piezomagnetoelastic structure [17] with two external magnets was studied, in which chaotic motion was observed outside the resonance frequency. It was further reported [18] that the softening response of a cantilever due to a magnetic attractor expands

Stochastic motions have been long observed with a pendulum in a repulsive magnetic field [19-20] In a generalization effort, the optimal relationship among the physical parameters for a coupling enhancement was provided in [16] [Cottone et al., 2009] using Duffing oscillator. Improvements for the non-linear system have been attributed to an advantage in the amplification of the vibration response from energy harvesters in the stochastic regime [17-

Here, we will first demonstrate how this capability can be used to improve power output from a broadband vibration source, having a 1/f frequency dependence (pink noise) [21]. Note that a 1/f vibration spectrum describes a vibration source in which the power spectral density of the vibration is inversely proportional to frequency. Since many naturally occurring vibration sources display a 1/f dependence, this provides evidence that the magnetic coupling could be used for more efficient energy harvesting in practical settings.

The second part of this chapter provides an in-depth study of the response of a magnetically coupled cantilever at different frequencies [22-23]. It is our observation that amplification of the cantilever output occurs not only under stochastic motion but also due to subharmonic and ultraharmonic resonance in the vicinity of the main resonant frequency. The partial solutions of subharmonic and ultraharmonic are intrinsically embedded in the magnetic coupled equation as derived in forced oscillations of weakly nonlinear systems [24]. For a particular weakly coupled cantilever experimented in this paper, maximum output is maintained at the resonant frequency through combination of ultra-harmonic components. In a singly parametric excited scan of voltage production with non-linear piezoelectric cantilever, four distinct types of efficiency improvements are observed, in which the signal is amplified above the linear cantilever operation: (1) ultraharmonic amplification below resonance; (2) stochastic amplifications in multi-frequency and multi-amplitude oscillations; (3) ultra-sub-harmonic amplification at multiple quarter frequencies; (4) sub-harmonic amplification at one-third frequencies. For data analysis, a 1-D non-linear system coupled

with piezoelectric charge production is modeled to illustrate the dynamic functions.

the response bandwidth and also increases the off resonant amplitude significantly.

Figure 1 shows the set-up for the magnetically coupled piezoelectric cantilever measurements. The cantilever is manufactured using commercially available unimorph piezoelectric discs composed of a 0.9 mm thick PZT layer deposited on a 1 mm thick brass shim (APC International, MFT-50T-1.9A1). The disc is cut into a 13 mm wide by 50 mm long strip, and clamped at one end to produce a 44 mm long cantilever. The PZT layer extends 25 mm along the length of the cantilever, and the remainder is brass only. The proof mass (including the magnet and an additional fixture that holds the magnet) weighs 2.4 gm, while the cantilever itself weighs 0.8 gm. The electrical leads are carefully soldered with thin lead wires (134 AWP, Vishay) to the top side of the PZT and the bottom side of the shim [21].

**Figure 1.** The experimental set-up for the magnetically coupled (non-linear) piezoelectric cantilever. The magnetic force is repulsive and bi-directional.

Vibration is generated by a shaker table (Labwork ET-126) driven by an amplified pink noise source (Labwork Pa-13 amplifier). The pink noise is generated numerically, with amplitude and crest factor set to -4dB and 1.41, respectively. The average shaker table acceleration is 7.5 m/s2, independent of the magnetic coupling. A custom Labview data acquisition program measures output voltage from the cantilever beam and the acceleration from the shaker table, once every second. The voltage peak to peak (Vpp) is measured by an oscilloscope (Agilent 54624A), and the dc voltage is detected with a digital multi-meter (YOGOGAWA 7561). A 5mm diameter round rare earth magnet (Radio Shack model 64-1895) is attached to the vibrating tip of the cantilever beam, while a similar opposing magnet is attached directly to the shaker table frame, with repulsive force. The distance between the magnets is adjusted to 5.5 mm, to make the magnetic force comparable to the spring force of the cantilever.

#### **2.2. Experiment results**

The voltage generated by the cantilever in response to the pink noise source is measured using three different circuits, (shown in Figures 2(a), 3(a), and 4(a)). In each case, the output from the coupled cantilever is compared with the output from the same cantilever in the uncoupled situation (with the opposing magnet removed). In Figure 2, the piezoelectric cantilever beam is wired directly to an oscilloscope with a 1 M Ohm input impedance and the peak-to-peak output voltage, Vpp is measured. As shown in Figure 2 (b) the cantilever output is seen to fluctuate as a function of time, reflecting the random nature of the vibrations. For much of the time, the output from the coupled and uncoupled cantilevers is similar. However, occasionally, very large voltage spikes are observed in the output from the coupled cantilever, that are not observed for the uncoupled case. The voltage peak to peak spans to 5.7 V (min. 0.7 V and max. 6.4 V) with the coupled setup and only 2.2 V (min. 0.9 V to max. 3 V) volts with the uncoupled cantilever. The overall RMS powers for the uncoupled cantilever are 3.95 µW and 4.85 µW for the coupled case. The ratio of the maximal voltage output from the coupled to the uncoupled is 2.1.

In Figure 3, the voltage generated by the piezoelectric cantilever beam is rectified, using 0.4 V forward biased diodes, and detected across a 22 µF capacitor and a 1 M Ohm resistor in parallel. As shown in Figure 3(b), the amplitude of the voltage output with this measurement circuit is most of the time higher in the coupled case than in the uncoupled case. This is because the RC decay time of the circuit is larger than the time between the large amplitude deflections of the cantilever. The average voltage measured across the capacitor or the voltage integration over time is approximately 50% higher in the coupled case.

**Figure 2.** (a) The open circuit measurement on Vpp directly from the piezoelectric cantilever, and (b) the higher swing voltage reflects the voltage generated by coupling setup with larger cantilever motions.

**Figure 3.** (a) The schematic of a rectified circuit with a 1 M Ohm resister, and (b) the fluctuations of the voltage indicate that more power being generated by the magnetic coupled cantilever.

In Figure 4, the rectified voltage is measured directly across the 22 µF capacitor without the 1 M Ohm resistor. As shown in Fig. 4(b), the voltage across the capacitor increases with time, until a maximum charging voltage is achieved. The maximum voltage measured across the capacitor is approximately 50% higher in the coupled case than in the uncoupled case. Note that there is a time delay for the coupled cantilever to achieve a higher voltage than the uncoupled cantilever. This is due to the time passing before the first large amplitude deflection occurs. The random nature of the motion means that this time will vary from run to run, however, on average the coupled cantilever output will be consistently higher than the uncoupled output. Note that in addition to producing more power, the higher voltage output enables circuit operation without a step-up transformer, eliminating the power loss in the transformer.

**Figure 4.** (a) The schematic of the storage circuit, and (b) DC voltage output measured on the storage capacitor indicating more charge is stored with the magnetic coupling setup.

#### **2.3. Discussion**

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uncoupled is 2.1.

case.

**2.2. Experiment results** 

The voltage generated by the cantilever in response to the pink noise source is measured using three different circuits, (shown in Figures 2(a), 3(a), and 4(a)). In each case, the output from the coupled cantilever is compared with the output from the same cantilever in the uncoupled situation (with the opposing magnet removed). In Figure 2, the piezoelectric cantilever beam is wired directly to an oscilloscope with a 1 M Ohm input impedance and the peak-to-peak output voltage, Vpp is measured. As shown in Figure 2 (b) the cantilever output is seen to fluctuate as a function of time, reflecting the random nature of the vibrations. For much of the time, the output from the coupled and uncoupled cantilevers is similar. However, occasionally, very large voltage spikes are observed in the output from the coupled cantilever, that are not observed for the uncoupled case. The voltage peak to peak spans to 5.7 V (min. 0.7 V and max. 6.4 V) with the coupled setup and only 2.2 V (min. 0.9 V to max. 3 V) volts with the uncoupled cantilever. The overall RMS powers for the uncoupled cantilever are 3.95 µW and 4.85 µW for the coupled case. The ratio of the maximal voltage output from the coupled to the

In Figure 3, the voltage generated by the piezoelectric cantilever beam is rectified, using 0.4 V forward biased diodes, and detected across a 22 µF capacitor and a 1 M Ohm resistor in parallel. As shown in Figure 3(b), the amplitude of the voltage output with this measurement circuit is most of the time higher in the coupled case than in the uncoupled case. This is because the RC decay time of the circuit is larger than the time between the large amplitude deflections of the cantilever. The average voltage measured across the capacitor or the voltage integration over time is approximately 50% higher in the coupled

**Figure 2.** (a) The open circuit measurement on Vpp directly from the piezoelectric cantilever, and (b) the higher swing voltage reflects the voltage generated by coupling setup with larger cantilever motions.

It is instructive to compare the force exerted on the cantilever in the coupled and uncoupled cases. To do this, an empirical measure of the magnetic force is obtained using the experimental set-up shown in Figure 5.

**Figure 5.** The magnetic force component function, Fz, is determined by the electronic scale versus the manual deflection of the cantilever.

The opposing magnet is mounted onto a measurement scale, and the position of the magnetized cantilever is manipulated by pushing up and down at the end of a cantilever beam, simulating flexure movement. The deflection z is measured using a micrometer, while the reading on the scale provides the force between the two magnets. The details of the force measurments were shown in [22]. Only the magnetic force in the z direction, Fz, contributes to the resultant spring force. At z=0, the force is zero in the z direction because the two magnetic forces only repel each other in the longitudinal direction. Fz increases as the angles between the two magnets increase until the overlap between the two magnets is zero. At this point, Fz decreases with increasing distance because the force is inversely proportional to the distance squared.

The spring force, the magnetic force and the resultant force (spring plus magnetic) are plotted in Figure 6,

**Figure 6.** The plot shows the magnitude of the magnetic forces exerted on the cantilever beam, the spring forces and the resultant forces.

The resultant force is significantly reduced compared to the bare spring force near z=0. The coupled system has three equilibrium points where the resultant force is zero, compared to the single equilibrium point of the bare spring force. Because the resultant force in the region of the three equilibrium points is relatively small, transitions between the three points occurs relatively easily. Note that the middle equilibrium is unstable, therefore when the piezoelectric cantilever is set up for the coupling experiment, the cantilever is off the equilibrium point toward ground in static state as shown in Figure 1. In Figure 7 the potential energy is plotted for both the uncoupled and coupled systems. The potential energy is calculated by direct integration of the force with respect to the displacement, z. This gives for the uncoupled case, and for the coupled case. For the coupled case, the resultant potential is raised, with two local minima symmetric to z=0. This double well structure allows easy movement of the cantilever beam even when excited by non resonant forces. Once it passes the local high potential, it drifts to the other side of the balance, resulting in an increased total deflection distance. This can be seen by considering the possible motion of the cantilever beam having a kinetic energy, h, which is large enough to surmount the potential barrier at z=0. With the same random acceleration background the coupled cantilever can travel further distance than the uncoupled one. The voltage output, which depends on the movement of the cantilever, therefore, increases. The ratio of the maximum displacement in the coupled and uncoupled systems determined from Figure 7 is 2.4. This is comparable to the ratio of maximum voltage output in the coupled and uncoupled systems, which was seen in Figure 2 (b), at 2.1.

288 Small-Scale Energy Harvesting

manual deflection of the cantilever.

proportional to the distance squared.

spring forces and the resultant forces.

plotted in Figure 6,

**Figure 5.** The magnetic force component function, Fz, is determined by the electronic scale versus the

The opposing magnet is mounted onto a measurement scale, and the position of the magnetized cantilever is manipulated by pushing up and down at the end of a cantilever beam, simulating flexure movement. The deflection z is measured using a micrometer, while the reading on the scale provides the force between the two magnets. The details of the force measurments were shown in [22]. Only the magnetic force in the z direction, Fz, contributes to the resultant spring force. At z=0, the force is zero in the z direction because the two magnetic forces only repel each other in the longitudinal direction. Fz increases as the angles between the two magnets increase until the overlap between the two magnets is zero. At this point, Fz decreases with increasing distance because the force is inversely

The spring force, the magnetic force and the resultant force (spring plus magnetic) are

**Figure 6.** The plot shows the magnitude of the magnetic forces exerted on the cantilever beam, the

**Figure 7.** The direct integration from the measured forces function in Fig. 6 leads to the magnetic potential, spring potential and the resultant spring potential. The responding range in the coupled and the uncoupled cantilever is defined by the same potential height, h.

The magnetic coupling (although a passive force requiring no energy) introduces a symmetric force which acts in the opposite direction to the spring force around z=0. Being comparable in magnitude to the spring force, the magnetic force compensates the spring potential, and introduces a double valley in the potential energy profile. Under the influence of the modified spring potential, the magnetically coupled cantilever responds to a random vibration source (like the pink noise) by moving chaotically between the two minima in the potential energy profile. As compared with the non-chaotic motion of the uncoupled cantilever around the single z=0 potential minimum, this produces larger cantilever deflection and more voltage output from the piezoelectric cantilever. The oscillations around the resonance frequency are unstable and chaotic, but persistent. The modified spring potential is higher, and flatter than the bare spring potential, making the magnetic coupled cantilever easier to excite in the random frequency region. The experiments show that the ratio of the open circuit peak to peak voltage output and the potential well are closely related. Future work includes the design and implementation of modified potential wells and further analysis of the gain due to the modified potential wells.

### **3. Resonance broadening in broad band spectrum**

### **3.1. Experiment setup**

The experiment set up is the same as Figure 1. In all measurements, the shaker table acceleration is set to approximately 4.2 m/s2 at resonant freqeuncy, and the frequency swept from 0 to 30 Hz in 0.5 Hz steps. The opposing magnet fitted at the free end of the cantilever supplies a symmetrical, repulsive force about the balance of the cantilever during vibration. The horizontal separation between the magnets (designated by ) is adjusted to be approximately between 6 to 6.5 mm. This separation is found to provide the best compensation for the spring force, and makes the effective restoring force as small as possible near the equilibrium point.

### **3.2. Experiment result with open circuit**

Figure 8 shows both the output of the piezoelectric cantilever as a function of shaker table vibration frequency for the linear and non-linear case. The voltage generated by the piezoelectric cantilever beam is directed measured by oscilloscope treated as an open circuit. At the resonance frequency (measured to be 9.5 Hz) the output of the cantilever was 53 V, and the peak height, resonance frequency and line width are all approximately the same for the linear and non-linear states (here linear refers to the non-coupled state, while non-linear refers to the magnetically coupled state). On either side of the main resonance, however, there is additional output observed for the non-linear cantilever, which is not observed in the linear state. As can be seen from a comparison of the linear and the non-linear runs, the overall amplitude profile of the non-linear run is much larger in the sense of a broadband distribution, although there are gaps between peaks in the overall pattern of the non-linear output.

Figure 9 shows the output of both the linear and non-liner cantilever measured as a function of time at selected frequency to illustrate the comparison of the linear and non-linear dynamics. The voltage output of the non-linear cantilever evolves with frequency, while being amplified close to the resonance frequency. The spectrum shows a variety of amplified motions and harmonics. For example, at a driving frequency as low as 6.5 Hz (between 6-7.5Hz) (Figure 9(a)) both the linear and non-linear cantilever motions follow the vibrations of the shaker table, producing periodic oscillations. The amplitude of the oscillations for the non-linear cantilever is 5 times larger than those for the linear cantilever, however. At the resonant frequency (Figure 9(b)) both linear and non-linear cantilevers oscillate at the driving frequency with equal amplitudes. At 13 Hz (Figure 9(c)) the linear cantilever motion continues to follow the vibrations of the shaker table, producing low amplitude periodic oscillations. The non-linear cantilever motion is aperiodic and has a magnitude which is on average 3 times larger than that of the linear cantilever. At 16 Hz (Figure 9 (d)) the non-linear cantilever produces a 3 times larger peak to peak amplitude than the linear cantilever, and shows multiple and periodic "half-way" vibrations. At 20Hz (Figure 9 (e)) the non-linear cantilever shows a 5 times larger amplitude at the frequency of 6.7Hz than the linear output at 20 Hz.

