**Analysis of Energy Harvesting Using Frequency Up-Conversion by Analytic Approximations**

Adam Wickenheiser

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52075

### **1. Introduction**

Energy harvesting is the process of capturing energy existing in the environment of a wireless device in order to power its electronics without the need to manually recharge the battery. By replenishing on-board energy storage autonomously, the need to recharge or replace the battery can be eliminated altogether, enabling devices to be placed in difficult-toreach areas. Vibration-based energy harvesting in particular has garnered much attention due to the ubiquity of vibrational energy in the environment, especially around machinery and vehicles (Roundy et al., 2003). Although several methods of electromechanical transduction from vibrations have been investigated, this chapter focuses on utilizing the piezoelectric effect.

Piezoelectric energy harvesters convert mechanical energy into electrical through the strain induced in the material by inertial loads. Typically, piezoelectric material is mounted on a structure that oscillates due to excitation of the host structure to which it is affixed. If a natural frequency of the structure is matched to the predominant excitation frequency, resonance occurs, where large strains in the piezoelectric material are induced by relatively small excitations. In order to take advantage of resonance, the natural frequency of the device must be matched to the predominant frequency component of the base excitation (Anderson & Wickenheiser, 2012). For many potential applications, ambient vibrations are low frequency, requiring longer length scales or a larger mass to match the resonance frequency to the excitation frequency (Roundy et al., 2003; Wickenheiser & Garcia, 2010a; Wickenheiser, 2011). In order to shrink the size and mass of these devices while reducing their natural frequencies, a variety of techniques have been investigated. Varying the cross sections along the beam length (Dietl & Garcia, 2010; Reissman et al., 2007; Roundy et al., 2005) and the ratio of tip mass to beam mass (Dietl & Garcia, 2010; Wickenheiser, 2011) have been shown to improve the electromechanical coupling (a factor in the energy conversion

© 2012 Wickenheiser, 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 Wickenheiser, 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.

rate) over a uniform cantilever beam design. Multi-beam structures can reduce the overall dimensions of the design by folding it in on itself while retaining a similar natural frequency to the original, straight configuration (Karami & Inman, 2011; Erturk et al., 2009); however, this requires a more complex analysis of the natural frequencies and mode shapes (Wickenheiser, 2012).

In resonant designs, minimizing the mechanical damping in the system enhances the power harvesting performance (Lefeuvre et al., 2005; Shu and Lien, 2006; Wickenheiser & Garcia, 2010c). Unfortunately, lightly damped systems are the most sensitive to discrepancies between the resonance and the driving frequencies. Several methods have been analyzed for tuning the stiffness of the vibrating beam in order to match a slowly varying base excitation frequency. (Challa et al., 2008) and (Reissman et al., 2009) have considered placing one or more magnets to either side of the tip mass to create either an attractive or repulsive force that changes the effective stiffness of the beam, thus allowing the natural frequencies to be adjusted to match the base excitation frequency. Similarly, (Mann & Sims, 2009) harvest energy from a magnet levitating in a cavity between two magnets; varying the spacing of the magnets changes the natural frequency of the levitation. (Leland & Wright, 2006) have proposed tuning the natural frequencies of the beam by applying an axial load; however, this technique has been found to increase the apparent mechanical damping in the structure. A similar concept has been developed for adjusting the pre-tension in extensional mode resonators (Morris et al., 2008). These methods can be considered "quasi-static" because the rate at which the natural frequencies can be tuned is often much slower than the vibration frequency. Thus, these methods are ideal if the base excitation is an approximately stationary process with frequencies concentrated in a narrow band.

The off-resonant response of these systems can be enhanced by destabilizing the relaxed state of the beam. A bi-stable cantilever beam can be created by adding a repelling magnet beyond a magnetic tip mass or by adding attracting magnets on either side. In this situation, the beam can be induced to jump from one well of attraction to the other either periodically, quasi-periodically, or chaotically, depending on the amplitude and frequency of the base excitation. Bi-stability can be realized with a "snap-through" mechanism, in which the mass moves perpendicularly to the elastic axis (Ramlan et al., 2010), using the aforementioned beam and magnetic set-up first analyzed by (Moon, 1978), and using an inherently bi-stable composite plate (Arrieta, 2010). This technique is suited for strong excitations that are able to drive the beam between the two potential wells; however, for low excitation levels the performance converges towards the linear system unless a perturbation is added to "kick" the system into the other well.

In this chapter, a technique known as frequency up-conversion is employed to generate strong off-resonant responses. This technique is based off a repetition of the bi-stable system to create a sequence of potential wells; the transition between them induces a "pluck" followed by a free response at the fundamental frequency. A similar concept has been pursued by Tieck et al. (2006), consisting of a rack placed transversely near the tip of the beam that would periodically pluck the beam as it vibrated. Other concepts utilizing mechanical rectification have been proposed for harvesting energy from buoy motion (Murray and Rastegar, 2009) and low-frequency, rotating machinery (Rastegar and Murray, 2008).

In the following sections, the equations of motion (EOMs) are derived for a uniform beam with magnetic tip mass under periodic base excitation. The eigenvalue problem for this design is then solved for the natural frequencies and mode shapes. The modal expansion is reduced to a single mode (the fundamental) in order to derive an approximate model for low frequencies well below the fundamental frequency. A simplification is derived based on neglecting the base excitation; this simplification leads to a model of the beam's excitation in terms of a sequence of plucks followed by free vibrations. A few simple case studies are presented to highlight the accuracy of this approximate model.

### **2. Derivation of electromechanical EOMs**

212 Small-Scale Energy Harvesting

(Wickenheiser, 2012).

the system into the other well.

rate) over a uniform cantilever beam design. Multi-beam structures can reduce the overall dimensions of the design by folding it in on itself while retaining a similar natural frequency to the original, straight configuration (Karami & Inman, 2011; Erturk et al., 2009); however, this requires a more complex analysis of the natural frequencies and mode shapes

In resonant designs, minimizing the mechanical damping in the system enhances the power harvesting performance (Lefeuvre et al., 2005; Shu and Lien, 2006; Wickenheiser & Garcia, 2010c). Unfortunately, lightly damped systems are the most sensitive to discrepancies between the resonance and the driving frequencies. Several methods have been analyzed for tuning the stiffness of the vibrating beam in order to match a slowly varying base excitation frequency. (Challa et al., 2008) and (Reissman et al., 2009) have considered placing one or more magnets to either side of the tip mass to create either an attractive or repulsive force that changes the effective stiffness of the beam, thus allowing the natural frequencies to be adjusted to match the base excitation frequency. Similarly, (Mann & Sims, 2009) harvest energy from a magnet levitating in a cavity between two magnets; varying the spacing of the magnets changes the natural frequency of the levitation. (Leland & Wright, 2006) have proposed tuning the natural frequencies of the beam by applying an axial load; however, this technique has been found to increase the apparent mechanical damping in the structure. A similar concept has been developed for adjusting the pre-tension in extensional mode resonators (Morris et al., 2008). These methods can be considered "quasi-static" because the rate at which the natural frequencies can be tuned is often much slower than the vibration frequency. Thus, these methods are ideal if the base excitation is an approximately

The off-resonant response of these systems can be enhanced by destabilizing the relaxed state of the beam. A bi-stable cantilever beam can be created by adding a repelling magnet beyond a magnetic tip mass or by adding attracting magnets on either side. In this situation, the beam can be induced to jump from one well of attraction to the other either periodically, quasi-periodically, or chaotically, depending on the amplitude and frequency of the base excitation. Bi-stability can be realized with a "snap-through" mechanism, in which the mass moves perpendicularly to the elastic axis (Ramlan et al., 2010), using the aforementioned beam and magnetic set-up first analyzed by (Moon, 1978), and using an inherently bi-stable composite plate (Arrieta, 2010). This technique is suited for strong excitations that are able to drive the beam between the two potential wells; however, for low excitation levels the performance converges towards the linear system unless a perturbation is added to "kick"

In this chapter, a technique known as frequency up-conversion is employed to generate strong off-resonant responses. This technique is based off a repetition of the bi-stable system to create a sequence of potential wells; the transition between them induces a "pluck" followed by a free response at the fundamental frequency. A similar concept has been pursued by Tieck et al. (2006), consisting of a rack placed transversely near the tip of the

stationary process with frequencies concentrated in a narrow band.

The layout of the piezoelectric, vibration-based energy harvester and the nearby magnetized structures used for mechanical rectification is presented in Fig. 1. For this study, a bimorph configuration is considered, in which piezoelectric layers are bonded to both sides of an inactive substructure. Other configurations, such as the unimorph, can be modeled with few modifications, as pointed out below. Electrodes are assumed to cover the upper and lower surfaces of each layer, and they are wired together in the "parallel" configuration, as depicted. In this configuration, the voltage drop across each layer is assumed to be the same, and the charge displaced by each layer is additive, much like capacitors in parallel. Because the piezoelectric layers are on opposite sides of the neutral axis, each layer experiences opposite strains; hence, they must be poled in the same direction to avoid charge cancellation. It is assumed that the electrodes and connecting wires have negligible resistance and that the resistivity of the piezoelectric material is significantly higher than that of the external circuitry; thus, the transducer impedance is assumed to be purely reactive.

A tip mass is connected to the free end of the beam, and its center of mass is displaced axially from the connection point by a distance *<sup>t</sup> d* . Tip masses are traditionally added to decrease the natural frequency of the beam and to increase the strain due to base excitation. In this situation, the tip mass is considered to be a permanent magnet and is attracted to ferromagnetic structures placed in a line parallel to the y-axis with spacing *md* between them. These structures are not magnets themselves; rather, they become magnetized due to the proximity of the magnetic tip mass. Thus, in this device, the tip mass is an active component of the excitation while fulfilling its passive role as just described.

In the following section, the EOMs for the electromechanical system presented in Fig. 1 are derived through force, moment, and charge balances while adopting the Euler-Bernoulli beam assumptions and linearized material constitutive equations. The approach taken herein is based on force and moment balances and is a generalization of the treatments by (Erturk & Inman, 2008; Söderkvist, 1990; Wickenheiser & Garcia, 2010c). It is assumed that each beam segment is uniform in cross section and material properties. Furthermore, the standard Euler-Bernoulli beam assumptions are adopted, including negligible rotary inertia and shear deformation (Inman, 2007). Subsequently, a solution consisting of a series of assumed modes is presented, and the EOMs are decoupled into modal dynamics equations. As will be demonstrated, only the first bending mode is excited significantly by the plucking of the magnetic force. Although higher modes can be excited by higher frequency base excitation, this study focuses primarily on base excitation frequencies well below the fundamental resonant frequency.

**Figure 1.** Layout and geometric parameters of cantilevered vibration energy harvester in parallel bimorph configuration with magnetic tip mass

#### **2.1. Electromechanical EOMs**

In this derivation, the states of the electromechanical system are the following: *w xt* , is the relative transverse deflection of the beam with respect to its base, *v t* is the voltage across the energy harvester as seen by the external circuit, and *i t* is the net current flowing into the external circuit. The input to the system is *yt* , the absolute transverse displacement of the base; therefore, *w xt y t* , is the absolute transverse deflection of the beam.

**Figure 2.** Free-body diagram of Euler-Bernoulli beam segment

Consider the free-body diagram shown in Fig. 2. Dropping higher order terms, balances of forces in the *y*-direction and moments yield

$$\begin{aligned} \frac{\partial V(\mathbf{x},t)}{\partial \mathbf{x}} + f(\mathbf{x},t) &= \left(\rho A\right)\_{\mathrm{eff}} \frac{\partial^2 w(\mathbf{x},t)}{\partial t^2} \\ \frac{\partial M(\mathbf{x},t)}{\partial \mathbf{x}} &= -V(\mathbf{x},t) \end{aligned} \tag{1}$$

where *V xt* , is the shear force, *Mx t*, is the internal moment generated by mechanical and electrical strain, *<sup>f</sup> x t*, is the externally applied force per unit length, and *eff A* is the mass per unit length (Inman, 2007). For the case of a bimorph beam segment, this term is given by

214 Small-Scale Energy Harvesting

fundamental resonant frequency.

*Rl*

*tv*

bimorph configuration with magnetic tip mass

*ty*

host structure

*ti*

*y*

*x*

**Figure 2.** Free-body diagram of Euler-Bernoulli beam segment

*x*

*M*

forces in the *y*-direction and moments yield

*y*

**2.1. Electromechanical EOMs** 

(Erturk & Inman, 2008; Söderkvist, 1990; Wickenheiser & Garcia, 2010c). It is assumed that each beam segment is uniform in cross section and material properties. Furthermore, the standard Euler-Bernoulli beam assumptions are adopted, including negligible rotary inertia and shear deformation (Inman, 2007). Subsequently, a solution consisting of a series of assumed modes is presented, and the EOMs are decoupled into modal dynamics equations. As will be demonstrated, only the first bending mode is excited significantly by the plucking of the magnetic force. Although higher modes can be excited by higher frequency base excitation, this study focuses primarily on base excitation frequencies well below the

**Figure 1.** Layout and geometric parameters of cantilevered vibration energy harvester in parallel

inactive substructure

*p t*

*L*

piezoelectric layers

the base; therefore, *w xt y t* , is the absolute transverse deflection of the beam.