290 Small-Scale Energy Harvesting

**3.1. Experiment setup** 

output.

possible near the equilibrium point.

**3.2. Experiment result with open circuit** 

potential energy profile. As compared with the non-chaotic motion of the uncoupled cantilever around the single z=0 potential minimum, this produces larger cantilever deflection and more voltage output from the piezoelectric cantilever. The oscillations around the resonance frequency are unstable and chaotic, but persistent. The modified spring potential is higher, and flatter than the bare spring potential, making the magnetic coupled cantilever easier to excite in the random frequency region. The experiments show that the ratio of the open circuit peak to peak voltage output and the potential well are closely related. Future work includes the design and implementation of modified potential

The experiment set up is the same as Figure 1. In all measurements, the shaker table acceleration is set to approximately 4.2 m/s2 at resonant freqeuncy, and the frequency swept from 0 to 30 Hz in 0.5 Hz steps. The opposing magnet fitted at the free end of the cantilever supplies a symmetrical, repulsive force about the balance of the cantilever during vibration. The horizontal separation between the magnets (designated by ) is adjusted to be approximately between 6 to 6.5 mm. This separation is found to provide the best compensation for the spring force, and makes the effective restoring force as small as

Figure 8 shows both the output of the piezoelectric cantilever as a function of shaker table vibration frequency for the linear and non-linear case. The voltage generated by the piezoelectric cantilever beam is directed measured by oscilloscope treated as an open circuit. At the resonance frequency (measured to be 9.5 Hz) the output of the cantilever was 53 V, and the peak height, resonance frequency and line width are all approximately the same for the linear and non-linear states (here linear refers to the non-coupled state, while non-linear refers to the magnetically coupled state). On either side of the main resonance, however, there is additional output observed for the non-linear cantilever, which is not observed in the linear state. As can be seen from a comparison of the linear and the non-linear runs, the overall amplitude profile of the non-linear run is much larger in the sense of a broadband distribution, although there are gaps between peaks in the overall pattern of the non-linear

Figure 9 shows the output of both the linear and non-liner cantilever measured as a function of time at selected frequency to illustrate the comparison of the linear and non-linear dynamics. The voltage output of the non-linear cantilever evolves with frequency, while being amplified close to the resonance frequency. The spectrum shows a variety of amplified motions and harmonics. For example, at a driving frequency as low as 6.5 Hz (between 6-7.5Hz) (Figure 9(a)) both the linear and non-linear cantilever motions follow the

wells and further analysis of the gain due to the modified potential wells.

**3. Resonance broadening in broad band spectrum** 

**Figure 8.** The voltage output (peak to peak) of the piezoelectric cantilever measured as a function of frequency (dash line for linear and solid line for non-linear state).

Note should be taken that there are two unexpected small peaks at 12.5 Hz and 17 Hz for the linear response. The peaks at 12.5 Hz and 17 Hz on the experiment data come from the torsion and standing wave oscillations. It is the result of how the piezoelectric cantilever was facilitated with magnet and its fixture as the proof mass. The cantilever is relatively thin and droops naturally due the weight of proof mass a few millimeters (as shown in Figure 1) to a curve. The L-shape fixture that holds the magnet was bolted with a screw on one side parallel to the brass shim. The magnet is then attached on the other side of the L-shape fixture, perpendicular to the brass shim in such way to make magnetic coupling. During the process, the cantilever was deformed and twisted slightly. As a result, the combined proof mass is slightly located off the center of the cantilever beam resulting in weight imbalance and torsion mode resonance. The fixture also creates an area where the free end is rigid with the fixture, which acts like a semi-fixed end, paving a way for a standing wave vibration when the cantilever is excited. Finite Element Analysis (FEA) simulating the structure and dimensions confirms that the first 3 modes of vibration include bending, torsion and standing wave oscillations.

**Figure 9.** The output of the linear (dash line) and non-linear (solid) system in the time domain: (a) 6.5 Hz (b) 9.5Hz at resonance; (c) 13 Hz ; (d) 16Hz ;(e) 20Hz

#### **3.3. Theoretical simulations**

292 Small-Scale Energy Harvesting

**Figure 9.** The output of the linear (dash line) and non-linear (solid) system in the time domain: (a) 6.5

Hz (b) 9.5Hz at resonance; (c) 13 Hz ; (d) 16Hz ;(e) 20Hz

The dynamics of the piezoelectric cantilever is modeled by a 1-D driven spring-mass system coupled with the piezoelectric effect under the influence of a magnetic force Fm(z) [17-18]:

$$
\dot{z}\,m\ddot{z} + d\dot{z} + kz + F\_m(z) + \sigma V = mA(\alpha)\cos(\alpha t),\tag{1}
$$

with mass m=0.0024 kg, damping coefficient d=0.0075 kg/sec, spring constant k=8.55 N/m, and angular frequency ω. Here, z is the vertical deflection of the cantilever, V is the generated voltage, σ=5x10-6 N/V is the coupling coefficient, and A is the acceleration of the shaker table (A=4.2 m/sec2 measured at resonance frequency). The voltage output is related to the deflection of the piezoelectric cantilever through:

$$
\dot{V} + \frac{1}{R\_l \mathcal{C}\_l} V + \theta \dot{\mathbf{z}} = 0 \tag{2}
$$

where R*l* is the equivalent resistance, C*l* is the equivalent capacitance and 1/ (R*l* C*l*)= 0.01 , and θ=1250 is the piezoelectric coupling coefficient in the measured circuit. The transverse magnetic force (in the z direction) is determined from the force between two magnetic dipoles (Kraftmakher, 2007):

$$F\_m(z, \eta) = \frac{-3\mu\_0 M^2 a (bz) (4\eta^2 - (bz)^2)}{4\pi (\left(bz\right)^2 + \eta^2)^{\frac{7}{2}}} \tag{3}$$

where M is the dipole magnetization, u0 is the permeability in air, and η is the horizontal separation between the magnets at z=0. The correction factors a and b are included to compensate for the flexure motion of cantilever and the magnetic force along the cantilever axis [16]. The magnetization M is determined by direct measurement of the axial force between the cantilever and a fixed magnet using a reference scale [22].

The solution to the coupled differential equations (1) and (2) is determined using Maple software to give the voltage output versus time for a given driving frequency, magnetic force function, and separation η. In order to fit our experiment data, the magnetic force Fm(z) was modified by a and b parameters and used for our calculation, where M = 0.011Am2, η = 6.5 mm, a = 1.04 and b = 1.21. As in the experiment, the output is calculated for t = 0 to 10 seconds, and the maximum peak-to-peak output over the last 2 seconds obtained. The result of the frequency domain is showed in Figure 10, which resembles the experimental result as seen in Figure 8.

Both the experiment and simulation figures show broadband vibration for the non-linear configuration between 6-20Hz. The simulation in Figures 11(a)-(e) reproduces many of the features observed in the experiment in Figures 9(a)-(e). The rest of Figures 11-15 reveals more about the complexity of the multiple harmonics in the non-linear systems. The simulations of the time domain with the corresponding frequency selected from experiment are shown in Figures 11(a)-15(a). Figures 11(b)-15(b) illustrate the velocity vs. voltage output of the piezoelectric cantilever in both the linear and non-linear cases. Figures 11(c)-15(c) are the Fourier transform of the coupled cantilever cases in Figures 11(a)-15(a), respectively, showing the compositions of frequency components for the non-linear states. The following section will discuss the multiple harmonic components directly derived from the non-linear dynamics simulations.

**Figure 10.** The simulated voltage output (peak to peak) of the piezoelectric cantilever is plotted in the frequency domain (dash line for linear and solid line non-linear).

### **3.4. Multiple harmonics analysis**

At a driving frequency of 6.5 Hz, as seen in Figure 11(a), both the linear and non-linear cantilever motion follow the vibrations of the shaker table, producing periodic oscillations. The amplitude of the oscillations for the coupled cantilever, however, is approximately 5 times larger than those for the linear cantilever, as seen in the experiment in Figure 9(a). The velocity vs. voltage in Figure 11(b) shows that the coupled cantilever has non-linear component in voltage production. Further analysis through Fourier transformation indicates that the non-linear cantilever shows the combination of the excited 6.5 Hz harmonic (dominant and high amplitude) and the 20 Hz ultraharmonic (3 times the excited frequency), as seen in Figure 11(c).

At the resonant frequency of 9.5 Hz (Figure 12(a)) both non-linear and linear cantilevers oscillate at the driving frequency with equal amplitude of voltage output. The responses for both the coupled and uncoupled cantilever at resonant frequency are almost identical in the voltage output. The velocity vs. voltage in Figure 12(b) shows a little non-linearity at 90o and -90o of the vibration cycles. Through Fourier transformation as seen in Figure 12(c), the nonlinear cantilever shows some components of vibration at the excited 9.5 Hz harmonic (dominant) and the 29 Hz ultraharmonic (3 times the excited frequency).

dynamics simulations.

of the piezoelectric cantilever in both the linear and non-linear cases. Figures 11(c)-15(c) are the Fourier transform of the coupled cantilever cases in Figures 11(a)-15(a), respectively, showing the compositions of frequency components for the non-linear states. The following section will discuss the multiple harmonic components directly derived from the non-linear

**Figure 10.** The simulated voltage output (peak to peak) of the piezoelectric cantilever is plotted in the

At a driving frequency of 6.5 Hz, as seen in Figure 11(a), both the linear and non-linear cantilever motion follow the vibrations of the shaker table, producing periodic oscillations. The amplitude of the oscillations for the coupled cantilever, however, is approximately 5 times larger than those for the linear cantilever, as seen in the experiment in Figure 9(a). The velocity vs. voltage in Figure 11(b) shows that the coupled cantilever has non-linear component in voltage production. Further analysis through Fourier transformation indicates that the non-linear cantilever shows the combination of the excited 6.5 Hz harmonic (dominant and high amplitude) and the 20 Hz ultraharmonic (3 times the excited

At the resonant frequency of 9.5 Hz (Figure 12(a)) both non-linear and linear cantilevers oscillate at the driving frequency with equal amplitude of voltage output. The responses for both the coupled and uncoupled cantilever at resonant frequency are almost identical in the voltage output. The velocity vs. voltage in Figure 12(b) shows a little non-linearity at 90o and -90o of the vibration cycles. Through Fourier transformation as seen in Figure 12(c), the nonlinear cantilever shows some components of vibration at the excited 9.5 Hz harmonic

(dominant) and the 29 Hz ultraharmonic (3 times the excited frequency).

frequency domain (dash line for linear and solid line non-linear).

**3.4. Multiple harmonics analysis** 

frequency), as seen in Figure 11(c).

**Figure 11.** The theoretical analysis of excited frequency at 6.5 Hz. (a) the time domain voltage output, dash line for linear and solid line for non-linear states; (b) the velocity vs. voltage output, dark line for linear and light line for non-linear state; (c) the Fourier transform of the non-linear state from the data of Figure 5(a).

**Figure 12.** The theoretical analysis of excited frequency at 9.5 Hz. (a) the time domain voltage output, dash line for linear and solid line for non-linear state; (b) the velocity vs. voltage output, light line for linear and dark line for non-linear state; (c) the Fourier transform of the non-linear state from the data of Figure 12(a).

The response for the non-linear cantilever is chaotic at 13 Hz as seen in Figure 13(a), but with average 3 folds larger magnitude than the linear one. The velocity vs. voltage relation in Figure 13(b) shows chaotic motions for the coupled cantilever. Using Fourier transformation for Figure 13(a) results in Figure 13(c), the coupled cantilever shows the linear response of a small portion of 13 Hz component combined with a large amplitude distribution at lower frequency that are attributed to the chaotic motion. Note that the small peaks at 12.5 Hz and 17 Hz are not observed in the simulation as seen and discussed in the experiment section. This small torsion and standing wave bending resonance are not accounted for by the simplified 1-D model used to simulate the spring mass damping model such as an ideal cantilever.

At 16Hz, the non-linear cantilever is periodic (Figure 14(a)) and is 3 times larger (peak to peak) in magnitude than the uncoupled one, with double prone of low frequency in the upper cycle. Apparently, it is and composed of different frequency and multiple haromonic motion, with large magnitude than the uncoupled motion. The evidence is also shown in the velocity vs. voltage relationship in Figure 14(b), where 3 different cyclic loops are identifiable. Fourier transformation from time data in Figure 14(a) proves that the non-linear cantilever delivers ultra-sub-harmonic vibration at n\*(16/4) Hz, where, n=integer in Figure 14(c).

**Figure 13.** The theoretical analysis of excited frequency at 13 Hz. (a) the time domain voltage output, dash line for linear and solid line for non-linear states; (b) the velocity vs. voltage output, light line for linear and dark line for non-linear state; (c) the Fourier transform of the non-linear state from the data of Figure 13(a).

such as an ideal cantilever.

14(c).

Figure 13(a).

The response for the non-linear cantilever is chaotic at 13 Hz as seen in Figure 13(a), but with average 3 folds larger magnitude than the linear one. The velocity vs. voltage relation in Figure 13(b) shows chaotic motions for the coupled cantilever. Using Fourier transformation for Figure 13(a) results in Figure 13(c), the coupled cantilever shows the linear response of a small portion of 13 Hz component combined with a large amplitude distribution at lower frequency that are attributed to the chaotic motion. Note that the small peaks at 12.5 Hz and 17 Hz are not observed in the simulation as seen and discussed in the experiment section. This small torsion and standing wave bending resonance are not accounted for by the simplified 1-D model used to simulate the spring mass damping model

At 16Hz, the non-linear cantilever is periodic (Figure 14(a)) and is 3 times larger (peak to peak) in magnitude than the uncoupled one, with double prone of low frequency in the upper cycle. Apparently, it is and composed of different frequency and multiple haromonic motion, with large magnitude than the uncoupled motion. The evidence is also shown in the velocity vs. voltage relationship in Figure 14(b), where 3 different cyclic loops are identifiable. Fourier transformation from time data in Figure 14(a) proves that the non-linear cantilever delivers ultra-sub-harmonic vibration at n\*(16/4) Hz, where, n=integer in Figure

**Figure 13.** The theoretical analysis of excited frequency at 13 Hz. (a) the time domain voltage output, dash line for linear and solid line for non-linear states; (b) the velocity vs. voltage output, light line for linear and dark line for non-linear state; (c) the Fourier transform of the non-linear state from the data of

**Figure 14.** The theoretical analysis of excited frequency at 16 Hz. (a) the time domain voltage output, dash line for linear and solid line for non-linear states; (b) the velocity vs. voltage output, light line for linear and dark line for non-linear state; (c) the Fourier transform of the non-linear state from the data of Figure 14(a).

At 20Hz, the response for the non-linear cantilever is periodic and also 3 folds larger peak to peak magnitude than the linear one as seen in Figure 15 (a). The velocity vs. voltage in Figure 15(b) shows some combination of cyclic motions for the non-linear cantilever. Through Fourier transformation, the coupled cantilever shows subharmonic at 6.7 Hz (dominant), excite frequency/3, and 20 Hz in Figure 15(c).

The combination of the stochastic and various harmonic features have three to five folds greater voltage production than the linear standard narrow band piezoelectric cantilever. Together with the un-damped resonant response enhance the performance well beyond that of a standard energy harvester.

**Figure 15.** The theoretical analysis of excited frequency at 20 Hz. (a) the time domain voltage output, dash line for linear and solid line for non-linear states; (b) the velocity vs. voltage output, light line for linear and dark line for non-linear state; (c) the Fourier transform of the non-linear state from the data of Figure 15(a).