In this derivation, the states of the electromechanical system are the following: *w xt* , is the relative transverse deflection of the beam with respect to its base, *v t* is the voltage across the energy harvester as seen by the external circuit, and *i t* is the net current flowing into the external circuit. The input to the system is *yt* , the absolute transverse displacement of

*f*

*<sup>V</sup> dx <sup>x</sup>*

*<sup>V</sup> <sup>V</sup>* 

> *dx <sup>x</sup> <sup>M</sup> <sup>M</sup>*

magnetic tip mass

*tt* , *Im*

*t d*

*m d*

(stationary relative to host structure)

ferromagnetic structures

*st*

Consider the free-body diagram shown in Fig. 2. Dropping higher order terms, balances of

*dx*

$$\left(\rho A\right)\_{eff} = \frac{m}{L} = \frac{\rho\_s t\_s bL + 2\rho\_p t\_p bL}{L} = b\left(\rho\_s t\_s + 2\rho\_p t\_p\right) \tag{2}$$

where *m* is the mass of the beam (not counting the tip mass), *L* is its length, *b* is its width, *s* and *st* are the density and thickness of the substrate, and *<sup>p</sup>* and *pt* are the density and thickness of one of the piezoelectric layers, respectively. As can be seen in Eq. (2), if the segment is monolithic, *eff A* is simply the product of the density of the material and the cross-sectional area. The externally applied force per unit length can be written as the sum of the distributed inertial force along the beam and the inertial force of the tip mass – which arise because the non-inertial frame of the base is taken as the reference – and the magnetic force applied at the center of the tip mass:

$$f\left(\mathbf{x},t\right) = -\left(\rho A\right)\_{\mathrm{eff}} \frac{d^2y\left(t\right)}{dt} - m\_t \frac{d^2y\left(t\right)}{dt} \delta\left(\mathbf{x} - L\right) - f\_{\mathrm{mag}}\left(t\right) \delta\left(\mathbf{x} - L\right) \tag{3}$$

where *mt* is the mass of the tip mass, *f mag t* is the magnetic force, and is the Dirac delta function. In this study, the magnetic force is assumed to be purely in the y-direction. Although there is a stiffening effect due to the axial attractive force, it is considered negligible. The negative sign on the magnetic force indicates that it is an attractive force.

The internal bending moment is the net contribution of the stresses in the axial direction in the beam. The stress within the piezoelectric layers is found from the linearized constitutive equations

$$\begin{aligned} T\_1 &= \boldsymbol{c}\_{11}^E \boldsymbol{S}\_1 - \boldsymbol{e}\_{31} \boldsymbol{E}\_3 \\ D\_3 &= \boldsymbol{e}\_{31} \boldsymbol{S}\_1 + \boldsymbol{\varepsilon}\_{33}^S \boldsymbol{E}\_3 \end{aligned} \tag{4}$$

where *T* is stress, *S* is strain, *E* is electric field, *D* is electric displacement, *c* is Young's Modulus, *e* is piezoelectric constant, and is dielectric constant. The subscripts indicate the direction of perturbation; in the cantilever configuration shown in Fig. 1, 1 corresponds to axial and 3 corresponds to transverse. The superscript *<sup>E</sup>* indicates a linearization at constant electric field, and the superscript *<sup>S</sup>* indicates a linearization at constant strain (IEEE, 1987). The use of Eq. (4) assumes the hypothesis of plane stress, which is reasonable since the beams are not directly loaded in the other directions, and small deflections. The stress within the substrate layer is given simply by the linear stress-strain relationship 1 11, 1 *<sup>s</sup> T cS* , where 11,*<sup>s</sup> c* is Young's Modulus of the substrate material in the axial direction. Since deformations are assumed small, the axial strain is the same as the case of pure bending, which is given by 2 2 <sup>1</sup> *S y w xt x* , (Beer & Johnson, 1992), and the transverse electric field is assumed constant and equal to <sup>3</sup> *<sup>p</sup> E vt t* , where *v t* is the voltage across the electrodes, and the top and bottom layer have opposite signs due to the parallel configuration wiring. (This approximation is reasonable given the thinness of the layers.)

Consider the case of a bimorph beam. The bending moment along the length of the beam is

 2 22 11 1 2 22 2 2 2 22 22 2 2 22 11 11, 11 2 2 2 <sup>31</sup> <sup>31</sup> 2 2 , , *s s s p s p s s s s s p s p s s s s p s p s t t t t t t t t t t mag t t t t t E E s t t t t t t t t t <sup>t</sup> p p M x t T bydy T bydy T bydy dyt m dH x L f t dH x L dt w xt c by dy c by dy c by dy x e e bydy bydy t t* 2 <sup>2</sup> <sup>3</sup> <sup>3</sup> <sup>2</sup> 11, 11 2 31 2 , <sup>2</sup> 12 12 2 *t t mag t E p ps s s p eff s p t t mag t vt Hx Hx L dyt m dH x L f t dH x L dt t t tt w xt c b cb t x EI e bt t vt Hx Hx L dyt m dH x L f t dH x dt L* (5)

where *H* is the Heaviside step function. In Eq. (5), the constant multiplying the 2 2 *w xt x* , term is defined as *eff EI* , the effective bending stiffness. (Note that if the beam segment is monolithic, this constant is simply the product of the Young's Modulus and the moment of inertia.) The constant multiplying the *v t* term is defined as , the electromechanical coupling coefficient. Substituting Eq. (5) into Eq. (1) yields

$$\begin{aligned} \left(\rho A\right)\_{\mathrm{eff}} & \frac{\partial^2 w(\mathbf{x}, t)}{\partial t^2} + \left(EI\right)\_{\mathrm{eff}} \frac{\partial^4 w(\mathbf{x}, t)}{\partial \mathbf{x}^4} + \mathcal{B} \left| \frac{d\delta(\mathbf{x})}{dx} - \frac{d\delta(\mathbf{x} - L)}{dx} \right| v(t) = \\ & - \left(\rho A\right)\_{\mathrm{eff}} \frac{d^2 y(t)}{dt} - \left[ m\_t \frac{d^2 y(t)}{dt} + f\_{\mathrm{mag}}\left(t\right) \right] \left[ \delta\left(\mathbf{x} - L\right) + d\_t \frac{d\delta\left(\mathbf{x} - L\right)}{dx} \right] \end{aligned} \tag{6}$$

which is the transverse mechanical EOM for the beam.

The electrical EOM can be found by integrating the electric displacement over the surface of the electrodes, yielding the net charge *q t* (IEEE, 1987):

$$\begin{split} q\left(t\right) &= \iint \limits\_{\text{layer}} D\_3 dA - \iint \limits\_{\text{layer}} D\_3 dA \\ &= b \int\_0^L \left[ \frac{1}{t\_p} \int\_{t\_s/2}^{t\_s/2 + t\_p} -e\_{31} y \frac{\partial^2 w(x, t)}{\partial x^2} dy - \frac{e\_{33}^S}{t\_p} v(t) \right] dx \\ &- b \int\_0^L \left[ \frac{1}{t\_p} \int\_{-t\_s/2 - t\_p}^{-t\_s/2} -e\_{31} y \frac{\partial^2 w(x, t)}{\partial x^2} dy + \frac{e\_{33}^S}{t\_p} v(t) \right] dx \\ &= \underbrace{-e\_{31} b \left(t\_s + t\_p \right)}\_{\mathcal{G}} \frac{\partial w\left(x, t\right)}{\partial x} \bigg|\_{x = L} - \underbrace{2 \frac{e\_{33}^S bL}{t\_p}}\_{\tilde{C}\_0} v\Big| \tag{7} \end{split} \tag{7}$$

where the constant multiplying the *v t* term is defined as *C*<sup>0</sup> , the net clamped capacitance of the segment. Eqs. (6–7) provide a coupled system of equations; these can be solved by relating the voltage *v t* to the charge *q t* through the external electronic interface.

#### **2.2. Modal decoupling**

216 Small-Scale Energy Harvesting

bending, which is given by 2 2

2

2

31

*s p*

2

,

stress within the substrate layer is given simply by the linear stress-strain relationship 1 11, 1 *<sup>s</sup> T cS* , where 11,*<sup>s</sup> c* is Young's Modulus of the substrate material in the axial direction. Since deformations are assumed small, the axial strain is the same as the case of pure

electric field is assumed constant and equal to <sup>3</sup> *<sup>p</sup> E vt t* , where *v t* is the voltage across the electrodes, and the top and bottom layer have opposite signs due to the parallel configuration wiring. (This approximation is reasonable given the thinness of the layers.)

Consider the case of a bimorph beam. The bending moment along the length of the beam is

*s s s p s p s s*

*t t t t E E s t t t t*

 

2 22 22 2

2 22 11 11, 11 2

*c by dy c by dy c by dy*

2 22 11 1 2 22

*t t t t t t t t*

*s s s p s p s s*

*dyt m dH x L f t dH x L dt*

2 2 <sup>31</sup> <sup>31</sup> 2 2

*t t mag t*

*e bt t vt Hx Hx L*

*EI*

12 12 2

 

*s p*

*c b cb t*

*E p ps s*

*dyt m dH x L f t dH x L dt*

*eff*

electromechanical coupling coefficient. Substituting Eq. (5) into Eq. (1) yields

2 4

, ,

2 4

2 2

*eff eff*

*dyt m dH x L f t dH x dt*

*t t mag t*

*e e bydy bydy t t*

<sup>2</sup> <sup>3</sup> <sup>3</sup> <sup>2</sup> 11, 11 2

, <sup>2</sup>

where *H* is the Heaviside step function. In Eq. (5), the constant multiplying the 2 2 *w xt x* , term is defined as *eff EI* , the effective bending stiffness. (Note that if the beam segment is monolithic, this constant is simply the product of the Young's Modulus

and the moment of inertia.) The constant multiplying the *v t* term is defined as

*w xt w xt d x d x L A EI v t t x dx dx*

*dyt dyt dxL A m f t xL d dt dt dx*

 

*t mag t eff*

*t t tt w xt*

*t t mag t*

*M x t T bydy T bydy T bydy*

*s s p s p s*

*t t t t t <sup>t</sup> p p*

 

<sup>1</sup> *S y w xt x* , (Beer & Johnson, 1992), and the transverse

*x*

*L*

 

 

*vt Hx Hx L*

*w xt*

*x*

,

(5)

, the

(6)

2

The system of coupled equations (6–7) can be solved by assuming that the transverse deflection of the beam can be written as a convergent series expansion of eigenfunctions, i.e.

$$\text{cov}\left(\mathbf{x}, t\right) = \sum\_{i=1}^{\infty} \phi\_i\left(\mathbf{x}\right) \eta\_i\left(t\right) \tag{8}$$

where *<sup>i</sup> x* is the *i*th transverse mode shape function, and *<sup>i</sup> t* is the *i*th modal displacement. Given the configuration in Fig. 1 with a tip mass having a nontrivial mass *mt* and moment of inertia *<sup>t</sup> I* , the eigenvalues *<sup>i</sup>* corresponding to the mode shapes must satisfy

$$\left[F\_{\text{eff}} - \frac{m\_t}{\left(\rho A\right)\_{\text{eff}}L} \mathcal{k}F\_{\text{cp}} - \frac{I\_t + m\_t d\_t^2}{\left(\rho A\right)\_{\text{eff}}L^3} \mathcal{k}^3 F\_{\text{cr}} + \frac{I\_t m\_t}{\left(\rho A\right)\_{\text{eff}}^2 L^4} \mathcal{k}^4 F\_{\text{cc}} - \frac{2m\_t d\_t}{\left(\rho A\right)\_{\text{eff}}L^2} \mathcal{k}^2 \sin\lambda \sinh\lambda = 0 \tag{9}$$

where 1 cos cosh *cf F* are the clamped-free, sin cosh cos sinh *cp F* are the clamped-pinned, sin cosh cos sinh *cr F* are the clamped-rolling, and 1 cos cosh *cc F* are the clamped-clamped eigenvalue terms, respectively (Oguamanam, 2003). The mode shape functions are given by

$$\begin{aligned} \phi\_i(\mathbf{x}) &= \cos\left(\boldsymbol{\lambda}\_i \frac{\mathbf{x}}{L}\right) - \cosh\left(\boldsymbol{\lambda}\_i \frac{\mathbf{x}}{L}\right) \\ &+ \frac{\sin\left(\boldsymbol{\lambda}\_i - \sinh\boldsymbol{\lambda}\_i\right) - \frac{m\_t}{\left(\rho\boldsymbol{\rho}\boldsymbol{\lambda}\_{\text{eff}}\right)L} \left[\boldsymbol{\lambda}\_i^2 \frac{d\_t}{L} \left(\sin\boldsymbol{\lambda}\_i + \sinh\boldsymbol{\lambda}\_i\right) - \boldsymbol{\lambda}\_i \left(\cos\boldsymbol{\lambda}\_i - \cosh\boldsymbol{\lambda}\_i\right)\right] \\ &+ \frac{\cos\left(\boldsymbol{\lambda}\_i + \cosh\boldsymbol{\lambda}\_i\right) - \frac{m\_t}{\left(\rho\boldsymbol{\rho}\boldsymbol{\lambda}\_{\text{eff}}\right)L} \left[\boldsymbol{\lambda}\_i^2 \frac{d\_t}{L} \left(\cos\boldsymbol{\lambda}\_i - \cosh\boldsymbol{\lambda}\_i\right) + \boldsymbol{\lambda}\_i \left(\sin\boldsymbol{\lambda}\_i - \sinh\boldsymbol{\lambda}\_i\right)\right] \\ &\times \left[\sin\left(\boldsymbol{\lambda}\_i \frac{\mathbf{x}}{L}\right) - \sinh\left(\boldsymbol{\lambda}\_i \frac{\mathbf{x}}{L}\right)\right] \end{aligned} \tag{10}$$

These functions may be scaled arbitrarily and still be admissible, and in the present case are done to satisfy the following orthogonality condition:

$$\begin{aligned} \left[ \int\_{0}^{L} \left( \rho A \right)\_{\text{eff}} \phi\_{i} \left( \mathbf{x} \right) \phi\_{j} \left( \mathbf{x} \right) d\mathbf{x} + m\_{t} \phi\_{i} \left( \mathbf{L} \right) \phi\_{j} \left( \mathbf{L} \right) + m\_{t} d\_{t} \left[ \frac{d \phi\_{i} \left( \mathbf{x} \right)}{d \mathbf{x}} \phi\_{j} \left( \mathbf{x} \right) + \phi\_{i} \left( \mathbf{x} \right) \frac{d \phi\_{j} \left( \mathbf{x} \right)}{d \mathbf{x}} \right]\_{\mathbf{x} = L} \\ + \left( I\_{t} + m\_{t} d\_{t}^{2} \right) \left[ \frac{d \phi\_{i} \left( \mathbf{x} \right)}{d \mathbf{x}} \frac{d \phi\_{j} \left( \mathbf{x} \right)}{d \mathbf{x}} \right]\_{\mathbf{x} = L} = \delta\_{ij} \end{aligned} \tag{11}$$

where *ij* is the Kronecker delta.

Substituting (8) into (6) and applying the orthogonality condition (11) results in

$$\begin{aligned} \frac{d^2 \eta\_k(t)}{dt^2} + 2\zeta\_k \rho\_k \frac{d \eta\_k(t)}{dt} + \alpha\_k^2 \eta\_k(t) + \Theta\_k \upsilon(t) &= \\ - \left(\rho A\right)\_{\text{eff}} \chi\_k \frac{d^2 \mathcal{y}(t)}{dt^2} - \beta\_k \left[ m\_t \frac{d^2 \mathcal{y}(t)}{dt} + f\_{\text{mag}}(t) \right] \end{aligned} \tag{12}$$

at which point a modal damping term has been inserted. The *k*th modal short-circuit (i.e. *v t* 0 ) natural frequency *<sup>k</sup>* is given by

$$\alpha\_k = \sqrt{\frac{\mathcal{A}\_k^4 \left(EI\right)\_{eff}}{\left(\rho A\right)\_{eff} L^4}}\tag{13}$$

for Euler-Bernoulli beams. Eq. (12) constitutes the EOM for the *k*th transverse vibrational mode. The modal influence coefficients appearing in Eq. (12) are given by

$$\Theta\_k = \mathcal{G} \frac{d\phi\_k\left(\mathbf{x}\right)}{d\mathbf{x}}\bigg|\_{\mathbf{x}=L} \quad \mathcal{V}\_k = \int\_0^L \phi\_k\left(\mathbf{x}\right) d\mathbf{x} \,, \ \mathcal{B}\_k = \phi\_k\left(L\right) + d\_t \frac{d\phi\_k\left(\mathbf{x}\right)}{d\mathbf{x}}\bigg|\_{\mathbf{x}=L} \tag{14}$$

*k* is the modal electromechanical coupling coefficient, *<sup>k</sup>* is the modal influence coefficient of the distributed inertial force along the beam, and *<sup>k</sup>* is the modal influence coefficient of the concentrated force at the tip. A similar decoupling of the electrical EOM (7) yields

218 Small-Scale Energy Harvesting

0

where *ij* 

*x*

cos cosh

 

*ii i*

sin sinh

is the Kronecker delta.

 

2

 

*I md*

*v t* 0 ) natural frequency

*i i*

done to satisfy the following orthogonality condition:

*i j t tt ij*

2

2

*d x d x*

*dx dx*

*x x L L*

*x x*

*L L*

 

> 

 

 

 

> 

*d x d x*

*t vt*

*dx dx*

 

(13)

 

 

> 

> >

 

> 

> > *x L*

(11)

(12)

is the modal influence

is the modal influence

(10)

2

*m d A L L m d A L L*

 

> 

*eff*

*eff*

*x L*

Substituting (8) into (6) and applying the orthogonality condition (11) results in

 

2

*k*

*k k k kk t*

 

mode. The modal influence coefficients appearing in Eq. (12) are given by

 

*k* is the modal electromechanical coupling coefficient, *<sup>k</sup>*

coefficient of the distributed inertial force along the beam, and *<sup>k</sup>*

2 *k k*

*dt dt*

*<sup>k</sup>* is given by

*d t dt*

 

sin sinh sin sinh cos cosh

*t t i i i i ii i i*

 

*t t i i i i ii i i*

 

cos cosh cos cosh sin sinh

These functions may be scaled arbitrarily and still be admissible, and in the present case are

*k k kk k*

 

2

at which point a modal damping term has been inserted. The *k*th modal short-circuit (i.e.

for Euler-Bernoulli beams. Eq. (12) constitutes the EOM for the *k*th transverse vibrational

4

*EI A L*

*eff*

 <sup>0</sup> , , *<sup>L</sup> k k*

*d x d x*

*dx dx*

*x L x L*

 (14)

 

*x dx L d*

4 *k eff*

*k k t mag eff*

*dyt dyt <sup>A</sup> m ft dt dt*

2 2

 

*L i j i j ti j tt j i eff*

*A x x dx m L L m d x x*

2

$$\eta\left(t\right) = \sum\_{i=1}^{\phi} \Theta\_i r\_i\left(t\right) - \mathbb{C}\_0 v\left(t\right) \tag{15}$$

It remains to write the applied magnetic force *f mag t* in terms of the modal coordinates. Due to the assumption of a symmetrical tip mass, this force is applied at its centroid, as shown in Fig. 3. It is further assumed that the ferromagnetic structures are placed uniformly with spacing *md* and that distant structures do not influence the magnetic force (a reasonable assumption given the <sup>3</sup> 1 *r* dependency). Additionally, the rotation of the tip mass is assumed small compared to its absolute translation (base motion + relative deflection), and so its effect on the magnitude of the magnetic force is ignored. Thus, the magnetic force is approximately sinusoidal with wavelength *md* , and so it can be written in the form

$$f\_{m\text{ag}}\left(t\right) = F\_{m\text{ag}}\sin\left[\frac{2\pi}{d\_m}\left(w\left(x\_{m'}, t\right) + y\left(t\right)\right)\right] \tag{16}$$

where *mx* is the *x*-coordinate of the tip mass centroid. The magnitude of this force *mag F* is a complicated function of the material properties and geometry of the tip mass and the ferromagnetic structures that is beyond the scope of this work (see Moon, 1978; Stanton et al., 2010). *mag F* is normalized by the maximum static tip load the beam can support without failing. In this study, a maximum strain of 0.1% is chosen, resulting in a maximum static tip load of

*EI*

*eff*

**Figure 3.** Tip mass coordinates used for locating the centroid in terms of modal coordinates.

The position of the tip mass centroid, shown in Fig. 3, can be written in terms of the modal coordinates:

$$\text{cov}\left(\mathbf{x}\_{m},t\right) \approx \text{w}\left(L,t\right) + d\_{t} \left.\frac{\partial \text{w}\left(\mathbf{x},t\right)}{\partial \mathbf{x}}\right|\_{\mathbf{x}=L} = \sum\_{i=1}^{\sigma} \beta\_{i} \eta\_{i}\left(t\right) \tag{18}$$

#### **3. Estimates of expected power harvested**

#### **3.1. Linear case, frequency domain**

In order to establish a baseline against which the effects of the magnetic force can be compared, in this section the magnetic interactions are not considered, i.e. 0 *mag F* . The most prevalent (e.g. duToit et al., 2005; Lefeuvre et al., 2005; Liao & Sodano, 2008; Shu & Lien, 2006) assumption of constant-amplitude, sinusoidal base excitation forms the basis for analysis of more complex periodic forcing. In this study, it is assumed that the base acceleration is a weakly stationary random process. This general framework includes the special cases of harmonic (single or multiple frequencies), white noise, band-limited noise, and periodic in mean square processes (Anderson & Wickenheiser, 2012; Lin, 1967). The average power dissipated by the load after transients have died out is given by

$$E\left[P\left(t\to\infty\right)\right] = \frac{E\left[\upsilon^2\left(t\to\infty\right)\right]}{R\_l} = \frac{R\_{vv}\left(0\right)}{R\_l} = \frac{1}{R\_l}\int\_{-\infty}^{\infty} \left| H\left(\rho o\right) \right|^2 \Phi\_{AA}\left(\rho o\right) d\rho \tag{19}$$

where *E* is the expectation operator, *Rvv* is the autocorrelation function of the voltage, *H* is the frequency transfer function of the energy harvester between acceleration and voltage, and *AA* is the spectral density of the base acceleration (Lin, 1967). Since the system is assumed to be stable, the power output is seen to approach a weakly stationary process as *t* .

In order to calculate the frequency transfer function, it is assumed that the electrical load can be represented by a resistor with value *Rl* . Eq. (15) can then be rewritten as

$$\frac{dv\left(t\right)}{R\_l} = i\left(t\right) = \frac{dq\left(t\right)}{dt} = \sum\_{i=1}^{n} \Theta\_i \frac{d\eta\_i\left(t\right)}{dt} - \mathcal{C}\_0 \frac{dv\left(t\right)}{dt} \tag{20}$$

Since the system of equations (12,19) is linear, the modal responses *<sup>k</sup> t* and output voltage *v t* are sinusoidal at the driving frequency of the base excitation. The frequency transfer function between base displacement and voltage can be derived from the EOMs, yielding

$$\frac{V(o)}{Y(o)} = \frac{R\_l \sum\_{j=1}^{o} \Theta\_j^2 \frac{\mathrm{i}\left[\left(\rho A\right)\_{\mathrm{eff}} \gamma\_j + m\_t \beta\_j\right] o^3}{o\_j^2 - o^2 + i2\zeta\_j' o o\_j o o}}{i R\_l \mathrm{C}\_0 o + 1 + R\_l \sum\_{j=1}^{o} \Theta\_j^2 \frac{\mathrm{i} o o}{o\_j^2 - o^2 + i2\zeta\_j' o o\_j o}}\tag{21}$$

where *i* 1 and is the base excitation frequency (Wickenheiser & Garcia, 2010b). The frequency transfer function between base acceleration and displacement is simply <sup>2</sup> *Y A* 1 . In this study, however, only the fundamental mode is assumed to be excited; hence, the *j* subscript is dropped and the fundamental natural frequency is written as *n* .

In order to use Eq. (18), an estimate of the spectral density of the base acceleration *AA* is required. An overview of spectral density estimation methods can be found in (Porat, 1994), any of which can provide an approximation of the base excitation signal of the form

$$\frac{d^2y\left(t\right)}{dt^2} = a\left(t\right) \approx \sum\_{k=1}^{N} A\_k \cos\left(\phi\_k t + \phi\_k\right) \tag{22}$$

where the component amplitudes *Ak* , frequencies *<sup>k</sup>* , and phase angles *<sup>k</sup>* are obtained from the spectral density estimate. The number of terms needed *N* is often determined by a user-defined error tolerance used to capture the "quality" of the signal approximation in some optimal manner. The spectral density is then given by

$$\begin{split} \left(\Phi\_{AA}\left(a\right)\right) &= \frac{1}{2\pi} \int\_{-\infty}^{\infty} R\_{AA}\left(t\right) e^{-i\alpha t} dt = \frac{1}{2\pi} \int\_{-\infty}^{\infty} E\left[a\left(t\_0 + t\right) \overline{a\left(t\_0\right)}\right] e^{-i\alpha t} dt \\ &= \frac{1}{2\pi} \int\_{-\infty}^{\infty} \lim\_{T \to \infty} \frac{1}{T} \int\_{0}^{T} \left[a\left(t\_0 + t\right) \overline{a\left(t\_0\right)}\right] dt\_0 e^{-i\alpha t} dt \end{split} \tag{23}$$

Consider first the case 2 *N* , where the base excitation is composed of the sum of two sinusoids. Then, without loss of generality,

$$a\left(t\right) = \underbrace{A\_1 \cos\left(\alpha\_1 t\right)}\_{a\_1(t)} + \underbrace{A\_2 \cos\left(\alpha\_2 t + \varphi\right)}\_{a\_2(t)} = \frac{A\_1}{2}\left(e^{i\alpha\_1 t} + e^{-i\alpha\_1 t}\right) + \frac{A\_2}{2}\left(e^{i\left(\alpha\_2 t + \varphi\right)} + e^{-i\left(\alpha\_2 t + \varphi\right)}\right) \tag{24}$$

Then

220 Small-Scale Energy Harvesting

**3. Estimates of expected power harvested** 

where *E* is the expectation operator, *Rvv*

 

**3.1. Linear case, frequency domain** 

coordinates:

voltage, *H*

yielding

acceleration and voltage, and *AA*

weakly stationary process as *t* .

The position of the tip mass centroid, shown in Fig. 3, can be written in terms of the modal

*w xt w x t w Lt d t x*

, , , *<sup>m</sup> <sup>t</sup> i i*

In order to establish a baseline against which the effects of the magnetic force can be compared, in this section the magnetic interactions are not considered, i.e. 0 *mag F* . The most prevalent (e.g. duToit et al., 2005; Lefeuvre et al., 2005; Liao & Sodano, 2008; Shu & Lien, 2006) assumption of constant-amplitude, sinusoidal base excitation forms the basis for analysis of more complex periodic forcing. In this study, it is assumed that the base acceleration is a weakly stationary random process. This general framework includes the special cases of harmonic (single or multiple frequencies), white noise, band-limited noise, and periodic in mean square processes (Anderson & Wickenheiser, 2012; Lin, 1967). The

> <sup>2</sup> <sup>0</sup> <sup>1</sup> <sup>2</sup> *vv*

> > *l ll*

1967). Since the system is assumed to be stable, the power output is seen to approach a

In order to calculate the frequency transfer function, it is assumed that the electrical load can

*v t dq t d t dv t i t <sup>C</sup> R dt dt dt* 

1

voltage *v t* are sinusoidal at the driving frequency of the base excitation. The frequency transfer function between base displacement and voltage can be derived from the EOMs,

 

*EPt H d R RR*

average power dissipated by the load after transients have died out is given by

*Ev t R*

be represented by a resistor with value *Rl* . Eq. (15) can then be rewritten as

*l i*

Since the system of equations (12,19) is linear, the modal responses *<sup>k</sup>*

2

1

*l j*

*R V i*

1

*<sup>Y</sup> <sup>i</sup> iR C R*

*l lj*

 

1

 

(18)

*AA*

is the autocorrelation function of the

 

*t* and output

(21)

is the spectral density of the base acceleration (Lin,

0

(20)

2

 

*i*

(19)

is the frequency transfer function of the energy harvester between

*i i*

<sup>3</sup>

*j tj eff*

 

2

 

*j j j j*

2 2

*iA m*

 

*j j j j*

2 0 2 2 1

*<sup>i</sup> x L*

$$\begin{split} R\_{AA}\left(t\right) &= R\_{A\_1A\_1}\left(t\right) + R\_{A\_2A\_2}\left(t\right) + \frac{A\_1A\_2}{4} \lim\_{T \to \infty} \frac{1}{T} \int\_0^T \left( e^{\frac{i}{\hbar}\left(a\_2(t\_0+t) + \rho\right)} + e^{-\frac{i}{\hbar}\left(a\_2(t\_0+t) + \rho\right)} \right) \left( e^{-i a\_1t\_0} + e^{i a\_1t\_0} \right) dt\_0 \\ &+ \frac{A\_1A\_2}{4} \lim\_{T \to \infty} \frac{1}{T} \int\_0^T \left( e^{\frac{i}{\hbar}\left(t\_0 + t\right)} + e^{-i a\_1\left(t\_0 + t\right)} \right) \left( e^{-i \left(a\_2t\_0 + \rho\right)} + e^{i \left(a\_2t\_0 + \rho\right)} \right) dt\_0 \end{split} \tag{25}$$