#### **3.5. Experience result with storage capacitor**

Figure 16 (a) shows the output of the other PZT cantilever with similar specs as a function of shaker table vibration frequency for the case where the opposing magnet is fixed to the shaker table. The voltage generated by the piezoelectric cantilever beam is rectified, and detected across a 22 µF capacitor and 1 M Ohm resistor in parallel, using the circuit shown in Figure 3 (a). The results from two measurement runs in the coupled state are shown, together with the output of the cantilever measured in the uncoupled state. (This is obtained by removing the opposing magnet.) At the resonance frequency, (measured to be approximately 10 Hz) the output of the cantilever exceeds 16 V, and the peak height, resonance frequency and linewidth are all approximately the same for the coupled and uncoupled states. On either side of the main resonance, however, there are additional output observed for the coupled cantilever, which is not observed in the uncoupled state. As can be seen from a comparison of the two coupled runs, the frequency distribution of the peaks are the result of the multiple harmonics, as predicted in the open circuit.

**Figure 16.** Voltage output of the piezoelectric cantilever as a function of shaker table frequency for **(a)**  single cantilever **(b)** double cantilever. Integrated voltage output as a function of frequency for **(c)**  single cantilever and **(d)** double cantilever.

Also measured was a double cantilever system, (as shown in Fig. 16(b)), in which the second magnet is connected to an opposing cantilever (having resonant frequency of around 60Hz) rather than to a fixed point. As shown in Fig. 16 (b), the results are similar to the single cantilever system, except that the double cantilever system shows a larger overall increase in off-resonance output. The overall improvement in the harvesting efficiency can be illustrated by plotting the integrated voltage output of the cantilever beam as a function of frequency. For both the single (Fig. 16 (c)) and double (Fig. 16 (d)) cantilever systems, the integrated voltage output over the 0-30 Hz bandwidth shows a substantial increase in the coupled versus the uncoupled case. The total improvement is 31%-87%, with some variation between measurement runs.

#### **4. Conclusion**

298 Small-Scale Energy Harvesting

Figure 15(a).

**3.5. Experience result with storage capacitor** 

the result of the multiple harmonics, as predicted in the open circuit.

**Figure 15.** The theoretical analysis of excited frequency at 20 Hz. (a) the time domain voltage output, dash line for linear and solid line for non-linear states; (b) the velocity vs. voltage output, light line for linear and dark line for non-linear state; (c) the Fourier transform of the non-linear state from the data of

Figure 16 (a) shows the output of the other PZT cantilever with similar specs as a function of shaker table vibration frequency for the case where the opposing magnet is fixed to the shaker table. The voltage generated by the piezoelectric cantilever beam is rectified, and detected across a 22 µF capacitor and 1 M Ohm resistor in parallel, using the circuit shown in Figure 3 (a). The results from two measurement runs in the coupled state are shown, together with the output of the cantilever measured in the uncoupled state. (This is obtained by removing the opposing magnet.) At the resonance frequency, (measured to be approximately 10 Hz) the output of the cantilever exceeds 16 V, and the peak height, resonance frequency and linewidth are all approximately the same for the coupled and uncoupled states. On either side of the main resonance, however, there are additional output observed for the coupled cantilever, which is not observed in the uncoupled state. As can be seen from a comparison of the two coupled runs, the frequency distribution of the peaks are

Piezoelectric cantilevers have been widely studied for energy scavenging applications, but suffer from poor output power outside of a narrow frequency range near the cantilever resonance. In this chapter, we have demonstrated how power output can be enhanced by applying a simple passive external force. When a symmetrical and repulsive magnetic force is applied to a piezoelectric cantilever beam to compensate the cantilever spring force, this lowers the spring potential and increases the output when driven by a random pink noise vibrational source. The principle may be applied to other vibration energy harvesting devices such as electromagnetic and capacitive types in random naturally pink noise environments.

In the parametrically excited piezoelectric cantilever experiments, linear and non-linear performances were compared. Overall, four distinct types of efficiency improvements appear in the non-linear configuration, in which the signal is amplified above the linear cantilever response: low frequency ultraharmonic amplification; stochastic amplifications in multi-frequency and multi-amplitude oscillations; ultra-sub-harmonic amplification at multiple quarter frequencies; subharmonic amplification at one-third frequencies. Taken together, the stochastic, sub-harmonic and ultra-harmonic response produces an average of three to five-fold increase in voltage production. For energy harvesting purposes, the combination of the four features together with the un-damped resonant response enhances the performance well beyond that of a standard energy harvester. Furthermore, an analytical model of the bi-stable dynamics produces results consistent with those observed experimentally. The simulation tool could be deployed in the future investigation for non-linear energy harvester design for broadband and beyond natural harmonic applications.

## **Author details**

Ji-Tzuoh Lin\* and Bruce William Alphenaar *Department of Electrical and Computer Engineering, University of Louisville, Louisville, KY, USA* 

Barclay Lee *Department of Bioengineering, California Institute of Technology, Pasadena, CA, USA* 

## **Acknowledgement**

The effort was funded by the Department Of Energy DE-FC26-06NT42795 and the U.S. Navy under Contract DAAB07-03-D-B010/TO-0198. Technical program oversight under the Navy contract was provided by Naval Surface Warfare Center, Crane Division.

### **5. References**


<sup>\*</sup> Corresponding Autor

[4] Kulah H. and Najafi K. 2004, An electromagnetic micro power generator for lowfrequency environmental vibrations, 17th IEEE Int. conf. Micro Electro Mechanical System, 1004 (MEMS '04) pp 237-40

300 Small-Scale Energy Harvesting

harmonic applications.

**Acknowledgement** 

Anal. Eng. Des. 38 115-24

Mater. Struct. 13 1131

Trans. Power Electron. 18 696-703

**5. References** 

Corresponding Autor

 \*

and Bruce William Alphenaar

**Author details** 

Ji-Tzuoh Lin\*

Barclay Lee

environments.

vibrational source. The principle may be applied to other vibration energy harvesting devices such as electromagnetic and capacitive types in random naturally pink noise

In the parametrically excited piezoelectric cantilever experiments, linear and non-linear performances were compared. Overall, four distinct types of efficiency improvements appear in the non-linear configuration, in which the signal is amplified above the linear cantilever response: low frequency ultraharmonic amplification; stochastic amplifications in multi-frequency and multi-amplitude oscillations; ultra-sub-harmonic amplification at multiple quarter frequencies; subharmonic amplification at one-third frequencies. Taken together, the stochastic, sub-harmonic and ultra-harmonic response produces an average of three to five-fold increase in voltage production. For energy harvesting purposes, the combination of the four features together with the un-damped resonant response enhances the performance well beyond that of a standard energy harvester. Furthermore, an analytical model of the bi-stable dynamics produces results consistent with those observed experimentally. The simulation tool could be deployed in the future investigation for non-linear energy harvester design for broadband and beyond natural

*Department of Electrical and Computer Engineering, University of Louisville, Louisville, KY, USA* 

The effort was funded by the Department Of Energy DE-FC26-06NT42795 and the U.S. Navy under Contract DAAB07-03-D-B010/TO-0198. Technical program oversight under the

[1] Elvin NG, Elvin A and Choi D 2003 A self-powered damage detection sensor J. Strain

[2] Ottman G K Hofmann H F and Lesieutre G A 2003 Optimized piezoelectric energy harvesting circuit using step-down converter in discontinuous conduction mode IEEE

[3] Roundy S 2004 A piezoelectric vibration based generator for wireless electronics, Smart

*Department of Bioengineering, California Institute of Technology, Pasadena, CA, USA* 

Navy contract was provided by Naval Surface Warfare Center, Crane Division.


**Chapter 0 Chapter 13**

## **Modeling Aspects of Nonlinear Energy Harvesting for Increased Bandwidth**

Marcus Neubauer, Jens Twiefel, Henrik Westermann and Jörg Wallaschek

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52232

### **1. Introduction**

302 Small-Scale Energy Harvesting

(11)

045012

Phys. 2007; 28, 409

Systems,( accepted May 4th on line) 2012

[19] Duchesne, B., C.W. Fischer, C.G. Gray, and K. R. Jeffrey. "Chaos In The Motion of An Inverted Pendulum: An Undergraduate Laboratory Experiment," Am. J. Phys. 1991; 59

[20] A. Siahmakoun, V. A. French, and J. Patterson. "Nonlinear Dynamics of A Sinusoidally

[21] Ji-Tzuoh Lin and Bruce Alphenaar. "Enhancement of Energy Harvested from a Random Vibration Source by Magnetic Coupling of a Piezoelectric Cantilever," Journal of

[22] Ji-Tzuoh Lin, Barclay Lee, and Bruce Alphenaar. "Magnetic Coupling of Piezoelectric Cantilever for Enhanced Energy Harvesting Efficiency" Smart Mater. Struct. 2010; 19

[23] Ji-Tzuoh Lin, Kevin Walsh and Bruce Alphenaar. Enhanced Stochastic, Subharmonic and Ultraharmonic Energy Harvesting, Journal of Intelligent Material and Structure

[24] A. Prosperetti "Subharmonics and Ultraharmonics in The Forced Oscillations of Weakly

[25] Kraftmakher, Y. "magnetic Field of A Dipole And the Dipole-Dipole Interaction," Eur. J.

Nonlinear Systems." American Journal of Physics Vol. 44 No. 6 1976

Driven Pendulum In a Propulsive Magnetic Field," Am. J. Phys. 1997; 65 (5)

Intelligent Material Systems and Structures, 2010; Vol. 21 Issue 13, 1337-1341

Over the last years, the field of energy harvesting has become a promising technique as power supply of autonomous electronic devices. Those use the surrounding energy, such as vibrations, temperature gradients or radiation, for conversion in electrical energy. Mechanical vibrations are an attractive source due to their high availability in technical environments, thus numerous research groups are working on this topic. The most important conversion methods for ambient vibrations are electromagnetic, electrostatic and piezoelectric. All those techniques have been successfully demonstrated in the past. [11] provides an overview of the basics in energy harvesting.

Due to the fact, that vibration energy harvester generates the most energy when the generator is excited at its resonance frequency, the converter needs to be tuned to the main external frequency of the individual environment. If the excitation frequency shifts, the performance of the generator may reduced drastically. In practical use the vibration of an environment may vary in a large spectrum. To overcome this disadvantage researchers work hard to increase the working bandwidth of an energy harvester.

This chapter is a contribution to the current state of the art for modeling broadband energy harvesting generators. In the first part the electromechanical model is derived in terms of using lumped parameters. The system is based on a piezoelectric bimorph structure. The coupled differential equations for the case of a simple electrical circuit are derived and furthermore the possibility to enhance the energy extraction is analyzed. The use of generator arrays to archive a high power outputs in a wide frequency range is discussed. In detail, the Synchronized Switch Harvesting on Inductor (SSHI) technique is studied. Further the modeling of the promising piezomagnetoelastic energy harvesting technique is covered in the last part.

©2012 Westermann et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Westermann et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **2. Linear piezoelectric energy harvesting system**

This section is devoted to the modeling of a linear piezoelectric bimorph for energy harvesting. This will be the basis for the proposed nonlinear techniques with enhanced bandwidth duscussed in the following sections.

#### **2.1. Modeling of piezomechanical structures**

Fundamental for the energy harvesting techniques presented in the following is the piezoelectric bimorph. In the following, the general modeling of piezomechanical structures is given, and further on applied to the case of a bending bimorph.

The calculations are based on the potential energy stored in a piezoelement,

$$
\delta M = \frac{1}{2} \int\_{V} \left( T\_i \mathbf{S}\_i + D\_3 E\_3 \right) \mathbf{d}V\_{\prime} \tag{1}
$$

where *T*, *S*, *D*, *E* represent the mechanical stress and strain as well as the electrical displacement and field. *V* is the volume of the piezoelement. A one-dimensional strain distribution within the piezoelement in axis direction *i* is assumed. According to [6] the axis of polarization is defined as *x*3. Therefore the transversal effect is represented by *i* = 1, where the mechanical strain is normal to the direction of polarization, and the longitudinal effect by *i* = 3, where the mechanical strain is in the direction of polarization. After some mathematical calculations, see [9] the energy can be written as

$$\mathcal{U} = \frac{1}{2} \frac{\mathcal{Q}\_{\text{P}}^{2}}{\mathcal{C}\_{\text{P}}} + \frac{1}{2} \frac{1}{s\_{\text{ii}}^{\text{E}}} \int\_{V} \left[ S\_{i}^{2} + \frac{k\_{3i}^{2}}{1 - k\_{3i}^{2}} \Delta S\_{i,3}^{2} \right] \mathbf{d}V + \frac{1}{2} \frac{\alpha^{2}}{\mathcal{C}\_{\text{P}}} \left( \ell\_{i} \bar{S}\_{i} \right)^{2} - \frac{1}{2} 2 \frac{\alpha}{\mathcal{C}\_{\text{P}}} \ell\_{i} \bar{S}\_{i} Q\_{\text{P}}.\tag{2}$$

The electrical charge at the electrodes is termed *Q*p, while the mechanical compliance in axis direction *i* is given as *sii*. Further on, the material coupling of the piezoelement is given by *k*3*<sup>i</sup>* = *d*<sup>2</sup> 3*i* / *s*E *iiε*T 33 . In the following, the stiffness *c*<sup>p</sup> in *xi* direction, the capacitance *C*<sup>p</sup> of the piezoceramics and the piezoelectric force factor *α* are introduced as

$$c\_{\rm P} = \frac{1}{s\_{ii}^{\rm E}} \frac{V}{\ell\_i^2},$$

$$\mathcal{C}\_{\rm P} = \varepsilon\_{33}^{\rm T} \left(1 - k\_{3i}^2\right) \frac{A\_{\rm el}}{\ell\_3},$$

$$\alpha = \frac{d\_{3i}}{s\_{ii}^{\rm E}} \frac{A\_{\rm el}}{\ell\_i}.\tag{3}$$

They depend on the area of electrodes *A*el, the piezoelectric constant *d*3*i*, the permittivity *ε*<sup>33</sup> as well as the geometry of the piezoelement (length <sup>3</sup> between electrodes and length *<sup>i</sup>* of the piezoelement in direction of mechanical strain).

The energy terms in Equation 2 can be classified as the stored electrical energy, the stored mechanical energy and the converted energy. For convenience of the following calculations,

**Figure 1.** Piezoelement with uniaxial strain distribution.

2 Will-be-set-by-IN-TECH

This section is devoted to the modeling of a linear piezoelectric bimorph for energy harvesting. This will be the basis for the proposed nonlinear techniques with enhanced bandwidth

Fundamental for the energy harvesting techniques presented in the following is the piezoelectric bimorph. In the following, the general modeling of piezomechanical structures

where *T*, *S*, *D*, *E* represent the mechanical stress and strain as well as the electrical displacement and field. *V* is the volume of the piezoelement. A one-dimensional strain distribution within the piezoelement in axis direction *i* is assumed. According to [6] the axis of polarization is defined as *x*3. Therefore the transversal effect is represented by *i* = 1, where the mechanical strain is normal to the direction of polarization, and the longitudinal effect by *i* = 3, where the mechanical strain is in the direction of polarization. After some mathematical

The electrical charge at the electrodes is termed *Q*p, while the mechanical compliance in axis direction *i* is given as *sii*. Further on, the material coupling of the piezoelement is given by

> *V* 2 *i* ,

> > *A*el *i*

They depend on the area of electrodes *A*el, the piezoelectric constant *d*3*i*, the permittivity *ε*<sup>33</sup> as well as the geometry of the piezoelement (length <sup>3</sup> between electrodes and length *<sup>i</sup>* of the

The energy terms in Equation 2 can be classified as the stored electrical energy, the stored mechanical energy and the converted energy. For convenience of the following calculations,

. In the following, the stiffness *c*<sup>p</sup> in *xi* direction, the capacitance *C*<sup>p</sup> of the

(*TiSi* + *D*3*E*3) d*V*, (1)

. (3)

*iQ*p. (2)

**2. Linear piezoelectric energy harvesting system**

is given, and further on applied to the case of a bending bimorph.

The calculations are based on the potential energy stored in a piezoelement,

*V*

*k*2 3*i* <sup>1</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> 3*i* Δ*S*<sup>2</sup> *i*,3 d*V* + 1 2 *α*2 *C*p (*iS*¯ *i*) <sup>2</sup> <sup>−</sup> <sup>1</sup> 2 2 *α C*p *iS*¯

*<sup>c</sup>*<sup>p</sup> <sup>=</sup> <sup>1</sup> *s*E *ii*

*C*<sup>p</sup> = *ε* T 33 <sup>1</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> 3*i A*el 3 ,

*<sup>α</sup>* <sup>=</sup> *<sup>d</sup>*3*<sup>i</sup> s*E *ii*

*<sup>U</sup>* <sup>=</sup> <sup>1</sup> 2 

**2.1. Modeling of piezomechanical structures**

calculations, see [9] the energy can be written as

piezoelement in direction of mechanical strain).

 *S*2 *<sup>i</sup>* +

piezoceramics and the piezoelectric force factor *α* are introduced as

*V*

*<sup>U</sup>* <sup>=</sup> <sup>1</sup> 2 *Q*2 p *C*p + 1 2 1 *s*E *ii*

*k*3*<sup>i</sup>* = *d*<sup>2</sup> 3*i* / *s*E *iiε*T 33 

duscussed in the following sections.

the mechanical strain is split into the mean value *S*¯ *<sup>i</sup>*, the mean strain *S*¯ *<sup>i</sup>*,3 along the *x*<sup>3</sup> axis (between the electrodes), and the difference Δ*Si*,3 between the actual strain and *S*¯ *i*,3,

$$\bar{S}\_{i,3}(\mathbf{x}\_1, \mathbf{x}\_2) = \frac{\int^{\ell\_3} S\_i(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) d\mathbf{x}\_3}{\ell\_3}, \quad \Delta S\_{i,3} = S\_i - \bar{S}\_{i,3}. \tag{4}$$

The reason for this representation is that piezoelectric systems with a homogeneous strain distribution is readliy described by Δ*Si*,3 = 0, which strongly simplifies the calculations. Additionally, the influence of an uneven strain distribution can be seen in the term Δ*Si*,3. See Figure 1 for an illustration of these definitions.

In case of a continuous system it is reasonable to discretize it for the further analysis. The mechanical deformation is then described by *n* degrees of freedom (DOF) *qi*, while the charge *Q*p is the electrical DOF,

$$\mathbf{q} = \begin{bmatrix} \mathbf{q}\_{\text{mech}} \\ \mathbf{Q}\_{\text{P}} \end{bmatrix}. \tag{5}$$

Each mechanical DOF is associated with a global mode shape, which defines the mechanical strain distribution *Si* within the piezoelectric volume (and the rest of the mechanical system). The overall strain distribution is then the sum of all mode shapes. In order to rewrite the energy term in Equation 2, the term <sup>−</sup>*iS*¯ *<sup>i</sup>*, which represents the mean deformation of the piezoelement in *xi* direction, will be represented by

$$-\ell\_i \bar{S}\_i = \sum\_{k=1}^n \kappa\_k q\_k. \tag{6}$$

Here, a mechanical coupling vector κ is introduced. In that form, the energy terms in Equation 2 can be rewritten as

**Figure 2.** Piezoelectric bimorph.

$$-\frac{1}{2}2\frac{\boldsymbol{\alpha}}{\mathbf{C}\_{\rm{P}}}\ell\_{i}\bar{\mathbf{S}}\_{i}\mathbf{Q}\_{\rm{P}}=\frac{1}{2}\mathbf{q}^{\rm{T}}\begin{bmatrix}\mathbf{0} & \frac{\boldsymbol{\alpha}}{\mathbf{C}\_{\rm{P}}}\kappa\\ \frac{\boldsymbol{\alpha}}{\mathbf{C}\_{\rm{P}}}\kappa^{\rm{T}}&\mathbf{0}\end{bmatrix}\mathbf{q}$$

$$\frac{1}{2}\frac{\boldsymbol{\alpha}^{2}}{\mathbf{C}\_{\rm{P}}}\left(\ell\_{i}\bar{\mathbf{S}}\_{i}\right)^{2}=\frac{1}{2}\mathbf{q}^{\rm{T}}\begin{bmatrix}\frac{\boldsymbol{\alpha}^{2}}{\mathbf{C}\_{\rm{P}}}\kappa\kappa\boldsymbol{\kappa}^{\rm{T}}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}\mathbf{q}$$

$$\frac{1}{2}\frac{1}{\bar{s}\_{\rm{i}i}^{\rm{E}}}\int\_{\mathcal{V}}\left[S\_{i}^{2}+\frac{k\_{3i}^{2}}{1-k\_{3i}^{2}}\Delta S\_{i,3}^{2}\right]dV=\frac{1}{2}\mathbf{q}^{\rm{T}}\begin{bmatrix}\mathbf{C}\_{\rm{mech}}\mathbf{0}\\ \mathbf{0}&0\end{bmatrix}\mathbf{q}.\tag{7}$$

The potential energy can be rewritten as

$$
\mathcal{U} = \frac{1}{2} \mathbf{q}^{\mathrm{T}} \mathbf{C} \mathbf{q}\_{\prime} \tag{8}
$$

so that the stiffness matrix of the system follows as

$$\mathbf{C} = \begin{bmatrix} \mathbf{C}\_{\text{mech}} + \frac{a^2}{\mathsf{C}\_p} \kappa \kappa^T \frac{a}{\mathsf{C}\_p} \kappa \\\ \frac{a}{\mathsf{C}\_p} \kappa^T & \frac{1}{\mathsf{C}\_p} \end{bmatrix}. \tag{9}$$

The term **C**mech represents the 'mechanical' stiffness matrix of the piezoelement, which can be deduced in the same way as standard mechanical systems.

#### **2.2. Piezoelectric bimorph**

Now we can apply the above obtained results for the piezoelectric bimorph. We are considering the general case of a piezoelectric layer which has a distance *e* to the neutral axis of the beam. The coordinate axes are defined in such a way that the origin is at contact between the piezoelectric and the substrate layers at the clamped end, see Figure 2. The *x*<sup>1</sup> axis is in beam direction and the deformations occur in *x*<sup>3</sup> direction, which is also the direction of polaziration. With Euler-Bernoulli assumptions the strain is only applied in *x*<sup>1</sup> direction. That means the transversal effect of the piezoceramics is utilized. The strain terms described in the previous section then read for the case of the clamped beam:

4 Will-be-set-by-IN-TECH

piezoelectric layer

**q**. (7)

substrate layer

**0** *<sup>α</sup> <sup>C</sup>*<sup>p</sup> κ ⎤ ⎦ **q**

⎤ ⎦ **q**

�

**q**T**Cq**, (8)

⎦ . (9)

*α <sup>C</sup>*pκ<sup>T</sup> <sup>0</sup>

� *C*mech **0**

**0** 0

*w*(*x*<sup>1</sup> *e* )

*iQ*<sup>p</sup> <sup>=</sup> <sup>1</sup> 2 **q**T ⎡ ⎣

<sup>d</sup>*<sup>V</sup>* <sup>=</sup> <sup>1</sup> 2 **q**T

*<sup>C</sup>*<sup>p</sup> κκ<sup>T</sup> *<sup>α</sup>*

*<sup>C</sup>*pκ<sup>T</sup> <sup>1</sup>

*<sup>C</sup>*<sup>p</sup> κ

⎤

*C*<sup>p</sup>

*x*1

−1 2 2 *α C*p *iS*¯

> 1 2 *α*2 *C*p (*iS*¯ *i*) <sup>2</sup> <sup>=</sup> <sup>1</sup> 2 **q**T ⎡ ⎣ *α*2 *<sup>C</sup>*pκκ<sup>T</sup> **<sup>0</sup> 0** 0

*<sup>U</sup>* <sup>=</sup> <sup>1</sup> 2

**<sup>C</sup>**mech <sup>+</sup> *<sup>α</sup>*<sup>2</sup>

*α*

The term **C**mech represents the 'mechanical' stiffness matrix of the piezoelement, which can be

Now we can apply the above obtained results for the piezoelectric bimorph. We are considering the general case of a piezoelectric layer which has a distance *e* to the neutral axis of the beam. The coordinate axes are defined in such a way that the origin is at contact between the piezoelectric and the substrate layers at the clamped end, see Figure 2. The *x*<sup>1</sup> axis is in beam direction and the deformations occur in *x*<sup>3</sup> direction, which is also the direction of polaziration. With Euler-Bernoulli assumptions the strain is only applied in *x*<sup>1</sup> direction. That means the transversal effect of the piezoceramics is utilized. The strain terms described in the

*k*2 3*i* <sup>1</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> 3*i* Δ*S*<sup>2</sup> *i*,3 �

**C** =

deduced in the same way as standard mechanical systems.

⎡ ⎣

*x*2

*x*3

1 2 1 *s*E *ii*

The potential energy can be rewritten as

**2.2. Piezoelectric bimorph**

�

� *S*2 *<sup>i</sup>* +

*V*

so that the stiffness matrix of the system follows as

**Figure 2.** Piezoelectric bimorph.

$$
\begin{split}
\bar{S}\_{1,3} &= \frac{\int^{\bar{\ell}\_3} \mathbf{S}\_1 \mathbf{dx}\_3}{\ell\_3} = -\left(e + \frac{\ell\_3}{2}\right) w'''(\mathbf{x}\_1, t), \\
\Delta S\_{1,3} &= S\_1 - \bar{S}\_{1,3} = \left(\frac{\ell\_3}{2} - \mathbf{x}\_3\right) w'''(\mathbf{x}\_1, t),
\end{split}
$$

$$
\bar{S}\_1 = -\left(e + \frac{\ell\_3}{2}\right) \frac{\int^{\ell\_1} w''(\mathbf{x}\_1, t) \mathbf{dx}\_1}{\ell\_1} = -\left(e + \frac{\ell\_3}{2}\right) \frac{w'(\ell\_1) - w'(0)}{\ell\_1}.
\tag{10}
$$

The bending of the beam is described by *w*(*x*1, *t*). This term will be split into the part depending on coordinate *x*<sup>1</sup> and the part depending on time *t*,

$$w(\mathbf{x}\_1, t) = \mathcal{W}(\mathbf{x}\_1) q(t), \quad w''(\mathbf{x}\_1, t) = \mathcal{W}''(\mathbf{x}\_1) q(t). \tag{11}$$

In this example, only one mechanical degree of freedom is used to describe the vibrations. This is typically a reasonable approximation when the system vibrates close to one of its eigenfrequencies. In this way, the general stiffness matrix according to Equation 9 reduces to

$$\mathbf{C} = \begin{bmatrix} c\_{\text{mech}} + \frac{\boldsymbol{a}^2}{\mathsf{C}\_{\text{p}}} \boldsymbol{\kappa}^2 & \frac{\boldsymbol{a}}{\mathsf{C}\_{\text{p}}} \boldsymbol{\kappa} \\ & \frac{\boldsymbol{a}}{\mathsf{C}\_{\text{p}}} \boldsymbol{\kappa} & \frac{1}{\mathsf{C}\_{\text{p}}} \end{bmatrix} \tag{12}$$

and the mechanical coupling is written as

$$\kappa = \left( e + \frac{\ell\_3}{2} \right) \left( \mathcal{W}'(\ell\_1) - \mathcal{W}'(0) \right), \tag{13}$$

while the piezoelectric coupling reads

$$\frac{a}{\mathcal{C}\_{\rm p}} = \frac{k\_{31}^2}{1 - k\_{31}^2} \frac{1}{d\_{31}} \frac{\ell\_3}{\ell\_1}, \quad \frac{a^2}{\mathcal{C}\_{\rm p}} = \frac{1}{s\_{11}^E} \frac{k\_{31}^2}{1 - k\_{31}^2} \frac{\ell\_2 \ell\_3}{\ell\_1}.\tag{14}$$

Terms of the kind *<sup>α</sup>*<sup>2</sup> *<sup>C</sup>*<sup>p</sup> *<sup>κ</sup>*2, which determine the increase in eigenfrequencies between short circuit electrodes and isolated electrodes are obtained as

$$\frac{a^2}{C\_p} \kappa^2 = \frac{1}{s\_{11}^E} \frac{k\_{31}^2}{1 - k\_{31}^2} \frac{\ell\_2 \ell\_3}{\ell\_1} \left(\varepsilon + \frac{\ell\_3}{2}\right)^2 \left(W'(\ell\_1) - W'(0)\right)^2. \tag{15}$$

For a better understanding of these terms, it is useful to introduce the area moment of inertia *I*PZT of the piezoceramics around its own center of gravity and the moment of inertia *I*nF around the neutral axis of the beam. Additionally, also the difference *<sup>I</sup>*nF <sup>−</sup> *<sup>I</sup>*PZT is included

#### 6 Will-be-set-by-IN-TECH 308 Small-Scale Energy Harvesting

in the results,

$$I^{\rm PZT} = \frac{\ell\_2 \ell\_3^3}{12},$$

$$I^{\rm nF} = \frac{\ell\_2 \ell\_3^3}{12} + \left(e + \frac{\ell\_3}{2}\right)^2 \ell\_2 \ell\_3.$$

$$I^{\rm nF} - I^{\rm PZT} = \left(e + \frac{\ell\_3}{2}\right)^2 \ell\_2 \ell\_3. \tag{16}$$

With these definitions, the coupling terms can be expressed as

$$\frac{a^2}{C\_\mathrm{p}}\kappa^2 = \frac{1}{s\_{11}^\mathrm{E}}\frac{k\_{31}^2}{1-k\_{31}^2}\frac{I^{\mathrm{nF}}-I^{\mathrm{PZT}}}{\ell\_1}\left(\mathcal{W}'(\ell\_1)-\mathcal{W}'(0)\right)^2. \tag{17}$$

This result can be used to discuss different geometries and types of bimorphs. Obviously a beam that consists only of piezoelectric material does not have any coupling at all, because the distance *<sup>e</sup>* is exactly one half of the thickness of the piezoelectric layer, *<sup>e</sup>* <sup>=</sup> <sup>−</sup> �<sup>3</sup> <sup>2</sup> . This means the term *<sup>I</sup>*nF <sup>−</sup> *<sup>I</sup>*PZT vanishes. Contrary to this, a beam which is made of two identical piezoelectric layer is represented by *e* = 0 because of the symmetry, and a coupling exists. However, yet more efficient is the design of bimorphs or trimorphs with a substrate layer, which moves the neutral axis away from the surface of the piezoelectric layer. This results in a positive value *e* > 0. The best type is a symmetric trimorph with identical piezoelectric layers on both sides of the substrate layer. Here the distance equals half of the substrate layer. More details about the optimization of bimorphs can be found in [10].

In general, the piezomechanical system can be described by the following differential equations,

$$
\begin{bmatrix} m\_{\text{mech}} \ 0 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} \ddot{q} \\ \ddot{Q}\_{\text{p}} \end{bmatrix} + \begin{bmatrix} c\_{\text{mech}} + \frac{a^2}{\mathsf{C}\_{\text{p}}} \kappa^2 \ \frac{\kappa}{\mathsf{C}\_{\text{p}}} \kappa \\ \frac{a}{\mathsf{C}\_{\text{p}}} \kappa & \frac{1}{\mathsf{C}\_{\text{p}}} \end{bmatrix} \begin{bmatrix} q \\ Q\_{\text{p}} \end{bmatrix} = \begin{bmatrix} F(t) \\ -u\_{\text{p}}(t) \end{bmatrix},\tag{18}
$$

with the modal mass *m*mech, the external force *F*(*t*) and the voltage *u*p(*t*) at the electrodes of the piezoelement.