Integrating the first term in the integrand and taking the limit yields

$$\begin{split} \lim\_{T \to \infty} \frac{1}{T} \Big|\_{0}^{T} e^{\frac{i}{\hbar} \alpha\_{2} (t\_{0} + t) + \varphi} \Big|\_{0}^{-i\alpha\_{1} t\_{0}} dt\_{0} &= \lim\_{T \to \infty} \frac{1}{T} \frac{1}{i \left( \alpha\_{2} - \alpha\_{1} \right)} \Big( e^{\frac{i}{\hbar} \left( \alpha\_{2} - \alpha\_{1} \right) T + \alpha\_{2} t + \varphi} \Big) - e^{i \left( \alpha\_{2} t + \varphi \right)} \\ &\le \lim\_{T \to \infty} \frac{1}{T} \frac{1}{\left| \alpha\_{2} - \alpha\_{1} \right|} \Big( 1 + 1 \Big) = 0 \end{split} \tag{26}$$

Each of the other integrated terms also averages out to 0 in the long run; hence,

$$R\_{AA}\left(t\right) = R\_{A\_1A\_1}\left(t\right) + R\_{A\_2A\_2}\left(t\right) \tag{27}$$

Then, by mathematical induction,

$$R\_{AA}\left(t\right) = \sum\_{k=1}^{N} R\_{A\_kA\_k}\left(t\right) \tag{28}$$

Using this result in Eq. (23) gives

$$\begin{split} \left(\Phi\_{AA}\left(\boldsymbol{\alpha}\right) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} R\_{AA}\left(t\right) e^{-i\boldsymbol{\alpha}t} dt &= \frac{1}{2\pi} \int\_{-\infty}^{\infty} e^{-i\boldsymbol{\alpha}t} \sum\_{k=1}^{N} \frac{A\_k^2}{4} \left(e^{i\boldsymbol{\alpha}\_k t} + e^{-i\boldsymbol{\alpha}\_k t}\right) dt \\ &= \sum\_{k=1}^{N} \frac{A\_k^2}{4} \left[\delta\left(\boldsymbol{\alpha} - \boldsymbol{\alpha}\_k\right) + \delta\left(\boldsymbol{\alpha} + \boldsymbol{\alpha}\_k\right)\right] \end{split} \tag{29}$$

Using Eqs. (19,29), the average power harvested can be simplified:

$$E\left[P\left(t\to\infty\right)\right] = \frac{1}{2R\_l} \sum\_{k=1}^{N} \frac{A\_k^2}{\alpha\_k^4} \left| H\left(o\_k\right) \right|^2\tag{30}$$

Eq. (30) indicates that the frequency transfer function *H* need only be evaluated at the component frequencies of the base acceleration. This equation can be rewritten in the form

$$E\left[P\left(t\to\infty\right)\right] = \sum\_{k=1}^{N} A\_k^2 \mathbb{C}\_k\tag{31}$$

where *Ck* can be interpreted as the gain of the harmonic of frequency *<sup>k</sup>* . This gain is given by the formula

$$\mathbf{C}\_{k} = \frac{\left[\left(\rho A\right)\_{\text{eff}}\,\nu + m\_{t}\beta\right]^{2}\alpha k\_{e}^{2}}{2\alpha\mathbf{o}\_{n}} \frac{\Omega\_{k}^{2}}{\Lambda\_{k}\left(\mathrm{i}\Omega\_{k}\right)\overline{\Lambda\_{k}\left(\mathrm{i}\Omega\_{k}\right)}}\tag{32}$$

where

$$
\Lambda\_k \left( i\Omega\_k \right) = \alpha \left( i\Omega\_k \right)^3 + \left( 2\zeta a + 1 \right) \left( i\Omega\_k \right)^2 + \left( a + 2\zeta + a k\_e^2 \right) \left( i\Omega\_k \right) + 1 \tag{33}
$$

The following non-dimensional parameters are employed in Eqs. (32-33):

$$\Omega\_k = \frac{\alpha\_k}{\alpha\_n}, \, k\_\varepsilon^2 = \frac{\Theta^2}{\mathcal{C}\_0 \alpha\_n^2}, \, \alpha = R\_l \mathcal{C}\_0 \alpha\_n \tag{34}$$

where *k* is the ratio of the frequency of the acceleration component to the fundamental natural frequency, <sup>2</sup> *<sup>e</sup> k* is the modal electromechanical coupling coefficient, and is the ratio of the load resistance to the modal impedance.

#### **3.2. Nonlinear case, time domain**

222 Small-Scale Energy Harvesting

by the formula

natural frequency, <sup>2</sup>

where

Then, by mathematical induction,

Using this result in Eq. (23) gives

*AA A A A A* 1 1 2 2

 <sup>1</sup> *k k*

 

*N k*

*l k k*

1

*k*

*k k*

<sup>2</sup> <sup>2</sup>

 

2 0

*<sup>e</sup> k* is the modal electromechanical coupling coefficient, and

*n k kk k*

*N*

 

1 2

component frequencies of the base acceleration. This equation can be rewritten in the form

<sup>2</sup>

2

3 2 <sup>2</sup> 21 2 1 *kk k <sup>k</sup> e k i i*

2

*k R C C*

0 , , *<sup>k</sup> k e l n n n*

where *k* is the ratio of the frequency of the acceleration component to the fundamental

*t e eff <sup>k</sup>*

*E Pt AC*

*<sup>A</sup> EPt <sup>H</sup> R*

*k k*

2 24

1 1

 

2

 

Using Eqs. (19,29), the average power harvested can be simplified:

where *Ck* can be interpreted as the gain of the harmonic of frequency

2

 *i*

2

The following non-dimensional parameters are employed in Eqs. (32-33):

  

*A mk*

 

*k*

ratio of the load resistance to the modal impedance.

*C*

4

Eq. (30) indicates that the frequency transfer function *H*

*A*

1

*k*

*AA AA*

*N k*

*N AA A A k Rt R t* 

*R tR tR t* (27)

2

1

*k*

<sup>2</sup> <sup>2</sup>

*k*

(30)

(31)

*i i*

(34)

4 1

*<sup>A</sup> R t e dt e e e dt*

 

*N i t i t k it it*

(28)

*k k*

 

need only be evaluated at the

*<sup>k</sup>* . This gain is given

(32)

*k i* (33)

is the

(29)

In the presence of the magnetic field, the EOMs become nonlinear, and the analysis based off of the frequency transfer function detailed in the previous section is no longer valid. Instead, the vibrations induced by the spatially periodic magnetic field are interpreted as a series of plucks that occur each time the tip mass crosses an unstable equilibrium point between the ferrous structures. Each pluck is followed by a free response – underdamped in this case – superposed on the relatively slow base motion. An example response showing these two superposed motions is depicted in Fig. 4. The free response at the fundamental frequency of the beam, as opposed to the frequency of the base motion, drives the majority of the energy harvested.

**Figure 4.** Absolute base and tip displacements: max 0.75 *mag F F* , *dm* 5 mm , *ytY t* sin , *Y* 15 mm , 2 Hz . The shaded areas are the basins of attraction of the stable equilibria (Wickenheiser & Garcia, 2010b).

To analyze the energy harvested from a pluck, first the effect of the magnetic field strength on the free response is considered. To simplify the analysis, the inertial force due to base excitation is assumed to be negligible, and the effect of the energy dissipated by the resistor is approximated by an additional damping term. Hence, the total effective modal damping ratio is written as *eff e* , the sum of the mechanical and electrical damping. The electrical damping term can be accurately approximated as

$$
\zeta\_e = \frac{k\_e^2}{2\sqrt{1 - k\_e^2}} \frac{a}{1 + a^2} \tag{35}
$$

in the case of steady-state oscillations (Davis & Lesieutre, 1995); this formula is validated for free oscillations in the sequel. Using this damping model, the modal EOM, Eq. (12), can be written as

$$\frac{d^2\eta\left(t\right)}{dt^2} + 2\zeta\_{c\not\!\!/} \alpha\_n \frac{d\eta\left(t\right)}{dt} + \alpha\_n^2 \eta\left(t\right) = -F\_{\text{mag}}\beta \sin\left[\frac{2\pi}{d\_m} \left(\beta \eta\left(t\right) + y\left(t\right)\right)\right] \tag{36}$$

Linearizing Eq. (36) about the point *<sup>m</sup> y t kd* , *t* 0 , where *k* is an integer, gives

$$\frac{d^2\eta\left(t\right)}{dt^2} + 2\zeta\_{eff}\alpha\_n \frac{d\eta\left(t\right)}{dt} + \left(\alpha\_n^2 + \frac{2\pi\beta^2}{d\_m}F\_{mag}\right)\eta\left(t\right) = 0\tag{36a}$$

Thus, the term in parentheses is the square of the effective natural frequency, <sup>2</sup> *n eff* , .

The amplitude of each pluck, and hence, the initial condition of the free response, is determined by the location of the unstable equilibrium between each pair of ferrous structures. In equilibrium,

$$
\rho \alpha\_n^2 \eta \left( t \right) = -F\_{\text{mag}} \beta \sin \left[ \frac{2\pi}{d\_m} (\beta \eta \left( t \right) + y \left( t \right)) \right] \tag{37}
$$

again assuming that the effect of the electromechanical coupling is negligible. First, consider the case when the base is moving upward, i.e. *y t* 0 . In this case, the pluck occurs when

$$\sin\left[\frac{2\pi}{d\_m}(\beta\eta(t) + y(t))\right] = 1\tag{38}$$

When this condition occurs, any more vertical motion of the tip results in a decreased downward magnetic force. At this point, the beam has passed over a local maximum in the magnetic potential, and it begins accelerating towards the next stable equilibrium. The response after cresting the potential hill is approximated by Eq. (36). By plugging Eq. (38) into Eq. (37), the amplitude of the pluck can be found:

$$
\eta\_0 = \frac{-F\_{\text{mag}}\beta}{\alpha\_n^2} \tag{39}
$$

If the times of the plucks are denoted *kt* , then solving Eq. (38) for *kt* , and using the fact that sin *k k y tY t* , yields

$$t\_k t\_k = \frac{1}{o\nu} \sin^{-1} \left( \frac{4k-3}{4} \frac{d\_m}{Y} + \frac{F\_{\text{mag}} \beta^2}{o o\_n^2 Y} \right)\_{\prime}, \ k = 1, \ldots, N \quad \text{where} \quad \left( N - 1 \right) d\_m < Y \le N d\_m \tag{40}$$

A similar formula can be derived for the pluck times when *y t* 0 :

$$t\_{k+N} = \frac{\pi}{\alpha o} - \frac{1}{\alpha o} \sin^{-1} \left( \frac{4\left(N - k + 1\right) - 1}{4} \frac{d\_m}{Y} - \frac{F\_{\text{mag}} \beta^2}{o o\_n^2 Y} \right), \ k = 1, \ldots, N \tag{41}$$

By examining Eq. (36), the (in this case) underdamped free response can be found to be

Analysis of Energy Harvesting Using Frequency Up-Conversion by Analytic Approximations 225

$$\boldsymbol{\eta}\left(t\right) = \eta\_0 \boldsymbol{e}^{-\boldsymbol{\zeta}\_{\text{eff}} \boldsymbol{\alpha}\_{\text{eff}} t} \cos\left(\boldsymbol{\alpha}\_{d, \text{eff}} t\right) \tag{42}$$

where <sup>2</sup> , , 1 *d eff n eff eff* is the effective damped natural frequency, and the initial amplitude 0 is given by Eq. (39). This solution can now be plugged into Eq. (20) to find the voltage response *v t* after the pluck. Assuming that the voltage is 0 at the time of the pluck, the solution is given by

$$w(t) = -X\_1 e^{-t/R\_1 C\_0} + X\_1 e^{-\zeta\_{rf} \alpha\_{nf} t} \cos \left(\alpha\_{d,eff} t \right) + X\_2 e^{-\zeta\_{gf} \alpha\_{nf} t} \sin \left(\alpha\_{d,eff} t \right) \tag{43}$$

where

224 Small-Scale Energy Harvesting

2

structures. In equilibrium,

sin *k k y tY t*

, yields

1

*k N*

 

2

*d t dt*

 

2

 

into Eq. (37), the amplitude of the pluck can be found:

Linearizing Eq. (36) about the point *<sup>m</sup> y t kd* ,

 

*d t dt*

2

<sup>2</sup> 2 0 *eff n <sup>n</sup> mag*

*m*

 

*t F t yt dt dt <sup>d</sup>*

 

The amplitude of each pluck, and hence, the initial condition of the free response, is determined by the location of the unstable equilibrium between each pair of ferrous

> <sup>2</sup> <sup>2</sup> sin *<sup>n</sup> mag m t F t yt <sup>d</sup>*

again assuming that the effect of the electromechanical coupling is negligible. First, consider the case when the base is moving upward, i.e. *y t* 0 . In this case, the pluck occurs when

<sup>2</sup> sin <sup>1</sup>

When this condition occurs, any more vertical motion of the tip results in a decreased downward magnetic force. At this point, the beam has passed over a local maximum in the magnetic potential, and it begins accelerating towards the next stable equilibrium. The response after cresting the potential hill is approximated by Eq. (36). By plugging Eq. (38)

> 0 2 *mag n*

If the times of the plucks are denoted *kt* , then solving Eq. (38) for *kt* , and using the fact that

*k m m*

<sup>2</sup>

*N k d F t k N*

By examining Eq. (36), the (in this case) underdamped free response can be found to be

sin , 1, , <sup>4</sup> *m mag*

2

*n*

*Y Y*

*t k N N d Y Nd*

2

1 4 11

2 1 43 sin , 1, , where 1 <sup>4</sup>

*n*

*m mag*

*Y Y*

A similar formula can be derived for the pluck times when *y t* 0 :

1

*k d F*

*F* 

*t yt <sup>d</sup>*

 

 

*m*

 

*m*

*F t*

 

*t* 0 , where *k* is an integer, gives

(36)

(36a)

*n eff* , .