#### **2.3. Linear energy harvester**

Based on these results, the simplest and linear energy harvester can be modeled. In this case only a resistor is connected as an electrical load at the electrodes of the piezoelement. Therefore the voltage *u*<sup>p</sup> is dependent on the time derivative of charge *Q*˙ <sup>p</sup> and the differential equations of motions for this case including damping *d*mech read

$$
\begin{bmatrix} m\_{\text{mech}} \ 0 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} \ddot{\boldsymbol{q}} \\ \ddot{\boldsymbol{Q}}\_{\text{p}} \end{bmatrix} + \begin{bmatrix} d\_{\text{mech}} \ 0 \\ 0 \ \boldsymbol{R} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{q}} \\ \dot{\boldsymbol{Q}}\_{\text{p}} \end{bmatrix} + \begin{bmatrix} c\_{\text{mech}} + \frac{\boldsymbol{a}^{2}}{\mathsf{C}\_{\text{p}}} \boldsymbol{\kappa}^{2} \frac{\boldsymbol{\kappa}}{\mathsf{C}\_{\text{p}}} \kappa \\ \frac{\boldsymbol{a}}{\mathsf{C}\_{\text{p}}} \boldsymbol{\kappa} \end{bmatrix} \begin{bmatrix} \boldsymbol{q} \\ \boldsymbol{Q}\_{\text{p}} \end{bmatrix} = \begin{bmatrix} F(t) \\ 0 \end{bmatrix} \tag{19}
$$

We seek for the amplitudes of the stationary oscillations,

6 Will-be-set-by-IN-TECH

3 <sup>12</sup> ,

> � *e* + �3 2 �<sup>2</sup> �2�3,

> > �2�3. (16)

�<sup>2</sup> . (17)

<sup>2</sup> . This

*I*

*I*

PZT =

*k*2 31 <sup>1</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> 31

*I* nF <sup>−</sup> *<sup>I</sup>*

*<sup>κ</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> *s*E 11

*α*2 *C*p

� *m*mech 0

**2.3. Linear energy harvester**

� � *q*¨

*Q*¨ p

� +

0 0

� � *q*¨

*Q*¨ p

With these definitions, the coupling terms can be expressed as

details about the optimization of bimorphs can be found in [10].

� + ⎡ ⎣

equations of motions for this case including damping *d*mech read

0 *R*

� � *q*˙

*Q*˙ p

� + �

� *d*mech 0

PZT <sup>=</sup> �2�<sup>3</sup>

nF <sup>=</sup> �2�<sup>3</sup> 3 <sup>12</sup> <sup>+</sup>

> � *e* + �3 2 �<sup>2</sup>

*<sup>I</sup>*nF <sup>−</sup> *<sup>I</sup>*PZT �1

This result can be used to discuss different geometries and types of bimorphs. Obviously a beam that consists only of piezoelectric material does not have any coupling at all, because the distance *<sup>e</sup>* is exactly one half of the thickness of the piezoelectric layer, *<sup>e</sup>* <sup>=</sup> <sup>−</sup> �<sup>3</sup>

means the term *<sup>I</sup>*nF <sup>−</sup> *<sup>I</sup>*PZT vanishes. Contrary to this, a beam which is made of two identical piezoelectric layer is represented by *e* = 0 because of the symmetry, and a coupling exists. However, yet more efficient is the design of bimorphs or trimorphs with a substrate layer, which moves the neutral axis away from the surface of the piezoelectric layer. This results in a positive value *e* > 0. The best type is a symmetric trimorph with identical piezoelectric layers on both sides of the substrate layer. Here the distance equals half of the substrate layer. More

In general, the piezomechanical system can be described by the following differential

with the modal mass *m*mech, the external force *F*(*t*) and the voltage *u*p(*t*) at the electrodes of

Based on these results, the simplest and linear energy harvester can be modeled. In this case only a resistor is connected as an electrical load at the electrodes of the piezoelement. Therefore the voltage *u*<sup>p</sup> is dependent on the time derivative of charge *Q*˙ <sup>p</sup> and the differential

*<sup>C</sup>*<sup>p</sup> *<sup>κ</sup>*<sup>2</sup> *<sup>α</sup> <sup>C</sup>*<sup>p</sup> *κ*

*C*<sup>p</sup>

*<sup>c</sup>*mech <sup>+</sup> *<sup>α</sup>*<sup>2</sup>

*α <sup>C</sup>*<sup>p</sup> *<sup>κ</sup>* <sup>1</sup>

*<sup>C</sup>*<sup>p</sup> *<sup>κ</sup>*<sup>2</sup> *<sup>α</sup> <sup>C</sup>*<sup>p</sup> *κ*

*C*<sup>p</sup>

� � *q*

*Q*p

� = � *F*(*t*)

�

(19)

0

⎤ ⎦ � *q*

*Q*p

� = � *F*(*t*)

−*u*p(*t*)

�

, (18)

*<sup>c</sup>*mech <sup>+</sup> *<sup>α</sup>*<sup>2</sup>

*α <sup>C</sup>*<sup>p</sup> *<sup>κ</sup>* <sup>1</sup>

� *W*�

(�1) − *W*�

(0)

in the results,

equations,

the piezoelement.

� *m*mech 0

0 0

$$\mathbf{\hat{q}} = \left(-\Omega^2 \mathbf{M} + j\Omega \mathbf{D} + \mathbf{C}\right)^{-1} \begin{bmatrix} \hat{F} \\ 0 \end{bmatrix} \tag{20}$$

with the corresponding system matrices according to Equation 19. The amplitudes of the time signals are marked by a hat. With the stationary charge amplitude *Q*ˆ <sup>p</sup> the instantaneous power *p*(*t*) can be calculated,

$$p(t) = R\dot{i}\_\mathbf{p}^2(t) = R\Omega^2 \dot{Q}\_\mathbf{p}^2 \cos^2\left(\Omega t\right),\tag{21}$$

where *i*p is the current. The energy that is dissipated in the resistor will be treated as the stationary harvested energy *E*h,stat. It is then the integral of the power *p*,

$$E\_{\rm h,stat} = \int\_0^{\frac{2\pi}{\hbar}} p(t) \mathbf{d}t = \pi R \Omega \hat{Q}\_{\rm p}^2. \tag{22}$$

This result, normalized to force amplitude, is shown in Figure 3 versus the load resistance *R* and the excitation frequency Ω, normalized to the mechanical eigenfrequenzcy *ω*0. For this study, the following system parameters are used:

$$m\_{\text{mech}} = 0.005 \text{kg}\_{\prime}$$

$$d\_{\text{mech}} = 0.1212 \text{Ns} / \text{m}\_{\prime}$$

$$c\_{\text{mech}} = 341.2651 \text{N} / \text{m}\_{\prime}$$

$$\alpha \text{x} = 0.002,$$

$$\mathsf{C}\_{\text{P}} = 83.676 \mathsf{n} \text{F}. \tag{23}$$

Obviously the harvested energy is highly frequency dependent. Only in a narrow frequency range around the eigenfrequency the efficiency is high. Also the resistor must be tuned for maximum harvested energy. The system is less sensitive towards changes of the resistance, but one can show that the optimal resistance which is optimal for low coupling and/or high damping is obtained as

$$R\_{\rm opt} = \frac{1}{\Omega C\_{\rm p}} \tag{24}$$

#### **3. Array configuration**

Utilizing an array configuration for the extension of the generator bandwidth is a commonly used approach. The idea is simple and powerfull: multiple generator elements are tuned to slightly (a few %) different eigenfrequencies. A major factor is the connection to the electrical system; if each individual element uses its own bridge rectifier, the elements are electrically uncoupled and their output power simply can be summed up. However, individual bridge

**Figure 3.** Harvested energy *E*h,stat versus normalized excitation frequency Ω and load resistance *R*.

rectifiers are a comparable big effort, a huge number of elements and cables is needed, further the voltage drop over each diode is also summed up, so that the losses increase. As soon the individual elements are made of one part the electrodes are connected naturally, which results automatically in parallel connected elements. In such a configuration, Figure 4, the elements are electrically coupled.

This section investigates the effect of the electrical coupling on the performance of piezoelectric energy harvesting generators. Two configurations are investigated, the electrical parallel connection as well as the electrical serial connection. Both cases are utilizing two elements as most basic version of a piezoelectric array. The model is based on the linear generator model in Equation 19, therefore a fore excitation is assumed. We further assume that the applied fore is equal on all elements, representing a common support. The resonance frequency tuning is made by an adoption of the modal mass of the elements (change of tip mass), all other parameters are assumed to be constant. The configurations including boundary conditions is given in Figure 4. Each element can be represented by the linear Equation 19. In the parallel configuration the two voltages are equal and *u*P1 = *u*P2 = 310 Small-Scale Energy Harvesting Modeling Aspects of Nonlinear Energy Harvesting for Increased Bandwidth <sup>9</sup> Modeling Aspects of Nonlinear Energy Harvesting for Increased Bandwidth 311

**Figure 4.** Schematic Circuit. Left: series configuration. Right: parallel configuration

8 Will-be-set-by-IN-TECH

0

0.7*ω*<sup>0</sup>

**Figure 3.** Harvested energy *E*h,stat versus normalized excitation frequency Ω and load resistance *R*.

rectifiers are a comparable big effort, a huge number of elements and cables is needed, further the voltage drop over each diode is also summed up, so that the losses increase. As soon the individual elements are made of one part the electrodes are connected naturally, which results automatically in parallel connected elements. In such a configuration, Figure 4, the elements

This section investigates the effect of the electrical coupling on the performance of piezoelectric energy harvesting generators. Two configurations are investigated, the electrical parallel connection as well as the electrical serial connection. Both cases are utilizing two elements as most basic version of a piezoelectric array. The model is based on the linear generator model in Equation 19, therefore a fore excitation is assumed. We further assume that the applied fore is equal on all elements, representing a common support. The resonance frequency tuning is made by an adoption of the modal mass of the elements (change of tip mass), all other parameters are assumed to be constant. The configurations including boundary conditions is given in Figure 4. Each element can be represented by the linear Equation 19. In the parallel configuration the two voltages are equal and *u*P1 = *u*P2 =

Excitation frequency Ω

*ω*0

1.3*ω*<sup>0</sup>

0.5

1

Resistance *R* [Ω]

1.5

2

are electrically coupled.

2.5

Harvested

 energy

*E*h,stat/ ˆ

*F*<sup>p</sup>

[J/N]

0.01

0.02

0

x 105

*R*<sup>L</sup> (*i*P1 + *i*P2 ) apply due to Kirchhoff's rules. Using both, the parallel configuration is described by

$$
\begin{bmatrix} m\_{\text{mech}} - \Delta m & 0 & 0 & 0 \\ 0 & m\_{\text{mech}} + \Delta m & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \ddot{q}\_1 \\ \ddot{q}\_2 \\ \ddot{Q}\_{\text{P}1} \\ \ddot{Q}\_{\text{P}2} \end{bmatrix} + \begin{bmatrix} d\_{\text{mech}} & 0 & 0 & 0 \\ 0 & d\_{\text{mech}} & 0 & 0 \\ 0 & 0 & R & R \\ 0 & 0 & R & R \end{bmatrix} \begin{bmatrix} \dot{q}\_1 \\ \dot{q}\_2 \\ \dot{Q}\_{\text{P}1} \\ \ddot{Q}\_{\text{P}2} \end{bmatrix} \\
$$

$$
\dots + \begin{bmatrix} c\_{\text{mech}} + \frac{a^2}{\text{C}\_{\text{p}}} \kappa^2 & 0 & \frac{a}{\text{C}\_{\text{p}}} & 0 \\ 0 & c\_{\text{mech}} + \frac{a^2}{\text{C}\_{\text{p}}} \kappa^2 & 0 & \frac{a}{\text{C}\_{\text{p}}} \\ \frac{a}{\text{C}\_{\text{p}}} \kappa & 0 & \frac{1}{\text{C}\_{\text{p}}} & 0 \\ 0 & \frac{a}{\text{C}\_{\text{p}}} \kappa & 0 & \frac{1}{\text{C}\_{\text{p}}} \end{bmatrix} \begin{bmatrix} q\_1 \\ q\_2 \\ Q\_{\text{P}1} \\ Q\_{\text{P}2} \end{bmatrix} = \begin{bmatrix} F(t) \\ F(t) \\ 0 \\ 0 \end{bmatrix}. \tag{25}
$$

The coupling between the elements is obviously seen in the damping matrix. In the series configuration, the current at both generators is equal. Therefore *u*P1 + *u*P2 = *R*L*i*P1 = *R*L*i*P2 is applied to couple the two generators:

$$
\begin{bmatrix} m\_{\text{mech}} - \Delta m & 0 & 0 \\ 0 & m\_{\text{mech}} + \Delta m & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \ddot{q}\_1 \\ \ddot{q}\_2 \\ \ddot{Q}\_P \end{bmatrix} + \begin{bmatrix} d\_{\text{mech}} & 0 & 0 \\ 0 & d\_{\text{mech}} & 0 \\ 0 & 0 & R \end{bmatrix} \begin{bmatrix} \dot{q}\_1 \\ \dot{q}\_2 \\ \dot{Q}\_P \end{bmatrix} + \dots
$$

$$
\begin{bmatrix} c\_{\text{mech}} + \frac{a^2}{\text{C}\_\text{p}} \kappa^2 & 0 & \frac{a}{\text{C}\_\text{p}} \kappa \\ 0 & c\_{\text{mech}} + \frac{a^2}{\text{C}\_\text{p}} \kappa^2 & \frac{a}{\text{C}\_\text{p}} \kappa \\ \frac{a}{\text{C}\_\text{p}} \kappa & \frac{a}{\text{C}\_\text{p}} \kappa & \frac{b}{\text{C}\_\text{p}} \end{bmatrix} \begin{bmatrix} q\_1 \\ q\_2 \\ Q\_\text{P} \end{bmatrix} = \begin{bmatrix} F(t) \\ F(t) \\ 0 \end{bmatrix} . \tag{26}
$$

Here the coupling is evidently in the stiffness matrix. Where all parameters correspond to the ones from the modeling section. The mass for frequency adoption is Δ*m* = 0.5*g*. For the steady state the system can be solved and the transfer functions can be determined analog to the single system. The total dissipated energy at the resistor is again given by

$$E\_{\rm h,stat} = \int\_0^{\frac{2\pi}{11}} u(t)i(t)dt. \tag{27}$$

Evaluating this equation for voltage and current over the load resistor gives the gained energy for the application, the result is depicted in Figure 5 for serial configuration and in Figure 6 for parallel configuration. For the in series connected generators the bandwidth is widened at high impedance loads and it is not significantly changed for low impedances. For the parallel configuration, a bandwidth expanded at low impedance loads and also not changed for high impedances.

**Figure 5.** Energy per period in serial configuration in mJ/N. Left: gained useable energy. Middle: element 1. Right: element2.

**Figure 6.** Energy per period in parallel configuration in mJ/N. Left: gained useable energy. Middle: element 1. Right: element2.

To explain why there is no widening of the bandwidth in serial connection at low impedances the Equation 27 is evaluated for both elements, using the individual voltage and the common current. Figure 5 show that the second element works as energy sink for low impedances, with the consequence, that the energy generated by element one is used to actuate the other one. Figure 7 shows this effect. Even with the overall maximum displacement of element two

312 Small-Scale Energy Harvesting Modeling Aspects of Nonlinear Energy Harvesting for Increased Bandwidth <sup>11</sup> Modeling Aspects of Nonlinear Energy Harvesting for Increased Bandwidth 313

**Figure 7.** Tip displacement of both elements in serial configuration in mm/N. Left: element 1. Right: element2.

**Figure 8.** Schematic of SSHI circuit.

10 Will-be-set-by-IN-TECH

Here the coupling is evidently in the stiffness matrix. Where all parameters correspond to the ones from the modeling section. The mass for frequency adoption is Δ*m* = 0.5*g*. For the steady state the system can be solved and the transfer functions can be determined analog to

> <sup>2</sup>*<sup>π</sup>* Ω 0

**Figure 5.** Energy per period in serial configuration in mJ/N. Left: gained useable energy. Middle:

**Figure 6.** Energy per period in parallel configuration in mJ/N. Left: gained useable energy. Middle:

To explain why there is no widening of the bandwidth in serial connection at low impedances the Equation 27 is evaluated for both elements, using the individual voltage and the common current. Figure 5 show that the second element works as energy sink for low impedances, with the consequence, that the energy generated by element one is used to actuate the other one. Figure 7 shows this effect. Even with the overall maximum displacement of element two

Evaluating this equation for voltage and current over the load resistor gives the gained energy for the application, the result is depicted in Figure 5 for serial configuration and in Figure 6 for parallel configuration. For the in series connected generators the bandwidth is widened at high impedance loads and it is not significantly changed for low impedances. For the parallel configuration, a bandwidth expanded at low impedance loads and also not changed for high

*u*(*t*)*i*(*t*)d*t*. (27)

the single system. The total dissipated energy at the resistor is again given by

*E*h,stat =

impedances.

element 1. Right: element2.

element 1. Right: element2.

at low impedances the gained useable energy is low. The same effect but for high impedances is shown in Fig. 6 for the parallel configuration. In this case element one is the energy sink.

Concluding, the utilization of arrays with serials or parallel electrical coupling has only a major positiv effect on the bandwidth for a matched impedance, for unmatched impedances the coupling can be a drawback. The bandwidth can be enlarged with any further element, the mean energy output over bandwidth in general is higher if the mistuning of the resonance frequencies is smaller. For power and bandwidth comparison of generator arrays it is reasonable to keep the volume of active material constant.

### **4. Switching networks (SSHI)**

An important technique to enhance the energy extraction is to use nonlinear switching networks. In detail, the 'Synchronized Switch Harvesting on Inductor' (SSHI) technique is studied. Such networks are an active field of research [3, 4, 7]. The corresponding network is shown in Figure 8. This nonlinear electric circuit consists of a switching *LR*-branch, a rectifier and load capacitor *C*r. The load is again described as a resistor *R*L. Assuming a sinusoidal mechanical deformation of the piezoceramics, the switch is briefly closed on minima and maxima of the deformation. During these times, an oscillating electric circuit is formed, as the capacitive piezoelectric transducer and the inductance are connected. During this electrical

#### 12 Will-be-set-by-IN-TECH 314 Small-Scale Energy Harvesting

semi-period the switch is kept close and the voltage is inverted. As this electrical period time is generally much shorter than the mechanical one, this occurs nearly instantaneously. After inversion, the switch is opened again until the next deformation extremum. Consequently, the resulting voltage signal at the piezoelectrodes is nearly rectangular-shaped. Previous publications have proven the enhanced performance of SSHI circuits especially for systems with low piezoelectrical coupling.

The modeling and optimization of such networks is not straight forward, as the overall system is nonlinear. In the following we will present a modeling technique that is based on the harmonic balance method.

#### **4.1. Period response for harmonic excitation**

Firstly, the periodic response if the SSHI is studied. In order to simplify the results, the following approximations are defined,

$$L\_{\sf s} \to 0, \quad \mathcal{C}\_{\sf r} \to \infty. \tag{28}$$

and the electrical losses remain constant. In practical realizations, all approximations are appropriate. A small inductance value results in a fast inversion, and a large storage capacitor means that the voltage at the load is nearly constant. Both situations are typically wanted. Further on, the nonlinear system can be treated as a piecewise linear system.

In order to obtain the stationary voltage signal it is necessary to study one semi-period of the system and consider the stationarity condition, which means the signal repeats after each period. We define the time axis in such a way that for *t* = 0 the voltage was just inverted and the switch is opened. With the approximation *C*<sup>r</sup> → ∞ the voltage *u*<sup>L</sup> at the load capacitor is constant at *u*0. This value has to be calculated yet. Because of the loss resistances - described by the electrical damping ratio *<sup>ζ</sup>* - the voltage changes from <sup>±</sup>*u*<sup>0</sup> to <sup>∓</sup>*u*0e−*πζ* . That means the absolute value of the voltage *u*p at the piezoelectrodes after inversion is smaller than the voltage at the load *u*0. Therefore the recifier blocks, and the piezovoltage changes linearly with the piezodeformation *q*,

$$
\mu\_{\rm p}(t) = -\frac{\mathfrak{a}\kappa}{\mathbb{C}\_{\rm p}} q(t) + \mathbb{C}\_{\rm 0} \tag{29}
$$

with an integration constant *C*0. According to the definitions, the piezodeformation must have an extremum at *t* = 0, so that the time signal reads

$$\mathfrak{q}(t) = \mathfrak{q}\cos\left(\Omega t\right),\tag{30}$$

and for the voltage signal it follows

$$
\mu\_{\rm P}(t) = -\frac{a\kappa}{\mathcal{C}\_{\rm P}} \mathfrak{q} \cos\left(\Omega t\right) + \mu\_{0} \mathbf{e}^{-\pi \zeta} + \frac{a\kappa}{\mathcal{C}\_{\rm P}} \mathfrak{q}.\tag{31}
$$

The voltage amplitude *u*<sup>0</sup> is still unknown, but at least the time *t*<sup>1</sup> can be calculated, at which the voltage at the piezoelectrodes equals the voltage at the load, *u*p(*t* = *t*1) = *u*0:

$$t\_1 = \arccos\left[\frac{\mu\_0}{\frac{\alpha \chi}{C\_p} \hat{\mathcal{J}}} \left(\mathbf{e}^{-\pi \zeta} - 1\right) + 1\right] / \Omega. \tag{32}$$

At this time *t*<sup>1</sup> the rectifier changes from 'blocking' to 'conducting'. Practically this means that the piezovoltage remains constant at *u*<sup>0</sup> from *t*<sup>1</sup> until the end of the semi-period *T*/2.

12 Will-be-set-by-IN-TECH

semi-period the switch is kept close and the voltage is inverted. As this electrical period time is generally much shorter than the mechanical one, this occurs nearly instantaneously. After inversion, the switch is opened again until the next deformation extremum. Consequently, the resulting voltage signal at the piezoelectrodes is nearly rectangular-shaped. Previous publications have proven the enhanced performance of SSHI circuits especially for systems

The modeling and optimization of such networks is not straight forward, as the overall system is nonlinear. In the following we will present a modeling technique that is based on the

Firstly, the periodic response if the SSHI is studied. In order to simplify the results, the

and the electrical losses remain constant. In practical realizations, all approximations are appropriate. A small inductance value results in a fast inversion, and a large storage capacitor means that the voltage at the load is nearly constant. Both situations are typically wanted.

In order to obtain the stationary voltage signal it is necessary to study one semi-period of the system and consider the stationarity condition, which means the signal repeats after each period. We define the time axis in such a way that for *t* = 0 the voltage was just inverted and the switch is opened. With the approximation *C*<sup>r</sup> → ∞ the voltage *u*<sup>L</sup> at the load capacitor is constant at *u*0. This value has to be calculated yet. Because of the loss resistances - described by the electrical damping ratio *<sup>ζ</sup>* - the voltage changes from <sup>±</sup>*u*<sup>0</sup> to <sup>∓</sup>*u*0e−*πζ* . That means the absolute value of the voltage *u*p at the piezoelectrodes after inversion is smaller than the voltage at the load *u*0. Therefore the recifier blocks, and the piezovoltage changes linearly

Further on, the nonlinear system can be treated as a piecewise linear system.

*<sup>u</sup>*p(*t*) = <sup>−</sup> *ακ*

*C*p

with an integration constant *C*0. According to the definitions, the piezodeformation must have

The voltage amplitude *u*<sup>0</sup> is still unknown, but at least the time *t*<sup>1</sup> can be calculated, at which

*q*ˆ cos (Ω*t*) + *u*0e−*πζ* +

<sup>e</sup>−*πζ* <sup>−</sup> <sup>1</sup>

 + 1 

*L*<sup>s</sup> → 0, *C*<sup>r</sup> → ∞. (28)

*q*(*t*) + *C*0, (29)

*q*ˆ. (31)

/Ω. (32)

*q*(*t*) = *q*ˆ cos (Ω*t*), (30)

*ακ C*p

with low piezoelectrical coupling.

following approximations are defined,

**4.1. Period response for harmonic excitation**

harmonic balance method.

with the piezodeformation *q*,

and for the voltage signal it follows

an extremum at *t* = 0, so that the time signal reads

*<sup>u</sup>*p(*t*) = <sup>−</sup> *ακ*

*t*<sup>1</sup> = acos

*C*p

the voltage at the piezoelectrodes equals the voltage at the load, *u*p(*t* = *t*1) = *u*0:

 *u*0 *ακ <sup>C</sup>*<sup>p</sup> *q*ˆ  The piezovoltage *u*<sup>0</sup> can be calculated based on the energy balance. Therefore, the transferred energy *E*t, the harvested energy *E*<sup>h</sup> and the energy dissipated in the switching branch *E*<sup>s</sup> must be obtained. They read, respectively,

$$E\_t = -2\int\_0^{T/2} F\_\mathbf{P}(t)\dot{q}(t)\mathbf{d}t = 2\kappa \mathbf{x}\Omega \dot{q} \left[\int\_0^{t\_1} u\_\mathbf{P}(t)\sin\left(\Omega t\right)\mathbf{d}t + \int\_{t\_1}^{T/2} u\_0 \sin\left(\Omega t\right)\mathbf{d}t\right],$$

$$E\_\mathbf{h} = 2\int\_0^{T/2} \frac{u\_0^2}{R} \mathbf{d}t = 2\frac{\pi}{\Omega} \frac{u\_0^2}{R},$$

$$E\_\mathbf{s} = 2\frac{1}{2} \mathbb{C}\_\mathbf{P} u\_0^2 \left[1 - \left(\mathbf{e}^{-\pi\xi}\right)^2\right].\tag{33}$$

The transferred energy corresponds to the total energy that is shifted from the mechanical system into the piezoelectric system, while the other terms are the energies that are dissipated within the load resistor and the resistor of the switching branch. In stationary situation, the equality

$$E\_{\rm f} = E\_{\rm h} + E\_{\rm s} \tag{34}$$

holds. With this equation, finally the stationary voltage amplitude can be recalculated as

$$\mu\_0 = 2 \frac{\alpha \kappa}{\mathcal{C}\_\mathbb{P} \left(1 - e^{-\pi \zeta} \right) + \frac{\pi}{\Pi \mathcal{R}}} \not\!\!/ . \tag{35}$$

Figure 9 shows the time signal of the voltage *u*p. Inserting the stationary voltage amplitude into the energies in Equation 33 gives us the stationary energies,

$$E\_{\rm t,stat} = 4 \frac{a^2 \kappa^2}{\mathcal{C}\_\mathcal{P}} \frac{1 - \mathbf{e}^{-2\pi \tilde{\zeta}} + \frac{2\pi}{\mathcal{C}\_\mathcal{p} \Omega \mathcal{R}}}{\left(1 - \mathbf{e}^{-\pi \tilde{\zeta}} + \frac{\pi}{\mathcal{C}\_\mathcal{p} \Omega \mathcal{R}}\right)^2} \hat{q}^2,$$

$$E\_{\rm b,stat} = 4 \frac{a^2 \kappa^2}{\mathcal{C}\_\mathcal{P}} \frac{\frac{2\pi}{\mathcal{C}\_\mathcal{p} \Omega \mathcal{R}}}{\left(1 - \mathbf{e}^{-\pi \tilde{\zeta}} + \frac{\pi}{\mathcal{C}\_\mathcal{p} \Omega \mathcal{R}}\right)^2} \hat{q}^2,$$

$$E\_{\rm s,stat} = 4 \frac{a^2 \kappa^2}{\mathcal{C}\_\mathcal{P}} \frac{1 - \mathbf{e}^{-2\pi \tilde{\zeta}}}{\left(1 - \mathbf{e}^{-\pi \tilde{\zeta}} + \frac{\pi}{\mathcal{C}\_\mathcal{p} \Omega \mathcal{R}}\right)^2} \hat{q}^2. \tag{36}$$

All these energy terms have quadratic dependency with the force factor *α*, which should be maximized for a high energy conversion. Also the electrical damping ratio *ζ* in the switching branch should be low. They also grow quadratically with the vibration amplitude *q*ˆ of the oscillator.

However, for the force excited vibrations, the vibration amplitude is influenced by the harvesting device. The transferred energy *E*t,stat yields a damping effect upon the oscillator, which reduces the vibration amplitudes. In order to determine the vibration amplitudes,

**Figure 9.** Time signals of piezodeformation and voltage at the electrodes.

the harmonic balance method is applied in the following. In this technique the shunted piezoceramics is replaced by a spring - damper combination. Therefore the period - but not harmonic - force response *F*p(*t*) = −*αu*p(*t*) of the shunted piezoceramics is expressed by its Fourier-series,

$$u\_{\mathbb{P}}(t) = \frac{1}{2}a\_0 + \sum\_{i=1}^{\infty} \left( a\_i \cos \left( i\Omega t \right) + b\_i \sin \left( i\Omega t \right) \right) \,. \tag{37}$$

The Fourier-coefficients *ai*, *bi* are obtained by the periodic timesignal *u*p(*t*),

$$a\_{\rm i} = \frac{2}{T} \int\_{\rm c}^{c+T} u\_{\rm P}(t) \cos \left( i \Omega t \right) \mathrm{d}t; \quad b\_{\rm i} = \frac{2}{T} \int\_{\rm c}^{c+T} u\_{\rm P}(t) \sin \left( i \Omega t \right) \mathrm{d}t. \tag{38}$$

The idea of the proposed linearization techniques is to approximate the periodic voltage signal by its main harmonics,

$$u\_{\mathbb{P}}(t) \approx a\_1 \cos\left(\Omega t\right) + b\_1 \sin\left(\Omega t\right). \tag{39}$$

This harmonic force signal is also produced by a spring - damper combination with the following parameters,

$$c^\* = \frac{a\_1}{\hat{q}}, \quad d^\* = -\frac{b\_1}{\Omega \hat{q}}.\tag{40}$$

In general, these replacement parameters *c*∗, *d*∗ are frequency dependent. With these results, the stationary vibration amplitudes of the oscillator with shunted piezoceramics can be recalculated,

$$\mathfrak{q} = \frac{\hat{\mathbf{f}\_{\mathbf{P}}}}{|-m\Omega^2 + j\left(d + d^\*\right)\Omega + c + c^\*|} \tag{41}$$

**Figure 10.** Harvested energy *E*h,stat versus normalized excitation frequency Ω and load resistance *R*.

Finally this stationary vibration amplitude *q*ˆ can be inserted into Equation 36 for the harvested energy *E*h,stat. The resulting energy is shown in Figure 10 versus the excitation frequency Ω and the load resistance *R*. This figure can be compared with the linear resistance case in Figure 3. It again shows that most energy is harvested at the resonance frequency *ω*<sup>0</sup> of the oscillator, because the vibration amplitudes are highest in this case. But also the load resistor must be tuned correctly in order to achieve the maximum energy. Compared to the standard case the maximum amount of harvested energy is similar, while the frequency bandwidth with SSHI tends to be larger. However, the voltage at the load resistance with SSHI circuit is nearly constant which is wanted in most practical cases, while it is a harmonics oscillations with the linear resistance. Additionally the damping effect upon the mechanical structure is higher, because of the additional energy that is dissipated within the resistance of the switching branch.