(37)

(38)

(39)

(40)

(41)

2

2 2

Thus, the term in parentheses is the square of the effective natural frequency, <sup>2</sup>

*dt dt d*

 

 

<sup>2</sup> <sup>2</sup> sin *eff n <sup>n</sup> mag*

$$X\_1 = \frac{\left(\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha}\right)^2 - \boldsymbol{\zeta}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha}}{1 - 2\boldsymbol{\zeta}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha} + \left(\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha}\right)^2} \\ \frac{\boldsymbol{\Theta}}{\boldsymbol{\Omega}\_0} \boldsymbol{\eta}\_{0'} \, \boldsymbol{X}\_2 = \frac{\boldsymbol{\zeta}\_{\varepsilon\|\boldsymbol{\varepsilon}}\left(\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha}\right)^2 - \left(\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha}\right)^2 - \boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha}}{1 - 2\boldsymbol{\zeta}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha} + \left(\boldsymbol{\Omega}\_{\varepsilon\|\boldsymbol{\varepsilon}}\boldsymbol{\alpha}\right)^2} \sqrt{1 - \boldsymbol{\zeta}\_{\varepsilon\|\boldsymbol{\varepsilon}}^2} \frac{\boldsymbol{\Theta}}{\boldsymbol{\Omega}\_0} \boldsymbol{\eta}\_{0'}.$$

and *eff n eff n* , .

The energy harvested during the free vibrations can be adequately approximated by the following formula:

$$\begin{split} E(t) &= \frac{1}{R\_l} \int\_0^t v^2 \left( \tau \right) d\tau \approx \frac{1}{2R\_l} \int\_0^t \frac{\Theta^2 R\_l^2 \alpha\_{n,eff}^2}{1 - 2\zeta\_{eff}\Omega\_{eff}\alpha + \left(\Omega\_{eff}\alpha\right)^2} \eta\_0^2 e^{-2\zeta\_{eff}\alpha\_{n,eff}\tau} d\tau \\ &= \frac{\Theta^2 R\_l \alpha\_{n,eff}}{4\zeta\_{eff} - 8\zeta\_{eff}^2 \Omega\_{eff}\alpha + 4\zeta\_{eff} \left(\Omega\_{eff}\alpha\right)^2} \eta\_0^2 \left(1 - e^{-2\zeta\_{eff}\alpha\_{n,eff}\tau}\right) \end{split} \tag{44}$$

The accuracy of this approximate formula can be seen in Fig. 5. The results of the simulation of the original EOMs, Eqs. (12,20), are plotted using a solid line, whereas Eq. (44) is plotted using a dashed line. To arrive at Eq. (44), it is assumed that the initial transients and the oscillating terms in <sup>2</sup> *v t* integrate out to 0; hence, the result is a smooth exponential curve. Although instantaneously the approximate curve may not be accurate, it matches the overall growth of the exact solution. Hence, Eq. (44) is an accurate representation of the energy harvested from a free vibration with a non-zero initial deflection and a zero initial velocity.

For a sequence of plucks, which is what occurs with the frequency up-conversion technique, it is assumed that the plucks are instantaneous and that the deflection is "reset" to the value given by Eq. (39) after each pluck. Hence, the total energy harvested during a half cycle of the base excitation is

$$E\_{\text{total}} = \sum\_{k=1}^{N} E\left(t\_{k+1} - t\_k\right) \tag{45}$$

**Figure 5.** Energy harvested during one free vibration after an initial pluck, comparison between simulation using exact EOMs and approximation using Eq. (44). Parameters used are listed in Table 1.

#### **4. Simulated response to sinusoidal base excitation**

In this section, the response of the system to sinusoidal base excitation, *y tY t* sin , is presented in both the time and frequency domains. The geometry and material properties used in the following simulations are listed in Table 1. The tip mass *mt* is approximately one-third of the overall beam mass, and its moment of inertia *<sup>t</sup> I* is calculated assuming the mass is roughly cube-shaped. The resistor value chosen for this study is the optimal value for energy harvesting at the fundamental frequency in the limit of small electromechanical coupling, i.e. *R C* 1 0 ,1 *SC* (Wickenheiser & Garcia, 2010c).

The first three natural frequencies of the beam are ,1 34.1 Hz *SC* , ,2 271.6 Hz *SC* , and ,3 806.8 Hz *SC* . Since frequencies around and below the fundamental frequency are of interest in this study, a three-mode expansion of the beam displacement is deemed sufficient. Furthermore, since the magnetic force is applied at the tip of the beam, only the fundamental mode is significantly excited by the plucking.

The transfer functions for power harvested (normalized by 2 3 *Y* ) are plotted in Fig. 6. Five different values for the magnetic force strength *mag F* are plotted alongside the baseline case of an inactive tip. For the cases with a nonzero magnetic force, the transfer functions are derived numerically. The system is simulated for 50 cycles of base motion, and the relative tip deflection is averaged over the last 20 cycles of each run in order to minimize the effects of initial transients. This process is completed 10 times at every frequency, and the results are averaged.


Analysis of Energy Harvesting Using Frequency Up-Conversion by Analytic Approximations 227

**Table 1.** Geometry and material properties.

226 Small-Scale Energy Harvesting

coupling, i.e. *R C* 1 0 ,1

,3 806.8 Hz

are averaged.

**Figure 5.** Energy harvested during one free vibration after an initial pluck, comparison between simulation using exact EOMs and approximation using Eq. (44). Parameters used are listed in Table 1.

In this section, the response of the system to sinusoidal base excitation, *y tY t* sin

*SC* (Wickenheiser & Garcia, 2010c).

presented in both the time and frequency domains. The geometry and material properties used in the following simulations are listed in Table 1. The tip mass *mt* is approximately one-third of the overall beam mass, and its moment of inertia *<sup>t</sup> I* is calculated assuming the mass is roughly cube-shaped. The resistor value chosen for this study is the optimal value for energy harvesting at the fundamental frequency in the limit of small electromechanical

*SC* . Since frequencies around and below the fundamental frequency are of interest in this study, a three-mode expansion of the beam displacement is deemed sufficient. Furthermore, since the magnetic force is applied at the tip of the beam, only the

different values for the magnetic force strength *mag F* are plotted alongside the baseline case of an inactive tip. For the cases with a nonzero magnetic force, the transfer functions are derived numerically. The system is simulated for 50 cycles of base motion, and the relative tip deflection is averaged over the last 20 cycles of each run in order to minimize the effects of initial transients. This process is completed 10 times at every frequency, and the results

*SC* , ,2 271.6 Hz 

, is

*SC* , and

) are plotted in Fig. 6. Five

**4. Simulated response to sinusoidal base excitation** 

The first three natural frequencies of the beam are ,1 34.1 Hz

fundamental mode is significantly excited by the plucking.

The transfer functions for power harvested (normalized by 2 3 *Y*

The overall trend of the responses indicates that the magnet has an increasing effect as the base excitation frequency decreases to 0. As is discussed in the sequel, at low frequencies relative to the fundamental resonance, the response converges to a sequence of free responses. In this regime, the inertial forces are negligible, and so the disturbances due to the magnetic force are relatively large. The normalized power approaches half an order of magnitude below its resonance value as 0 due to the energy harvested from the plucks. As the driving frequency increases, the beam tip has less time to oscillate in each potential well, and, thus, its motion tends to converge towards the motion of the baseline case. As *SC*,1 , the frequency of the base motion approaches the frequency of the impulse response of the beam from the magnetic force. Hence, the time in which the beam is in free response decays to 0, and so all of the frequency response functions converge towards the baseline function, as shown in Fig. 5. A discussion of the variation in frequency response with respect to the magnet parameters *mag F* and *md* can be found in (Wickenheiser & Garcia, 2010b).

**Figure 6.** Normalized power harvested transfer function: 5 mm *md* , *Y* 15 mm (Wickenheiser & Garcia, 2010b).

Fig. 7 compares two methods of simulating the tip deflection response: using the original EOMs and assuming a series of undamped free responses give by Eq. (42). The most notable difference that can be seen from the figure is that the free response assumption does not take into account the base motion, which causes the solution to the EOMs to drift upward or downward depending on the sign of the base velocity. Another difference between the two models is that the assumed time of the plucks, indicated by the vertical lines and given by Eqs. (40,41), generally occur before the plucks in the actual solution. This happens because the beam has enough inertia to resist the pull of the next magnet, i.e. to overcome the potential well barrier between magnets. Only after the beam's velocity decays sufficiently does it become trapped in the next potential well in the sequence.

Fig. 8 depicts the voltage response during the same simulation that has been plotted in Fig. 7. A comparison of the two curves plotted shows an excellent agreement between the simulation results and the predicted voltage given by Eq. (43). There is a slight asymmetry in the curve representing the simulation results due to the base excitation. This discrepancy is much less pronounced than in Fig. 7 since the voltage is dominated by the velocity of the tip deflection, which is not affected directly by the base motion, unlike the tip position. The primary difference between the two curves in Fig. 8 is the error in predicting when the plucks occur, as discussed in the previous paragraph. This error causes an over-prediction in the number of cycles of free oscillation, but this error is small compared to the duration of the free response between plucks. The initial magnitude of the voltage free response is well predicted, however; this prediction is much more significant in the estimation of voltage given by Eq. (43).

228 Small-Scale Energy Harvesting

resonance value as 0

Garcia, 2010b).

The overall trend of the responses indicates that the magnet has an increasing effect as the base excitation frequency decreases to 0. As is discussed in the sequel, at low frequencies relative to the fundamental resonance, the response converges to a sequence of free responses. In this regime, the inertial forces are negligible, and so the disturbances due to the magnetic force are relatively large. The normalized power approaches half an order of magnitude below its

frequency increases, the beam tip has less time to oscillate in each potential well, and, thus, its

frequency of the base motion approaches the frequency of the impulse response of the beam from the magnetic force. Hence, the time in which the beam is in free response decays to 0, and so all of the frequency response functions converge towards the baseline function, as shown in Fig. 5. A discussion of the variation in frequency response with respect to the magnet

**Figure 6.** Normalized power harvested transfer function: 5 mm *md* , *Y* 15 mm (Wickenheiser &

does it become trapped in the next potential well in the sequence.

Fig. 7 compares two methods of simulating the tip deflection response: using the original EOMs and assuming a series of undamped free responses give by Eq. (42). The most notable difference that can be seen from the figure is that the free response assumption does not take into account the base motion, which causes the solution to the EOMs to drift upward or downward depending on the sign of the base velocity. Another difference between the two models is that the assumed time of the plucks, indicated by the vertical lines and given by Eqs. (40,41), generally occur before the plucks in the actual solution. This happens because the beam has enough inertia to resist the pull of the next magnet, i.e. to overcome the potential well barrier between magnets. Only after the beam's velocity decays sufficiently

motion tends to converge towards the motion of the baseline case. As

parameters *mag F* and *md* can be found in (Wickenheiser & Garcia, 2010b).

due to the energy harvested from the plucks. As the driving

 *SC*,1 , the

**Figure 7.** Comparison of simulated tip deflection from EOMs (solid) and a series of plucks, Eq. (42) (dashed). Vertical lines indicate the estimated times of plucking according to Eqs. (40,41).

Fig. 9 shows the comparison between the power frequency transfer functions of the simulation, the approximation by a series of plucks, and the linear system (without magnets). This plot is generated using the same procedure as the one used to produce Fig. 6. The most striking feature of this plot is that the simulation results are seen to converge to the approximation by a series of plucks for low frequencies and converge to the linear system approximation at frequencies approaching the fundamental resonance. As previously mentioned, at frequencies around the fundamental resonance, the beam is not allowed to vibrate freely because the pluck frequency exceeds its natural frequency. Hence, there is no exponential decay in the amplitude of the tip deflection between plucks. In this case, the forced response (i.e. particular solution) dominates the motion, and so the frequency transfer function of the nonlinear system approaches that of the linear system. At low frequencies, the base excitation term becomes negligible, and so the mechanical EOM reduces to Eq. (36), the basis for the series of plucks approximation. In this scenario, the magnetic force drives the excitation of the beam, whereas the inertial force due to the base excitation is negligible. This is manifested in the decrease in the linear response at low frequencies.

**Figure 8.** Comparison of simulated voltage from EOMs (solid) and a series of plucks, Eq. (43) (dashed).

**Figure 9.** Comparisons of normalized power frequency transfer functions between EOMs (solid), a series of plucks (dashed), and the linear system (dash-dot).

#### **5. Conclusions**

This chapter presents an accurate means of approximating the non-linear response of the frequency up-conversion technique as a series of free responses. This simplification is based on the assumption that the base excitation is negligible, and so it only holds at low frequencies compared to the fundamental resonance of the beam. This approximation, however, is useful in the design of energy harvesters utilizing this technique as it enables power to be generated at very low frequencies. This means that the device can be designed for a fundamental frequency much higher than the nominal base excitation frequency, which tends to result in smaller and lighter transducers. At low frequencies, the approximation derived herein is shown to agree well with the simulation results of the full non-linear equations of motion in terms of displacement, voltage, and power harvested. It is confirmed through analysis of the frequency transfer function that the non-linear system converges to the approximation by a series of free responses at low frequencies and to the linear system response at frequencies around the fundamental. Hence, a combination of analytical solutions can be used to predict the energy harvesting performance of this nonlinear device in lieu of simulation of the full dynamics equations.

### **Author details**

230 Small-Scale Energy Harvesting

frequencies.

reduces to Eq. (36), the basis for the series of plucks approximation. In this scenario, the magnetic force drives the excitation of the beam, whereas the inertial force due to the base excitation is negligible. This is manifested in the decrease in the linear response at low

**Figure 8.** Comparison of simulated voltage from EOMs (solid) and a series of plucks, Eq. (43) (dashed).

**Figure 9.** Comparisons of normalized power frequency transfer functions between EOMs (solid), a

This chapter presents an accurate means of approximating the non-linear response of the frequency up-conversion technique as a series of free responses. This simplification is based

series of plucks (dashed), and the linear system (dash-dot).

**5. Conclusions** 

Adam Wickenheiser *George Washington University, United States* 

### **6. References**

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Rastegar, J. & Murray, R. (2008). Novel Two-Stage Electrical Energy Generators for Highly Variable and Low-Speed Linear or Rotary Input Motion. Proceedings of ASME, ISBN 9780791843260.