#### **5. Piezomagnetoelastic energy harvesting**

14 Will-be-set-by-IN-TECH

Time *t*

the harmonic balance method is applied in the following. In this technique the shunted piezoceramics is replaced by a spring - damper combination. Therefore the period - but not harmonic - force response *F*p(*t*) = −*αu*p(*t*) of the shunted piezoceramics is expressed by its

The idea of the proposed linearization techniques is to approximate the periodic voltage signal

This harmonic force signal is also produced by a spring - damper combination with the

In general, these replacement parameters *c*∗, *d*∗ are frequency dependent. With these results, the stationary vibration amplitudes of the oscillator with shunted piezoceramics can be

*<sup>q</sup>*<sup>ˆ</sup> , *<sup>d</sup>*<sup>∗</sup> <sup>=</sup> <sup>−</sup> *<sup>b</sup>*<sup>1</sup>

p

*T*

Ω*q*ˆ

 *c*+*T c*

*u*p(*t*) ≈ *a*<sup>1</sup> cos (Ω*t*) + *b*<sup>1</sup> sin (Ω*t*). (39)


**Figure 9.** Time signals of piezodeformation and voltage at the electrodes.

2 *a*<sup>0</sup> +

∞ ∑ *i*=1

*<sup>u</sup>*p(*t*) cos (*i*Ω*t*)d*t*; *bi* <sup>=</sup> <sup>2</sup>

*<sup>c</sup>*<sup>∗</sup> <sup>=</sup> *<sup>a</sup>*<sup>1</sup>

*<sup>q</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>F</sup>*<sup>ˆ</sup>

The Fourier-coefficients *ai*, *bi* are obtained by the periodic timesignal *u*p(*t*),

*<sup>u</sup>*p(*t*) = <sup>1</sup>

 *c*+*T c*

0 *t*<sup>1</sup> *T*/2 *T*

Voltage

−*u*<sup>0</sup>

*ai* <sup>=</sup> <sup>2</sup> *T*

by its main harmonics,

following parameters,

recalculated,

Fourier-series,

*u*p

*u*0

0

Displ. *q* Voltage *u*p

(*ai* cos (*i*Ω*t*) + *bi* sin (*i*Ω*t*)). (37)

*u*p(*t*) sin (*i*Ω*t*)d*t*. (38)

. (40)

One major drawback for energy harvesting systems is that conventional generators produce the maximum energy when the system is excited at its resonance frequency. If the excitation frequency shifts the output power is drastically reduced. To overcome this disadvantage multiple researchers work on different broadband techniques to widen the operational frequency range. The focus in this section is on piezomagnetoelastic energy harvesting strategies. The equations of motion (EOM) are derived in the previous section. The generator is based on a model with lumped parameters. The broadband response is achieved by using nonlinear magnetic forces. Piezomagnetoelastic generators are studied in a bunch of multiple research activities [1, 2, 13, 14]. In many approaches the system is modeled as Duffing oscillator. Usually the system parameters in the model are adjusted manually to match the amplitude or power response of the experiment. The aim in this work is to investigate an analytically approach to derive the duffing parameters out of the system parameters.

This section is organized as followed. The mechanical EOM is derived for the piezomagnetoelastic energy harvesting system. In the following the duffing parameters are derived with respect to the system and input parameters and the system dynamic is discussed. The last part shows the analytic solution for large orbit oscillations.

#### **5.1. Modeling of the piezomagnetoelastic energy harvesting system**

Figure 11 gives a schematic view of the piezomagnetoelastic system. The energy harvester consist of a cantilever with two piezoelectric patches mounted on each side of an inactive substructure. The system is excited by a harmonic force

$$F(t) = \hat{F}\sin(\omega t)\tag{42}$$

where *F*ˆ is the amplitude of the excitation and *ω* is the excitation frequency. A magnetic tip mass is attached to the free end of the beam. Another permanent magnet is stationary mounted near the free end. The magnets are oppositely poled so they exhibit a repulsive force. The nonlinear magnetic force leads to two stable equilibrium positions. Figure 11 shows both symmetric stable equilibrium positions. The tip displacement is given with *q* and the magnet spacing is *s*. The coupled mechanical and electrical differential equations are derived in the previous section

$$m\ddot{\eta} + d\dot{\eta} + c\eta - \frac{a}{\mathbb{C}\_{\mathbb{P}}} \mathbb{Q}\_{\mathbb{P}} + \frac{d\mathcal{U}\_{\text{mag}}}{dq} = F(t) \tag{43}$$

and

$$R\dot{Q}\_{\rm P} + \frac{1}{C\_{\rm P}}Q\_{\rm P} - \frac{\alpha}{C\_{\rm P}}q = 0.\tag{44}$$

Equation 43 and 44 are similar to 19 where *m* = *m*mech is the modal mass. The modal damping is *<sup>d</sup>* and *<sup>c</sup>* <sup>=</sup> *<sup>c</sup>*mech <sup>+</sup> *<sup>α</sup>*<sup>2</sup> *<sup>C</sup>*<sup>p</sup> *<sup>κ</sup>*<sup>2</sup> is the total mechanical stiffness. Additionally to the linear differential equations the nonlinear magnetic force leads to

$$\mathcal{U}\_{\rm mag} = \frac{\mu\_0}{4\pi} \nabla \frac{\mu\_{\rm A} \mathbf{r}\_{\rm AB}}{||\mathbf{r}\_{\rm AB}||\_2^3} \mu\_{\rm B} \tag{45}$$

where *μ*<sup>0</sup> is the permeability of free space, μ*<sup>A</sup>* and μ*<sup>B</sup>* are the magnetic dipole moment vectors. **r***AB* is the vector from the source of magnet B to magnet A. �·�<sup>2</sup> is the EUKLIDEAN norm and ∇ is the vector gradient defined with

$$\nabla \frac{1}{r^n} = \begin{bmatrix} \partial / \partial x \\ \partial / \partial y \\ \partial / \partial z \end{bmatrix} \frac{1}{r^n} = -\frac{n}{r^{n+1}} \begin{bmatrix} \mathbf{x} / r \\ y / r \\ z / r \end{bmatrix} = -\frac{n \mathbf{r}}{r^{n+2}}.\tag{46}$$

**Figure 11.** Schematic view of the piezomagnetoelastic energy harvesting system

The magnetic dipole moment vectors are written as

16 Will-be-set-by-IN-TECH

frequency shifts the output power is drastically reduced. To overcome this disadvantage multiple researchers work on different broadband techniques to widen the operational frequency range. The focus in this section is on piezomagnetoelastic energy harvesting strategies. The equations of motion (EOM) are derived in the previous section. The generator is based on a model with lumped parameters. The broadband response is achieved by using nonlinear magnetic forces. Piezomagnetoelastic generators are studied in a bunch of multiple research activities [1, 2, 13, 14]. In many approaches the system is modeled as Duffing oscillator. Usually the system parameters in the model are adjusted manually to match the amplitude or power response of the experiment. The aim in this work is to investigate an

analytically approach to derive the duffing parameters out of the system parameters.

The last part shows the analytic solution for large orbit oscillations.

substructure. The system is excited by a harmonic force

previous section

damping is *<sup>d</sup>* and *<sup>c</sup>* <sup>=</sup> *<sup>c</sup>*mech <sup>+</sup> *<sup>α</sup>*<sup>2</sup>

and ∇ is the vector gradient defined with

∇ 1 *<sup>r</sup><sup>n</sup>* <sup>=</sup>

and

**5.1. Modeling of the piezomagnetoelastic energy harvesting system**

*mq*¨ <sup>+</sup> *dq*˙ <sup>+</sup> *cq* <sup>−</sup> *<sup>α</sup>*

*RQ*˙ <sup>P</sup> + 1 *C*P

*<sup>U</sup>*mag <sup>=</sup> *<sup>μ</sup>*<sup>0</sup>

differential equations the nonlinear magnetic force leads to

⎡ ⎣

*∂*/*∂x ∂*/*∂y ∂*/*∂z*

⎤ ⎦ 1 *<sup>r</sup><sup>n</sup>* <sup>=</sup> <sup>−</sup> *<sup>n</sup>*

This section is organized as followed. The mechanical EOM is derived for the piezomagnetoelastic energy harvesting system. In the following the duffing parameters are derived with respect to the system and input parameters and the system dynamic is discussed.

Figure 11 gives a schematic view of the piezomagnetoelastic system. The energy harvester consist of a cantilever with two piezoelectric patches mounted on each side of an inactive

where *F*ˆ is the amplitude of the excitation and *ω* is the excitation frequency. A magnetic tip mass is attached to the free end of the beam. Another permanent magnet is stationary mounted near the free end. The magnets are oppositely poled so they exhibit a repulsive force. The nonlinear magnetic force leads to two stable equilibrium positions. Figure 11 shows both symmetric stable equilibrium positions. The tip displacement is given with *q* and the magnet spacing is *s*. The coupled mechanical and electrical differential equations are derived in the

> *C*P *Q*<sup>P</sup> +

> > *<sup>Q</sup>*<sup>P</sup> <sup>−</sup> *<sup>α</sup> C*P

> > > <sup>∇</sup> <sup>μ</sup>A**r**AB �**r**AB�<sup>3</sup> 2

> > > > *rn*+<sup>1</sup>

⎡ ⎣ *x*/*r y*/*r z*/*r*

⎤

<sup>⎦</sup> <sup>=</sup> <sup>−</sup> *<sup>n</sup>***<sup>r</sup>**

Equation 43 and 44 are similar to 19 where *m* = *m*mech is the modal mass. The modal

4*π*

where *μ*<sup>0</sup> is the permeability of free space, μ*<sup>A</sup>* and μ*<sup>B</sup>* are the magnetic dipole moment vectors. **r***AB* is the vector from the source of magnet B to magnet A. �·�<sup>2</sup> is the EUKLIDEAN norm

*dU*mag

*<sup>C</sup>*<sup>p</sup> *<sup>κ</sup>*<sup>2</sup> is the total mechanical stiffness. Additionally to the linear

*F*(*t*) = *F*ˆ sin(*ωt*) (42)

*dq* <sup>=</sup> *<sup>F</sup>*(*t*) (43)

*q* = 0. (44)

μ<sup>B</sup> (45)

*<sup>r</sup>n*+<sup>2</sup> . (46)

$$\mu\_A = M\_\text{A} V\_\text{A} \begin{bmatrix} \cos \phi \\ 0 \\ \sin \phi \end{bmatrix} \tag{47a}$$

$$
\mu\_B = M\_\text{B} V\_\text{B} \begin{bmatrix} -1 \\ 0 \\ 0 \end{bmatrix} \text{ \AA} \tag{47b}
$$

where *M*<sup>A</sup> and *M*<sup>B</sup> represents the vector sum of all microscopic magnetic moments within a ferromagnetic material and *V*<sup>A</sup> and *V*<sup>B</sup> are the volumes of the magnets. Details of the force between magnetic dipoles can be found in [16]. *φ* is the rotation angle at the magnet A.

The vector from the source of magnet B to magnet A is

$$\mathbf{r}\_{AB} = \begin{bmatrix} -\left(s + \frac{l\_\mathbf{A}}{2} + \frac{l\_\mathbf{B}}{2} + \left(l\_\mathbf{P} + \frac{l\_\mathbf{A}}{2}\right)(1 - \cos\phi)\right) \\\ 0 \\\ q \end{bmatrix} \tag{48}$$

where *l*<sup>A</sup> and *l*<sup>B</sup> are the length of the magnets and *l*<sup>P</sup> is the length of the beam in *x*-direction. Figure 12 a) presents the potential energy

$$\mathcal{U} = \mathcal{U}\_{\text{mech}} + \mathcal{U}\_{\text{mag}} = \frac{1}{2} c \eta^2 + \frac{\mu\_0}{4\pi} \left( \frac{\mu\_A \cdot \mu\_B}{r\_{AB}^3} - 3 \frac{(\mu\_A \cdot \mathbf{r}\_{AB}) \left(\mu\_B \cdot \mathbf{r}\_{AB}\right)}{r\_{AB}^5} \right) \tag{49}$$

where *U*mech is the mechanical potential energy and *c* is the equivalent stiffness of the beam. *U* is normalized to the potential energy *U*<sup>0</sup> for a magnet distance of *s* = 0.79*s*0. The tip displacement *q* is normalized to the tip displacement *q*<sup>0</sup> for *s* = 0. The derivation of the mechanical potential energy expression is shown in [15] and is proportional to *q*<sup>2</sup> for the first bending mode. *U*mag is given in Equation 45. The Figure shows the potential energy for different magnet distances with respect to *s*<sup>0</sup> which is the critical magnet distance where the potential energy change from two to one stable equilibrium position. The nonlinear magnetic

**Figure 12.** Potential energy and restoring force

force leads to an energy hump at zero tip displacement. A sharp peak results for a small spacing.

The system exhibits two stable equilibrium positions which are the local minimum positions in the potential energy and one unstable equilibrium for zero displacement (*s* < *s*0). The hump disappears for large *s* and there is only one stable equilibrium for zero displacement (*s* > *s*0).

The normalized restoring force

$$F\_{\text{res}} = \frac{d\mathcal{U}}{dq} \tag{50}$$

is additionally presented in Figure 12 b). *Fres* is normalized to the restoring force for a magnet spacing of *s* = 0.79*s*0. Equation 50 is a nonlinear function of the system parameters in particular a function of *s*.

The benefit of the magnet force is the nonlinearity in the system response. Due to the restoring force the system exhibits overhanging resonance curves which strongly depends on the parameters. For *s* < *s*<sup>0</sup> and low excitation energy that the system only oscillates around one equilibrium the EOM has a hardening stiffness so the resonance curve overhang to the left. A softening response is given for *s* > *s*<sup>0</sup> with a distorted peak to the right. Specially for a hardening stiffness the system exhibit two equilibrium positions and the system bounces between both positions if the energy of the excitation is high enough. These large orbit deflections generate the most energy and are most important for designing the system setup.

#### **5.2. Approximation with the Duffing oscillator**

The potential energy shown in Figure 12 a) is approximated as fourth order polynomial function

$$\mathcal{U}\mathcal{U}\_{\text{Duff}} = \frac{1}{4}\alpha\eta^4 + \frac{1}{2}\beta\eta^2 \tag{51}$$

in a bunch of different research activities. The nonlinear restoring force shown in Figure 12 b) is than approximated as the Duffing equation

$$F\_{\rm res,Duff} = \frac{d\mathcal{U}\_{\rm Duff}}{dq} = \alpha q^3 + \beta q \tag{52}$$

with the Duffing parameters *α* and *β*. The approach as Duffing equation is not only limited to piezomagnetoelastic energy harvesting techniques. [5] used this equation to model an electrostatic energy harvesting system and [8] approximate an electromagnetic energy generator. In the following *α* is the Duffing parameter before the cubic term instead of the coupling factor in Equation 43 to be conform to the most publications concerning the duffing equation. The uncoupled equation of motion (EOM) for the piezomagnetoelastic system in terms of the duffing oscillator is written as

$$
\alpha m \ddot{q} + d\dot{q} + \alpha q^3 + \beta q = F(t). \tag{53}
$$

The Duffing parameters are complicated functions of the beam, the piezo, the magnet parameters and in particular the magnet distance. It can be recognized in Figure 12 b) that the cubic part *αq*<sup>3</sup> is always positive for repulsive magnets because the function is always monotonically increasing for large *q* values. Only the linear part *βq* can be either positive or negative with respect to the magnet distance. For *s* < *s*<sup>0</sup> the parameter *β* is negative and the system exhibits two stable and one unstable equilibrium position. For *s* ≥ *s*<sup>0</sup> the system has only one stable equilibrium position and *β* is positive. *s*<sup>0</sup> is the critical magnet spacing.