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Physique IV, Vol. 128, pp. 177–186, ISSN 1155-4339.

Erturk, A.; Renno, J. M. & Inman, D. J. (2009). Modeling of piezoelectric energy harvesting from an L-shaped beam-mass structure with an application to UAVs. Journal of Intelligent Materials Systems and Structures, Vol. 20, pp. 529–544, ISSN

Inman, D. J. (2007). Engineering Vibration (3rd), Pearson, ISBN 0-13-228173-2, Upper Saddle

Karami, M. A. & Inman, D. J. (2011). Electromechanical Modeling of the Low-Frequency Zigzag Micro-Energy Harvester. Journal of Intelligent Material Systems and Structures,

Lefeuvre, E.; Badel, A.; Benayad, A.; Lebrun, L.; Richard, C. & Guyomar, D. (2005). A comparison between several approaches of piezoelectric energy harvesting. Journal De

Leland, E. S. & Wright, P. K. (2006) Resonance tuning of piezoelectric vibration energy scavenging generators using compressive axial preload. Smart Materials and Structures,

Liao, Y. & Sodano, H. A. (2008). Model of a single mode energy harvester and properties for optimal power generation. Smart Materials and Structures, Vol. 17, No. 6, 065026, ISSN

Lin, Y. K. (1967). Probabilistic Theory of Structural Dynamics, McGraw-Hill, ISBN 0-88-

Mann, B. P. & Sims, N. D. (2009). Energy harvesting from the nonlinear oscillations of magnetic levitation, Journal of Sound and Vibration, Vol. 319, pp. 515-530, ISSN 0022-

Moon, F. C. (1978). Problems in magneto-solid mechanics, Mechanics Today, Vol. 4, pp. 307-

Morris, D. J.; Youngsman, J. M.; Anderson, M. J. & Bahr, D. F. (2008). A resonant frequency tunable, extensional mode piezoelectric vibration harvesting mechanism, Smart

Murray, R. & Rastegar, J. (2009). Novel Two-Stage Piezoelectric-Based Ocean Wave Energy Harvesters for Moored or Unmoored Buoys. Proceedings of SPIE, ISSN 0277-786X, San

Oguamanam, D. C. D. (2003). Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. International Journal of Mechanical Sciences, Vol. 45, pp.

Porat, B. (1994). Digital Processing of Random Signals, Prentice-Hall, ISBN 0-48-646298-6,

Ramlan, R.; Brennan, M. J.; Mace, B. R. & Kovacic, I. (2010). Potential benefits of a non-linear stiffness in an energy harvesting device. Nonlinear Dynamics, Vol. 59, pp. 545–558,

Materials and Structures, Vol. 17, No. 6, 065021, ISSN 0964-1726.


Wickenheiser, A. M. & Garcia, E. (2010c). Power Optimization of Vibration Energy Harvesters Utilizing Passive and Active Circuits. Journal of Intelligent Materials Systems and Structures, Vol. 21, No. 13, pp. 1343–1361, ISSN 1045-389X.

### **Chapter 10**

## **Strategies for Wideband Mechanical Energy Harvester**

B. Ahmed Seddik, G. Despesse, S. Boisseau and E. Defay

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51898

### **1. Introduction**

234 Small-Scale Energy Harvesting

Wickenheiser, A. M. & Garcia, E. (2010c). Power Optimization of Vibration Energy Harvesters Utilizing Passive and Active Circuits. Journal of Intelligent Materials

Systems and Structures, Vol. 21, No. 13, pp. 1343–1361, ISSN 1045-389X.

The energy harvesting market expands day by day, this is mainly due to the number of the implemented low power sensors in different fields, such as: human body, building, car engine…etc. These sensors are in most cases powered by batteries, but the main drawback of this technique is the need of a continuous control of their state of charge, the recharge and the replacement which is in most cases expensive. Thus, in order to overcome these limitations, one of the most promising solutions is to harvest the surrounding energy beside the system to power. In our environment, we can find many types of recoverable energy, for example: mechanical energy, thermal energy and radiative energy (solar, infra-red, radiofrequency). This chapter is dedicated to mechanical energy and more particularly to mechanical vibration energy produced by cars, fridges, mechanical engines and so on. The mechanical to electrical converter can be electromagnetic, electrostatic or piezoelectric. In case of an electromagnetic conversion, the vibrations are used to create a relative movement between a coil and a permanent magnet. In case of an electrostatic transduction, the vibrations are used to create a variable capacitance. In case of a piezoelectric transduction, the vibrations are used to apply a mechanical stress on a piezoelectric material. Actually, a vibration energy harvester (VEH) features 3 main components as presented in the Figure 1.

© 2012 Despesse 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 Despesse 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.

The first stage of the conversion chain is a Mechanical to Mechanical (M2M) converter usually based on a mechanical resonator system. This converter translates the input vibration into a relative displacement between the resonator seismic mass and the vibration source. In addition, by using a resonant mechanism, the relative displacement amplitude can be larger compared to the vibration source displacement amplitude, increasing then the extracted mechanical power from the vibration source. Then, thanks to a dedicated Mechanical to Electrical (M2E) converter, which could be electromagnetic, electrostatic or piezoelectric, the amplified relative displacement is converted into electrical energy. Finally, an Electrical to Electrical (E2E) converter translates this electrical energy into a usable energy with a stable direct voltage able to supply an electrical circuit (3V for example). The efficiency of the harvester is tightly related to each stage of this chain. Moreover, as it can be noted from Figure 1, each stage has an effect on the other stages. Thus, the improvement of the VEH efficiency should take into account all the stages and also the relations between them. In what follows, more details are given for each stage.

#### **1.1. The mechanical to mechanical converter**

The first aim of this converter is to translate a vibration into a relative displacement able to actuate the mechanical input of the M2E converter. To make that, a seismic mass is required and the mechanical work that can be produced from the vibration is proportional to this mass and then to its size. This seismic mass is the main limitation in terms of power density capability for the main developed systems. In order to amplify the inertial effect of this seismic mass, it is necessary to use the resonance effect, which means using resonators. Actually, such devices are commonly modeled by a mass spring system connected to the vibration source and damped by the M2E converter and the mechanical losses. The efficiency of the M2M converter could be measured by the amplification gain of the vibration displacement amplitude (Q factor). However, the amplification gain is inversely proportional to the frequency bandwidth of the resonator making the system very sensitive to any change of the input vibration. This shift is commonly occurred especially in vibrations produced by car engine, in which case the frequency depends on the motor speed which is susceptible to change over time. In addition, the VEH resonant frequency is susceptible to change over time because of the aging of the materials. In fact, as the material of the harvester is subjected to a continuous mechanical stress, the mechanical stiffness will be altered during time and so the resonant frequency. To overcome this limitation many solutions have been proposed in literature, most of them are summarized in [1-2]. Actually, a few of these solutions allow an automatic adaptation without comprising the efficiency of the harvester neither the power balance of the VEH.

#### **1.2. The mechanical to electrical converter**

Once the vibrations are converted into an amplified relative displacement between two elements, this displacement is converted into electricity using electromagnetic, electrostatic or piezoelectric principle. The efficiency of this converter depends on its mechanical and electrical losses and also its good impedance matching with the mechanical source and electrical load. Many approaches have been developed in order to improve the efficiency of this converter, these approaches depend mainly on the type of the converter, most of them are linked to the system size and the vibration source characteristics [3-5].

### **1.3. The electrical to electrical converter**

236 Small-Scale Energy Harvesting

The first stage of the conversion chain is a Mechanical to Mechanical (M2M) converter usually based on a mechanical resonator system. This converter translates the input vibration into a relative displacement between the resonator seismic mass and the vibration source. In addition, by using a resonant mechanism, the relative displacement amplitude can be larger compared to the vibration source displacement amplitude, increasing then the extracted mechanical power from the vibration source. Then, thanks to a dedicated Mechanical to Electrical (M2E) converter, which could be electromagnetic, electrostatic or piezoelectric, the amplified relative displacement is converted into electrical energy. Finally, an Electrical to Electrical (E2E) converter translates this electrical energy into a usable energy with a stable direct voltage able to supply an electrical circuit (3V for example). The efficiency of the harvester is tightly related to each stage of this chain. Moreover, as it can be noted from Figure 1, each stage has an effect on the other stages. Thus, the improvement of the VEH efficiency should take into account all the stages and also the relations between

The first aim of this converter is to translate a vibration into a relative displacement able to actuate the mechanical input of the M2E converter. To make that, a seismic mass is required and the mechanical work that can be produced from the vibration is proportional to this mass and then to its size. This seismic mass is the main limitation in terms of power density capability for the main developed systems. In order to amplify the inertial effect of this seismic mass, it is necessary to use the resonance effect, which means using resonators. Actually, such devices are commonly modeled by a mass spring system connected to the vibration source and damped by the M2E converter and the mechanical losses. The efficiency of the M2M converter could be measured by the amplification gain of the vibration displacement amplitude (Q factor). However, the amplification gain is inversely proportional to the frequency bandwidth of the resonator making the system very sensitive to any change of the input vibration. This shift is commonly occurred especially in vibrations produced by car engine, in which case the frequency depends on the motor speed which is susceptible to change over time. In addition, the VEH resonant frequency is susceptible to change over time because of the aging of the materials. In fact, as the material of the harvester is subjected to a continuous mechanical stress, the mechanical stiffness will be altered during time and so the resonant frequency. To overcome this limitation many solutions have been proposed in literature, most of them are summarized in [1-2]. Actually, a few of these solutions allow an automatic adaptation without comprising

Once the vibrations are converted into an amplified relative displacement between two elements, this displacement is converted into electricity using electromagnetic, electrostatic or piezoelectric principle. The efficiency of this converter depends on its mechanical and electrical losses and also its good impedance matching with the mechanical source and electrical load. Many approaches have been developed in order to improve the efficiency of

them. In what follows, more details are given for each stage.

the efficiency of the harvester neither the power balance of the VEH.

**1.2. The mechanical to electrical converter** 

**1.1. The mechanical to mechanical converter** 

The maximum of the extracted electrical power is achieved when the electrical converted power is equal to the mechanical dissipated power in the mechanical structure. However, the mechanical damping depends on the used material, while the electrical damping depends on the converted power (electromechanical coupling of the structure) and the output electrical impedance. When the vibration frequency and amplitude are known and fixed, one can design a harvester to fit the optimized conditions (in terms of resonance and damping forces). However, when the vibration magnitude changes, this equality cannot be satisfied any more since the damping forces (mechanical and electrical) have different variation profile when the vibration amplitude changes. Consequently, the VEH efficiency is decreased. At the present time, only one study has been done in this perspective, which means adapting in real time the damping forces in order to maintain an optimum point of electrical energy extraction [6].

This brief introduction highlights two improvement areas. The first one consists to ensure the tracking of the vibration frequency, while the second one consists to adjust in real time the electrical damping force with respect to the mechanical one. This chapter covers these areas of VEH efficiency improvement. At the present time, more works have been done to cover the first area of investigation than the second one.

The next part of this chapter gives an overview of the main approaches developed in the state of the art to ensure a vibration frequency tracking. These approaches are classified according to the type of vibration source: mixed frequencies vibration or vibration with a main frequency that changes over time.

### **2. State of the art**

Many solutions have been proposed in the state of the art in order to overcome the system degradation related to the shift between the resonant frequency and the vibration one. The best way to make comparison between these solutions is to classify them according to the type of input vibration signal to which they could be subjected. Actually, two main types of vibration signals exist:

### **2.1. Vibration with multiple of harmonic at different frequencies**

Basically, for such signals, the energy of the vibration signal is spread over a wide bandwidth. Hence, using a one degree of freedom resonator will not harvest the energy efficiently even if the resonant frequency is included in the bandwidth of the vibration signal. This kind of signal exists in staircases, buildings, train rails...etc. Solutions developed to extract the maximum of energy from such vibrations spectrum are based on systems with a wide bandwidth. Hereafter the main techniques developed in this issue:

### *2.1.1. High electrical damping systems*

Despesse et al., [7] proposed an electrostatic converter with a high electrical coupling coefficient in order to broaden the resonance peak. The fabricated prototype is able to recover mechanical vibration below 100 Hz, with a global conversion efficiency of 60% at 50Hz. In fact, the main disadvantage of this structure is the quality factor of the converter, the resonance peak is broadened by increasing the electrical damping coefficient, the quality factor is then decreased and therefore, the quantity of the scavenged energy is relatively decreased when the input vibration frequency reaches the resonant frequency compared to a system with a high quality factor. To take advantage from this solution without decreasing permanently the quality factor and keep the same efficiency as a high quality factor system when the resonant frequency is equal to the vibration one, we should adjust in real time the electrical coupling and then the electrical damping. When the vibration input frequency is equal to the resonant frequency, the electrical coupling could be very low enabling a full resonant effect and when the vibration frequency shifts, the coupling could be increased to reach a higher output power. In fact, if the mechanical to electrical converter can reach a high electromechanical coupling, it is easy to temporarily decrease this coupling by mismatching the output electrical impedance.

### *2.1.2. Multi-modes systems*

Shahruz et al., [8] proposed to expand the bandwidth using a multi modal structure. This structure is composed of several cantilevers. Each one has a defined resonance frequency. However, for a given vibration frequency, there is only one cantilever excited at its resonance frequency and all the others generate only a few amount of energy which limits the power density of the whole system. Another solution has been proposed by Roundy et al., [9] similar to the previous one, which consists on using a mechanical resonator composed of 3 different proof masses and four springs, they predict that the bandwidth could be multiplied by 3; nevertheless, the functionality of this system has never been experimentally verified. However, in both cases the power density of the converter is reduced since the harvested energy is proportional to the seismic mass, and in such cases only one cantilever works efficiently. Nevertheless, it can be interesting to use this technique to harvest a main vibration frequency and its harmonics that can be significantly separated in frequency but well known in advance.

#### *2.1.3. Non-resonant system*

Yang et al., [10] proposed another idea to broaden the VEH bandwidth, this idea consists to design a harvester where the effect of the air damping can be controlled. The idea is quite similar to the first one, except that for the present approach, the mechanical damping is increased instead of the electrical one, decreasing then significantly the mechanical extracted power. Actually, this solution is always less interesting than a high quality factor VEH, the bandwidth is in fact just increased because the mechanical damping limits significantly the output power when the vibration frequency fits the resonant frequency and not because it increases the output power outside the resonant frequency.

### **2.2. Harmonic vibrations**

238 Small-Scale Energy Harvesting

*2.1.1. High electrical damping systems* 

mismatching the output electrical impedance.

*2.1.2. Multi-modes systems* 

*2.1.3. Non-resonant system* 

increases the output power outside the resonant frequency.

Despesse et al., [7] proposed an electrostatic converter with a high electrical coupling coefficient in order to broaden the resonance peak. The fabricated prototype is able to recover mechanical vibration below 100 Hz, with a global conversion efficiency of 60% at 50Hz. In fact, the main disadvantage of this structure is the quality factor of the converter, the resonance peak is broadened by increasing the electrical damping coefficient, the quality factor is then decreased and therefore, the quantity of the scavenged energy is relatively decreased when the input vibration frequency reaches the resonant frequency compared to a system with a high quality factor. To take advantage from this solution without decreasing permanently the quality factor and keep the same efficiency as a high quality factor system when the resonant frequency is equal to the vibration one, we should adjust in real time the electrical coupling and then the electrical damping. When the vibration input frequency is equal to the resonant frequency, the electrical coupling could be very low enabling a full resonant effect and when the vibration frequency shifts, the coupling could be increased to reach a higher output power. In fact, if the mechanical to electrical converter can reach a high electromechanical coupling, it is easy to temporarily decrease this coupling by

Shahruz et al., [8] proposed to expand the bandwidth using a multi modal structure. This structure is composed of several cantilevers. Each one has a defined resonance frequency. However, for a given vibration frequency, there is only one cantilever excited at its resonance frequency and all the others generate only a few amount of energy which limits the power density of the whole system. Another solution has been proposed by Roundy et al., [9] similar to the previous one, which consists on using a mechanical resonator composed of 3 different proof masses and four springs, they predict that the bandwidth could be multiplied by 3; nevertheless, the functionality of this system has never been experimentally verified. However, in both cases the power density of the converter is reduced since the harvested energy is proportional to the seismic mass, and in such cases only one cantilever works efficiently. Nevertheless, it can be interesting to use this technique to harvest a main vibration frequency and its harmonics that can be significantly separated in frequency but well known in advance.

Yang et al., [10] proposed another idea to broaden the VEH bandwidth, this idea consists to design a harvester where the effect of the air damping can be controlled. The idea is quite similar to the first one, except that for the present approach, the mechanical damping is increased instead of the electrical one, decreasing then significantly the mechanical extracted power. Actually, this solution is always less interesting than a high quality factor VEH, the bandwidth is in fact just increased because the mechanical damping limits significantly the output power when the vibration frequency fits the resonant frequency and not because it This is the most common type of vibrations, the energy of vibration is mostly concentrated around one determined frequency. However, there are two origins to the shift between the resonant frequency and the vibration one for such type of vibration:


Hence, solutions developed in this issue trend to minimize the shift between the resonant frequency and the vibration one by tuning the resonant frequency. To do so, two types of approaches have been developed:

### *2.2.1. Passive tuning of the resonant frequency:*

The passive way for tuning the resonant frequency means that the adaptation system does not need to be supplied by an external source of energy. Hereafter some techniques developed in this perspective:

*Using non linear spring* 

Marzencki et al., [11] has proposed an energy harvesting device employing the mechanical non linear strain stiffness using a clamped-clamped beam. For such systems the stiffness depends on the amplitude and the frequency of the vibration source. Hence, a well design of the structure could allow an adaptation between the vibration frequency and the resonant one. In this work, they report a tuning ratio of resonance frequency of over 36% for a clamped-clamped beam device with an input acceleration of 2g. This solution allows an efficient way to make a passive dynamic adaptation of the resonant frequency. However, there are some limits of this technique: a high resonant frequency tuning ratio requires a high acceleration. In addition, the frequency response of a non linear VEH has a hysteresis aspect, which means that when the frequency exceeds a specific value, the output power drops off dramatically, and it is not possible to come back to the previous point unless the vibration frequency decreases to the start frequency of the VEH. Furthermore, the principle efficiency is very dependent on the input vibration amplitude. For low amplitudes, there is a limited non-linear effect. Inversely, for large input displacements, the non-linear effect limits the relative displacement and then the output power.

*Manuel tuning of the resonant frequency* 

Leland et al., [12] suggests another technique for tuning the resonance frequency, it consists on applying *manually* an axial preload in order to change the resonance frequency and match the frequency of the vibration source. Using this technique, the developed system is able to adjust the resonance frequency of about 24% below its unloaded resonance frequency. This solution is viable only when the vibration frequency is known a priori and is not susceptible to change over time, since it allows fitting the right resonance frequency before implementation of the VEH.

#### *2.2.2. Active tuning of the resonant frequency:*

The active tuning means that the resonant frequency is adjusted in real time using an active process. The drawback of this technology is the power required to make the dynamic tuning of the resonant frequency. Hereafter, the main active tuning techniques:

*Application of a force* 

The application of a force means that the VEH is equipped with an actuator, magnetic [13], piezoelectric [14] or electrostatic actuator [15]. The main task of this actuator is to apply an additional force on the seismic mass in the same direction as the vibration one in order to affect the mechanical stiffness of the VEH (added or substituted a force to the spring force of the resonator). This force induces a change in the effective mechanical stiffness and then the resonant frequency. Among all the works made in this perspective only a few of them report a positive power balance between the output power delivered by the VEH and the energy required to adjust in real time the resonant frequency. Eichhorn et al., [16] have presented a smart and self-sufficient frequency tunable vibration energy harvester, they report a resonant frequency tuning ratio up to 26% and a consumption of the tuning system about10% of the VEH output power. However, the system could perform the frequency adjustment only once a 22s. Another system proposed by Lallart et al., [17] based on an original approach that permits increasing the effective bandwidth by a factor of 4 in terms of mechanical vibration with a positive power balance.

*Electrical load adaptation* 

The idea of this technique is to adjust the resonant frequency by adjusting the value of the electrical load coupled with the VEH. This technique has been used for electromagnetic harvester. Since the stiffness is related to the electrical current in the coil, by adjusting the electrical load, the current could be changed and then the resonant frequency as shown in Figure 2, [18]. However, this approach requires too much power for the implementation, about 1000 times the power generated by the VEH.

**Figure 2.** Method for adjusting the resonant frequency by adapting the electrical load principal scheme [18]

As it can be noted from this overview of the existed solutions, the main limit is either the lack of dynamic adaptability or the tuning ratio. To have a good dynamic of resonant frequency it is necessary to use active solutions. However, the main drawback of such techniques is the negative power balance between the output power from the converter part and the power required to make the tuning of the resonant frequency. The works of Eichorn [16] and Lallart [17] show the possibility to develop wideband system with a positive power balance, their results are for great interest especially for autonomous system development. However, in both cases the positive balance is achieved by making a compromise with either the frequency of tuning adjustment or the tuning ratio. This brings us to conclude that it remains a large quantity of work to perform in this perspective in order to achieve a complete wideband harvester, able to be implemented in environment where the frequency vibration frequency is susceptible to change over time regardless the decrease of the output power due to the shift between the input frequency and the resonant one. Among laboratories interested by developing a viable solution using active technique there is *CEA-Leti.* Three different solutions have been developed in this laboratory. The next part gives details of each of these developed solutions, the first solution is based on the amplification of the generated relative displacement at off resonance, this solution is applicable for both types of vibration described before. The second and the third solutions are based on active tuning of the resonant frequency and are more applicable for the second type of vibration.

### **3. Solutions developed by** *CEA-Leti*

240 Small-Scale Energy Harvesting

*Application of a force* 

*Electrical load adaptation* 

[18]

*2.2.2. Active tuning of the resonant frequency:* 

mechanical vibration with a positive power balance.

about 1000 times the power generated by the VEH.

The active tuning means that the resonant frequency is adjusted in real time using an active process. The drawback of this technology is the power required to make the dynamic tuning

The application of a force means that the VEH is equipped with an actuator, magnetic [13], piezoelectric [14] or electrostatic actuator [15]. The main task of this actuator is to apply an additional force on the seismic mass in the same direction as the vibration one in order to affect the mechanical stiffness of the VEH (added or substituted a force to the spring force of the resonator). This force induces a change in the effective mechanical stiffness and then the resonant frequency. Among all the works made in this perspective only a few of them report a positive power balance between the output power delivered by the VEH and the energy required to adjust in real time the resonant frequency. Eichhorn et al., [16] have presented a smart and self-sufficient frequency tunable vibration energy harvester, they report a resonant frequency tuning ratio up to 26% and a consumption of the tuning system about10% of the VEH output power. However, the system could perform the frequency adjustment only once a 22s. Another system proposed by Lallart et al., [17] based on an original approach that permits increasing the effective bandwidth by a factor of 4 in terms of

The idea of this technique is to adjust the resonant frequency by adjusting the value of the electrical load coupled with the VEH. This technique has been used for electromagnetic harvester. Since the stiffness is related to the electrical current in the coil, by adjusting the electrical load, the current could be changed and then the resonant frequency as shown in Figure 2, [18]. However, this approach requires too much power for the implementation,

**Figure 2.** Method for adjusting the resonant frequency by adapting the electrical load principal scheme

As it can be noted from this overview of the existed solutions, the main limit is either the lack of dynamic adaptability or the tuning ratio. To have a good dynamic of resonant frequency it is necessary to use active solutions. However, the main drawback of such

of the resonant frequency. Hereafter, the main active tuning techniques:

### **3.1. Amplification of relative displacement at off resonance (rebound technique) [19]**

In this part, a new approach for amplifying the relative movement of a cantilever system at off-resonance is presented. The aim is to broaden the resonance peak of resonators without compromising the quality factor of the system. The idea is to gather the resonance phenomenon conditions at off resonance. This could be done by adding a *rebound* mechanism to the VEH, when the speed of the vibration source reaches an extremum, the seismic mass is mechanically connected to the vibration source via a high stiffness spring, the movement of the seismic mass is then inverted and its speed is increased, which means a transfer of energy from the source to the VEH. This operation is called "*rebound*". This approach is useful for VEH operating in environments where vibrations could be spread over a wide bandwidth or characterized by one main frequency susceptible to change over time. In the following, the principle is presented in details by giving the modeling and optimization approaches.

#### *3.1.1. Description of the approach*

The original idea of this approach is based on the principal of the elastic collision theory. When two solid bodies enter into collision there is an exchange of mechanical energy between these two bodies. After collision, the small mass goes in the opposite direction with a higher speed as shown in Figure 3. In the case of VEH, the small mass will represent the seismic mass of the harvester, while the big one will represent the mass of the vibration source.

The following calculations are intended to estimate the final energy of the seismic mass m2 after the collision in order to compare it to its initial energy and the speed of m1 before the collision. The objective is to deduce the energy gain of m2 and the way to maximize it.

**Figure 3.** Illustration of the direct collision between two masses

The conservation of the kinetic energy before and after the collision leads to the following equations:

$$\begin{cases} m\_1 \vec{p}\_1 + m\_2 \vec{p}\_2 = m\_1 \vec{p}\_1^\prime + m\_2 \vec{p}\_2^\prime \\ m\_1 \upsilon\_1^2 + m\_2 \upsilon\_2^2 = m\_1 \upsilon\_1^2 + m\_2 \upsilon\_2^\prime \end{cases} \tag{1}$$

Where: *v1* and *v2* are the speed of the mass *m1* and *m2,*respectively, *p1* and *p2* are the quantity of movement (*mi*.*vi*),. This leads to the final speed *v'2* of the seismic mass:

$$
\vec{\upsilon}\_2^\prime = -\vec{\upsilon}\_2 + 2\vec{V}\_I \tag{2}
$$

Where *VI* is the speed of the center of inertia of the whole moving system giving by the following equation:

$$V\_I = \frac{m\_1 v\_1 + m\_2 v\_2}{m\_1 + m\_2} \tag{3}$$

By considering a mass *m2* much smaller than the mass *m1* the center of inertia speed becomes: *V v <sup>I</sup>* <sup>1</sup>

It can be seen that the mass *m2* changes its movement direction after the collision and goes back with a higher speed (*v'2*). A speed gain of twice the speed of the mass *m1* before the collision is reached. Considering the mass *m1* much greater than the mass *m2*, the achieved gain in terms of kinetic energy *E1*-*E'1* of the seismic mass *m2* is as follows:

*Before collision:* 

$$E\_1 = \frac{1}{2}m\_2v\_2^2\tag{4}$$

*After collision* 

242 Small-Scale Energy Harvesting

equations:

following equation:

becomes: *V v <sup>I</sup>* <sup>1</sup>

*Before collision:* 

**Figure 3.** Illustration of the direct collision between two masses

*m1*

*m1*

of movement (*mi*.*vi*),. This leads to the final speed *v'2* of the seismic mass:

The conservation of the kinetic energy before and after the collision leads to the following

*m2*

*m2*

11 22 1 1 22 22 2 2 11 22 1 1 2 2

Where: *v1* and *v2* are the speed of the mass *m1* and *m2,*respectively, *p1* and *p2* are the quantity

Where *VI* is the speed of the center of inertia of the whole moving system giving by the

*mv mv <sup>V</sup> m m*

By considering a mass *m2* much smaller than the mass *m1* the center of inertia speed

It can be seen that the mass *m2* changes its movement direction after the collision and goes back with a higher speed (*v'2*). A speed gain of twice the speed of the mass *m1* before the collision is reached. Considering the mass *m1* much greater than the mass *m2*, the achieved

> 1 22 1 2

2

11 22 1 2

*mp mp mp mp mv mv mv mv* 

'

*I*

gain in terms of kinetic energy *E1*-*E'1* of the seismic mass *m2* is as follows:

' ' ' '

2 2 <sup>2</sup> *<sup>I</sup> v vV* (2)

Before collision

After collision

(3)

*E mv* (4)

(1)

$$E\_1 = \frac{1}{2} m\_2 v\_2^{\prime 2} = \frac{1}{2} m\_2 v\_2^2 + 2 m\_2 v\_1^2 + 2 m\_2 v\_1 v\_2 \tag{5}$$

This energy gain is proportional to the square of the vibration source speed *v1* and to the product between the vibration source speed *v1* and the seismic mass speed *v2*. As soon as the seismic mass reaches a speed higher than the vibration source one, the energy gain becomes mainly related to the last term. Higher the initial speed (or initial energy) of the seismic mass is, higher the energy gain is, like a resonant mechanism on a half period.