The nonlinear restoring force *F*res shown in Figure 12 b) can be approximated as a third order Taylor serious given with

$$p(q) = k\_1 q^3 + k\_2 q^2 + k\_3 q + k\_4 \tag{54}$$

where *k*1−<sup>4</sup> are constants derived from the system parameters. By comparing Equation 54 and Equation 52 note that *k*<sup>1</sup> = *α* and *k*<sup>3</sup> = *β*. To calculate the constant parameters in Equation 54 it takes four constrains.

1. The restoring force is point symmetric:

18 Will-be-set-by-IN-TECH

*U*/*U*0

spacing.

(*s* > *s*0).

function

0

1/3

2/3

1

4/3

5/3

*q*/*q*<sup>0</sup> *q*/*q*<sup>0</sup>

force leads to an energy hump at zero tip displacement. A sharp peak results for a small

The system exhibits two stable equilibrium positions which are the local minimum positions in the potential energy and one unstable equilibrium for zero displacement (*s* < *s*0). The hump disappears for large *s* and there is only one stable equilibrium for zero displacement

*<sup>F</sup>*res <sup>=</sup> *dU*

is additionally presented in Figure 12 b). *Fres* is normalized to the restoring force for a magnet spacing of *s* = 0.79*s*0. Equation 50 is a nonlinear function of the system parameters in

The benefit of the magnet force is the nonlinearity in the system response. Due to the restoring force the system exhibits overhanging resonance curves which strongly depends on the parameters. For *s* < *s*<sup>0</sup> and low excitation energy that the system only oscillates around one equilibrium the EOM has a hardening stiffness so the resonance curve overhang to the left. A softening response is given for *s* > *s*<sup>0</sup> with a distorted peak to the right. Specially for a hardening stiffness the system exhibit two equilibrium positions and the system bounces between both positions if the energy of the excitation is high enough. These large orbit deflections generate the most energy and are most important for designing the system setup.

The potential energy shown in Figure 12 a) is approximated as fourth order polynomial

4 *αq*<sup>4</sup> + 1

*<sup>U</sup>*Duff <sup>=</sup> <sup>1</sup>



0

1

2

3


**5.2. Approximation with the Duffing oscillator**

**Figure 12.** Potential energy and restoring force

The normalized restoring force

particular a function of *s*.

a) b)

*F*res/*F*res0

0.66*s*<sup>0</sup> 0.79*s*<sup>0</sup> 1.00*s*<sup>0</sup> 1.31*s*<sup>0</sup>


*dq* (50)

<sup>2</sup> *<sup>β</sup>q*<sup>2</sup> (51)

$$p(0) = 0,\tag{55}$$

$$\frac{d^2p(0)}{dq^2} = 0.\tag{56}$$

2. The Taylor series must exhibit the same equilibrium positions:

$$p(q\_{\text{eq}}) = 0.\tag{57}$$

3. The oscillation must be suitable for small magnet distances *s* around one equilibrium position

$$\frac{dp(q\_{\rm eq})}{dq} = a,\tag{58}$$

**Figure 13.** Equilibrium positions and *β* with respect to the magnet distance *s*.

where *q*eq is the equilibrium position and *a* is the slope at *q*eq. *a* and *q*eq can be calculated from the original system. The solution for *k*1−<sup>4</sup> is

$$k\_1 = a = \frac{a}{2q\_{\rm eq}^2},\tag{59a}$$

$$k\_2 = 0,\tag{59b}$$

$$k\_3 = \beta = -\frac{a}{2},\tag{59c}$$

$$k\_4 = 0.\tag{59d}$$

Equation 53 becomes

$$m\ddot{\eta} + d\dot{\eta} + \frac{a}{2q\_{\text{eq}}^2} \eta^3 - \frac{a}{2} \eta = F(t). \tag{60}$$

Equation 60 is usable if the potential energy exhibits two equilibrium positions. If the magnet distance is larger than *s*<sup>0</sup> the Duffing parameters *α* and *β* are simply the liner and the cubic part of the third order Taylor series of Equation 50 and *α* and *β* are positive.

This behavior can be recognized in Figure 13 a) and b). The graphs are calculated by using the potential energy of two point dipols in Equation 49. Figure a) shows the equilibrium position over the magnet distance. The equilibrium position *q*eq is normalized to the known *q*0. The magnet distance is normalized to the critical magnet distance *s*0. Note that the distance between the two stable *q*eq becomes smaller for very small *s* because of the rotation angle *φ* of magnet A. The Figure 13 b) gives the modification of the Duffing parameter *β* with respect to the normalized known magnet distance. *β* is normalized to *β*<sup>0</sup> the linear Duffing parameter

**Figure 14.** Duffing approximation of the restoring force and displacement

for *s* = 0. For *s* < *s*<sup>0</sup> the system exhibit a negative linear restoring force due to the negative parameter *β*. *β* is positive for *s* > *s*0.

Figure 14 shows the approximation of Equation 50 with Equation 52 and the Duffing parameter known from Equation 59. The pictures a) show the restoring force derived from the force between two point dipoles and the Duffing equation for different magnet distances. The pictures b) gives the corresponding time variant tip displacement. It can be recognized that the Duffing equation approximate the restoring force very well for the different ranges of magnet spacing. The Duffing equation shows the hardening response for small magnet distances (*s* = 0.38*s*0) and it approximates the influence of the different attractors (*s* = 0.91*s*0). For *s* > *s*<sup>0</sup> the approximation is also very good.

#### **5.3. Solution for large orbit oscillations**

20 Will-be-set-by-IN-TECH

*s*/*s*<sup>0</sup> *s*/*s*<sup>0</sup>

where *q*eq is the equilibrium position and *a* is the slope at *q*eq. *a* and *q*eq can be calculated from

*<sup>k</sup>*<sup>1</sup> <sup>=</sup> *<sup>α</sup>* <sup>=</sup> *<sup>a</sup>*

*<sup>k</sup>*<sup>3</sup> <sup>=</sup> *<sup>β</sup>* <sup>=</sup> <sup>−</sup> *<sup>a</sup>*

*a* 2*q*<sup>2</sup> eq

Equation 60 is usable if the potential energy exhibits two equilibrium positions. If the magnet distance is larger than *s*<sup>0</sup> the Duffing parameters *α* and *β* are simply the liner and the cubic

This behavior can be recognized in Figure 13 a) and b). The graphs are calculated by using the potential energy of two point dipols in Equation 49. Figure a) shows the equilibrium position over the magnet distance. The equilibrium position *q*eq is normalized to the known *q*0. The magnet distance is normalized to the critical magnet distance *s*0. Note that the distance between the two stable *q*eq becomes smaller for very small *s* because of the rotation angle *φ* of magnet A. The Figure 13 b) gives the modification of the Duffing parameter *β* with respect to the normalized known magnet distance. *β* is normalized to *β*<sup>0</sup> the linear Duffing parameter

2*q*<sup>2</sup> eq

2

*<sup>q</sup>*<sup>3</sup> <sup>−</sup> *<sup>a</sup>* 2





0 0.25 0.5 0.75 1 1.25 1.5 -1

, (59a)

, (59c)

*q* = *F*(*t*). (60)

*k*<sup>2</sup> = 0, (59b)

*k*<sup>4</sup> = 0. (59d)

0.25 0.5 0.75 1 1.25 1.5

the original system. The solution for *k*1−<sup>4</sup> is

**Figure 13.** Equilibrium positions and *β* with respect to the magnet distance *s*.

*mq*¨ + *dq*˙ +

part of the third order Taylor series of Equation 50 and *α* and *β* are positive.


*β*/*β*0

y11x11

Equation 53 becomes



0

0.5

1

1.5

*q*/*q*0

In Section 5.1 and Section 5.2 the piezomagnetoelastic energy harvesting system is presented and an approximation with the Duffing oscillator is given. In the following Equation 60 is solved for large orbit oscillations so that system bounces between both symmetric equilibrium positions or around the only one for *s* > *s*0. The EOM can be written as

$$
\ddot{q} + 2D\omega\_0 \dot{q} + \epsilon q^3 + \text{sgn}(\beta)\omega\_0^2 q = f\_0 \cos(\omega t) \tag{61}
$$

where sgn(*β*) is either positive or negative for *β* > 0 or *β* < 0 with

$$D = \frac{d}{2\omega\_0 m'} \tag{62a}$$

$$
\omega\_0^2 = \frac{|\beta|}{m},
\tag{62b}
$$

$$
\epsilon = \frac{\alpha}{m'} \tag{62c}
$$

$$f\_0 = \frac{\hat{F}}{m}.\tag{62d}$$

The Equation 61 can be solved by applying the harmonic balance method. The harmonic balance method is well known and details can be found in [12]. The amplitude response is assumed as harmonic with the frequency of the excitation

$$
\mathfrak{q} = \mathfrak{q} \cos(\omega t - \mathfrak{q}) \tag{63}
$$

where *q*ˆ is the amplitude and *ϕ* is the phase of the tip displacement. Insert Equation 63 in Equation 61 and only consider terms with the excitation frequency leads to

$$\sqrt{\left[\left(\text{sgn}(\beta)\omega\_0^2 - \omega^2\right)\hat{\eta} + \frac{3}{4}\epsilon\hat{\eta}^3\right]^2 + 4D^2\omega\_0^2\omega^2\hat{\eta}^2}\cos\left(\omega t - \dots \right)$$

$$\left(\dots - \hat{\eta} + \arctan\left(\frac{2D\omega\_0\omega\hat{\eta}}{(\text{sgn}(\beta)\omega\_0^2 - \omega^2)\hat{\eta} + \frac{3}{4}\epsilon\hat{\eta}^3}\right)\right) = f\_0\cos(\omega t). \tag{64}$$

The Equation 61 is valid if the amplitude and the phase in Equation 63 solves the equations. The solution gives the frequency response with respect to the amplitude *q*ˆ

$$
\omega\_{1/2}^2 = \text{sgn}(\beta)\omega\_0^2 - 2D\omega\_0^2 + \frac{3}{4}\varepsilon\dot{\eta}^2 \pm \sqrt{\frac{f\_0^2}{\dot{\eta}^2} + 4D^2\omega\_0^2 \left(D^2\omega\_0^2 - \text{sgn}(\beta)\omega\_0^2 - \frac{3}{4}\varepsilon\dot{\eta}^2\right)}.\tag{65}$$

Figure 15 shows the analytical amplitude response given in Equation 65 and the numerical solution of the Duffing oscillator. The graph shows the solution for three different magnet distances. The frequency was slowly increased so the system remains a steady state response. One can recognize that the harmonic balance is well suited if the Duffing equation has a positive linear restoring force (*β* > 0). If the Duffing oscillator exhibits a negative restoring force and the system excitation delivers enough energy that the energy harvester bounces between both stable equilibrium positions than the harmonic balance predicts the influence of

**Figure 15.** Numerical and analytical results for the large orbit duffing equation

the attractors till the first jump. The system behavior after the first jump can not be predicted with this harmonic approximation.

### **6. Conclusions**

22 Will-be-set-by-IN-TECH

solved for large orbit oscillations so that system bounces between both symmetric equilibrium

*<sup>D</sup>* <sup>=</sup> *<sup>d</sup>*

*ω*2 <sup>0</sup> <sup>=</sup> <sup>|</sup>*β*<sup>|</sup>

*�* <sup>=</sup> *<sup>α</sup>*

*<sup>f</sup>*<sup>0</sup> <sup>=</sup> *<sup>F</sup>*<sup>ˆ</sup>

The Equation 61 can be solved by applying the harmonic balance method. The harmonic balance method is well known and details can be found in [12]. The amplitude response is

where *q*ˆ is the amplitude and *ϕ* is the phase of the tip displacement. Insert Equation 63 in

2*Dω*0*ωq*ˆ

The Equation 61 is valid if the amplitude and the phase in Equation 63 solves the equations.

Figure 15 shows the analytical amplitude response given in Equation 65 and the numerical solution of the Duffing oscillator. The graph shows the solution for three different magnet distances. The frequency was slowly increased so the system remains a steady state response. One can recognize that the harmonic balance is well suited if the Duffing equation has a positive linear restoring force (*β* > 0). If the Duffing oscillator exhibits a negative restoring force and the system excitation delivers enough energy that the energy harvester bounces between both stable equilibrium positions than the harmonic balance predicts the influence of

<sup>0</sup> <sup>−</sup> *<sup>ω</sup>*2)*q*<sup>ˆ</sup> <sup>+</sup> <sup>3</sup>

*<sup>q</sup>*ˆ<sup>2</sup> <sup>+</sup> <sup>4</sup>*D*2*ω*<sup>2</sup>

0 *D*2*ω*<sup>2</sup>

+ 4*D*2*ω*<sup>2</sup>

<sup>4</sup> *�q*ˆ<sup>3</sup>

Equation 61 and only consider terms with the excitation frequency leads to

(sgn(*β*)*ω*<sup>2</sup>

The solution gives the frequency response with respect to the amplitude *q*ˆ

<sup>0</sup> − *<sup>ω</sup>*<sup>2</sup> *q*ˆ + 3 4 *�q*ˆ<sup>3</sup> 2

<sup>0</sup>*q* = *f*<sup>0</sup> cos(*ωt*) (61)

<sup>2</sup>*ω*0*m*, (62a)

*<sup>m</sup>* , (62b)

*<sup>m</sup>*, (62c)

*<sup>m</sup>*. (62d)

*q* = *qcos* ˆ (*ωt* − *ϕ*) (63)

<sup>0</sup>*ω*2*q*ˆ<sup>2</sup> cos (*ω<sup>t</sup>* − ...

<sup>0</sup> <sup>−</sup> sgn(*β*)*ω*<sup>2</sup>

= *f*<sup>0</sup> cos(*ωt*). (64)

<sup>0</sup> <sup>−</sup> <sup>3</sup> 4 *�q*ˆ<sup>2</sup> 

. (65)

positions or around the only one for *s* > *s*0. The EOM can be written as

where sgn(*β*) is either positive or negative for *β* > 0 or *β* < 0 with

assumed as harmonic with the frequency of the excitation

sgn(*β*)*ω*<sup>2</sup>

... − *ϕ* + arctan

<sup>0</sup> <sup>−</sup> <sup>2</sup>*Dω*<sup>2</sup>

<sup>0</sup> + 3 4 *�q*ˆ 2 ± *f* 2 0

 

*ω*2

1/2 <sup>=</sup> sgn(*β*)*ω*<sup>2</sup>

*q*¨ + 2*Dω*0*q*˙ + *�q*<sup>3</sup> + sgn(*β*)*ω*<sup>2</sup>

This chapter presents different techniques to enlarge the frequency bandwidth of piezoelectric energy harvester. A precise modeling of piezomechanical structures is given, and the linear harvesting system is given as the reference. In detail a nonlinear switching SSHI circuit, nonlinear magnet forces and an array configuration of several bimorphs are discussed. For all cases, appropriate modeling techniques are presented that allow an efficient yet precise analysis. The nonlinear techniques alter the system dynamics especially for off-resonance vibrations. The magnet forces generate a bistable system, in which the bimorph oscillates between both equilibria. The SSHI circuit increase the energy conversion rate by producing a rectangular shaped voltage signal, while the array configuration tunes the individual bimorphs to slightly different frequencies, so that the energy conversion is distributed to a broader frequency range.

### **Author details**

Marcus Neubauer, Jens Twiefel, Henrik Westermann and Jörg Wallaschek *Leibniz University Hannover, Institute of Dynamics and Vibration Research, Appelstrasse 11, 30167 Hannover, Germany*

### **7. References**