This is the basic idea of the present approach. Let us investigate in more detail how it could be possible to implement this approach with a real VEH to amplify the movement of the seismic mass on a random type vibration source.

#### *3.1.2. Application of the rebound mechanism for VEH*

The process of rebound mechanism to implement with the VEH will help to extract more energy from the environment at off resonance for harmonic signals and even for random vibration. To understand the operating principle of this technique, we will look in more detail at the mechanical behavior of the VEH at or close to the resonance. Consider the equivalent model of a converter consisting of a spring with a stiffness *k*, attached from one side to a seismic mass *m,* and from the other side to the vibration source. The mass is also attached to the vibration source via damper as shown in Figure 4. At the resonance, the speed of the source and the effort imposed by the spring on the vibration source are in phase opposition. In other words, the source provides a mechanical work to cantilever and not the reverse. Hence, maximum of mechanical energy is transferred from the source to the resonant system (resonance phenomenon). The system is then able to extract more energy from the vibration source such that the relative displacement is larger (the force exerted by the spring is important), which means that the quality factor is high. However, at off resonance, the vibration source displacement and the effort imposed by the spring on the support are not synchronized, reducing then the average power transferred to the seismic mass. Hence, only a small amount of energy is absorbed by the mass spring system from the vibration source.

By considering the previous idea, more absorption of mechanical work from the environment at off resonance can be ensured by synchronizing a rebound of the seismic mass on the vibration source when the vibration source speed is in opposite direction with the seismic mass speed, the kinetic energy gain of the seismic mass is tightly related to the vibration source speed and to the initial seismic mass speed (as given by the equation (5)), it is more convenient to synchronize the rebound to a speed extremum of the vibration source. This will amplify the relative speed of the seismic mass and then the absorbed energy from the vibration source. The energy absorbed from the vibration source increases gradually, higher the seismic mass speed becomes, higher the extracted mechanical energy is. When the speed of the seismic mass is greater, the force exerted on the source during the rebound is higher and then mechanical work extracted from the vibration source is higher too. Obviously, there is a physical limit to this amplification mechanism, this limit is fixed by the damping coefficient of the structure; a structure with a high quality factor will allow more amplification and vice versa, like for a resonance mechanism. To validate the principle, the rebound mechanism is introduced to a mass spring harvester system as shown in Figure 4. The spring *k1* represents the system guidance that has minimal spring stiffness. The rebound is applied by connecting mechanically the vibration source to a the seismic mass via a large stiffness spring *k2*, This connection is ensured by using two actuators *Act1* and *Act2,* as shown in the figure below. When these actuators are activated, the mechanical stiffness of the resonant system is modified (from *k1* to *k1*+*k2*~*k2*). Hence, the system will feature two natural frequencies: *fr1,* related to the stiffness *k1* when the spring *k2* is not connected and *fr2,* related to the stiffness *k2*.when the spring *k2* is connected.

**Figure 4.** Equivalent VEH model with the rebound process

For a non damped collision, one can obtain a theoretical speed gain of the seismic mass twice the speed of the vibration source during the rebound, (as explained above). The next section gives an over view about the different techniques that could be used for actuating the rebound.

#### *3.1.3. Rebound mechanism choice*

Many types of mechanisms could be used for applying the rebound. This mechanism should be able to apply the rebound at any time by connecting on demand the seismic mass to the vibration source via a spring of stiffness *k2* significantly higher than *k1*. Hereafter the different types of mechanism that could be used:

*Thermal actuation* 

A current is applied in a thermal resistance when the vibration source speed reaches its maximum value; the thermal material expands and then makes a connection with the seismic mass. The main disadvantage of this technique is the reaction time of the material and its power consumption.

#### *Electromagnetic actuation*

244 Small-Scale Energy Harvesting

the stiffness *k2*.when the spring *k2* is connected.

**Figure 4.** Equivalent VEH model with the rebound process

different types of mechanism that could be used:

the rebound.

*Thermal actuation* 

*3.1.3. Rebound mechanism choice* 

is higher and then mechanical work extracted from the vibration source is higher too. Obviously, there is a physical limit to this amplification mechanism, this limit is fixed by the damping coefficient of the structure; a structure with a high quality factor will allow more amplification and vice versa, like for a resonance mechanism. To validate the principle, the rebound mechanism is introduced to a mass spring harvester system as shown in Figure 4. The spring *k1* represents the system guidance that has minimal spring stiffness. The rebound is applied by connecting mechanically the vibration source to a the seismic mass via a large stiffness spring *k2*, This connection is ensured by using two actuators *Act1* and *Act2,* as shown in the figure below. When these actuators are activated, the mechanical stiffness of the resonant system is modified (from *k1* to *k1*+*k2*~*k2*). Hence, the system will feature two natural frequencies: *fr1,* related to the stiffness *k1* when the spring *k2* is not connected and *fr2,* related to

For a non damped collision, one can obtain a theoretical speed gain of the seismic mass twice the speed of the vibration source during the rebound, (as explained above). The next section gives an over view about the different techniques that could be used for actuating

 *m* 

*k2b k2a*

*z(t) <sup>A</sup> Act1 ct2*

*b* 

*k2*

*y(t)* 

*k1* 

Many types of mechanisms could be used for applying the rebound. This mechanism should be able to apply the rebound at any time by connecting on demand the seismic mass to the vibration source via a spring of stiffness *k2* significantly higher than *k1*. Hereafter the It is also possible to actuate the rebound by using an electromagnetic actuator composed of a coil and a core; such a method is used for breaking motors. When the coil is powered, the magnetic core moves and blocks the seismic mass. The main disadvantage of this approach is the power consumption which is relatively high compared to the power that can be scavenged (<1 mW for a centimeter scale device).

*Piezoelectric actuation* 

A third solution is to use a piezoelectric actuator enabling a short displacement with a high effort in a good agreement with the need to efficiently pinch the seismic mass. Furthermore the reaction time is very short compared to the blocking time (100 µs to few ms) and its consumption is relatively low (capacitive mechanism).

The piezoelectric actuation has been adopted to implement the present idea thanks to its accuracy, time of response, low power consumption and its compatibility with the application. The piezoelectric actuators chosen are a linear actuators developed by CEDRAT Technology (APA400M) placed on each side of the seismic mass.

#### *3.1.4. Simulations results*

The time simulation diagram is presented in Figure 5. This diagram is composed of two working phases. The first one is used when the resonant frequency of the system is *fr1*  (rebound process deactivated), and the second phase is used when the resonant frequency is *fr2* (the rebound process activated).

First, it is supposed that the system starts oscillating at a random frequency, different than the resonant one. The algorithm starts computing the speed and the displacement of the seismic mass. When a maximum speed of the vibration source is detected, the rebound is activated, the simulation phase is then changed, the simulation jumps to the second phase. This latter remains a certain period of time (the maintain of the connection of *k2* with the seismic mass). After that, the system goes back to the first phase (seismic mass connected to the spring *k1* only). This operation is repeated as often as the vibration source reaches some maximum speed, which means twice a period of vibration. The transition from one phase to another updates the initial condition of the new phase from the final position at the previous phase (initial displacement and speed, of both the seismic mass and the vibration source).

This simulation process has been used in different conditions in terms of vibration frequency, of rebound time duration and mechanical quality factor. After a deep investigation, some criteria have been established allowing an optimal amplification of the seismic mass's relative displacement:

**Figure 5.** Rebound simulation diagram

*The rebound time duration* 

The simulation results show that the optimal rebound time duration is related to the resonant frequency *fr2* occurring during the rebound time:

$$
\Delta t = \frac{1}{2f\_{r2}}\tag{6}
$$

In fact, the rebound is a compression/decompression cycle of the spring *k2*, corresponding to the half of the resonant period occurring during the rebound time.

*The second resonant frequency* 

The simulation shows that the operating frequency bandwidth where there is a positive gain in terms of relative displacement is limited between *fr1* and *fr2*/2. Hence, the higher the distance between *fr1* and *fr2* is, the higher the bandwidth of the harvester is. To enlarge the operating frequency bandwidth, it is interesting to choose a large resonant frequency *fr2*. However, a high resonant frequency *fr2* implies short rebound duration, this will conduct to more difficulties to actuate the rebound. Thus, a tradeoff has been made between the bandwidth and the mechanical challenges to reduce the rebound time.

The first resonant frequency *fr1*, while *k2* is open, has been fixed at 50Hz and the second one, during the rebound, at 200Hz.

The Figure 6 below presents the seismic mass speed amplification reached at the following conditions:


**Table 1.** Simulation parameters for rebound system

246 Small-Scale Energy Harvesting

**Figure 5.** Rebound simulation diagram

resonant frequency *fr2* occurring during the rebound time:

Re-initialization

the half of the resonant period occurring during the rebound time.

bandwidth and the mechanical challenges to reduce the rebound time.

The simulation results show that the optimal rebound time duration is related to the

*t*

Initialization

Determination of

Maximum speed detection

Re-initialization

Determination of

Rebound time duration

(6)

Phase 1

Phase 2

*f*

In fact, the rebound is a compression/decompression cycle of the spring *k2*, corresponding to

The simulation shows that the operating frequency bandwidth where there is a positive gain in terms of relative displacement is limited between *fr1* and *fr2*/2. Hence, the higher the distance between *fr1* and *fr2* is, the higher the bandwidth of the harvester is. To enlarge the operating frequency bandwidth, it is interesting to choose a large resonant frequency *fr2*. However, a high resonant frequency *fr2* implies short rebound duration, this will conduct to more difficulties to actuate the rebound. Thus, a tradeoff has been made between the

*The rebound time duration* 

*The second resonant frequency* 

The figure below shows the transient behavior of the seismic mass speed for different quality factor values. The maximal amplification is related to the mechanical quality factor. Hence, higher the quality factor is, higher the amplification gain is.

**Figure 6.** Seismic mass speed amplification (*Vm*: Seismic mass speed and *Vs*: vibration source speed)

The next section gives some details about the electronic attended to ensure the control of the actuators.

#### *3.1.5. Drive electronic*

The aim of this electronic is to deliver the command signals to activate and deactivate the actuators (act1 and act2). These signals are provided after measuring and processing the acceleration signal of the vibration source. Hence, the setting of electronic components is based on the following specifications:


The selected electronic architecture is composed of 5 stages placed in series:

**Figure 7.** Synoptic scheme of the selected electronic

*Phase shift* 

It is worth to remind that the rebound time duration is extremely low compared to the vibration period. The present approach needs then an extreme accuracy in terms time of activating/deactivating the actuators. However, the rebound is actuated after processing the acceleration signal. Hence, this operation will introduce a delay on the command signals. In order to overcome this limitation, a phase shift is added in order to make compensation to the delay introduced by the electronic. Hence, the information arrives to the last stage at the right moment.

*Zero detection* 

Detecting a vibration source speed extremum is equivalent to detect the zero acceleration of the input vibration. The second stage is then attended to detect zero acceleration on the vibration source. This is done by comparing the acceleration signal with zero using a comparator, the output voltage of the comparator changes from 0 to *Vcc* (for positive speed maximum) or from *Vcc* to 0 Volt (for negative speed maximum).

*Pulse generation* 

The third stage is attended to generate a pulse of an accurate duration each time the signal delivered by the previous stage change of state (rising or falling edge).

*Power circuit* 

The power circuit contains the switches to power from an external source to the actuators used for processing to the rebound.

The presented electronic was developed and tested with the mechanical system; the next part presents the experimental results

#### *3.1.6. Experimental validation:*

The experimental setup is shown in Figure 8 below. The cantilever is represented by its equivalent model, which is composed of a mass, a spring and a damper system. All these components are enclosed in the casing which is mechanically connected to the vibration source.

The device *(a)* is a Laser vibrometer (type: LSV250) connected to a computer in order to measure the displacement magnitude of the seismic mass. The acceleration of the vibration source is measured by an accelerometer *(e),* the measured signal is provided to the electronic

Strategies for Wideband Mechanical Energy Harvester 249

**Figure 8.** Experimental setup scheme for the rebound technique validation

*(c)* described in the previous section. This electronic processes the measured signal and generates the driving signals to the actuators (APA400M) *(f)* for connecting or disconnecting the spring *k2* relaying the seismic mass to the vibration source.

The manufactured structure is shown in the Figure 9 followed by a brief definition of the different components.

**Figure 9.** Picture of the fabricated structure

248 Small-Scale Energy Harvesting

*Phase shift* 

right moment.

*Zero detection* 

*Pulse generation* 

*Power circuit* 

source.

used for processing to the rebound.

part presents the experimental results

*3.1.6. Experimental validation:* 

**Figure 7.** Synoptic scheme of the selected electronic

The selected electronic architecture is composed of 5 stages placed in series:

It is worth to remind that the rebound time duration is extremely low compared to the vibration period. The present approach needs then an extreme accuracy in terms time of activating/deactivating the actuators. However, the rebound is actuated after processing the acceleration signal. Hence, this operation will introduce a delay on the command signals. In order to overcome this limitation, a phase shift is added in order to make compensation to the delay introduced by the electronic. Hence, the information arrives to the last stage at the

Accelerometer Phase shift Zero detection Pulse generation Actuators

Detecting a vibration source speed extremum is equivalent to detect the zero acceleration of the input vibration. The second stage is then attended to detect zero acceleration on the vibration source. This is done by comparing the acceleration signal with zero using a comparator, the output voltage of the comparator changes from 0 to *Vcc* (for positive speed

The third stage is attended to generate a pulse of an accurate duration each time the signal

The power circuit contains the switches to power from an external source to the actuators

The presented electronic was developed and tested with the mechanical system; the next

The experimental setup is shown in Figure 8 below. The cantilever is represented by its equivalent model, which is composed of a mass, a spring and a damper system. All these components are enclosed in the casing which is mechanically connected to the vibration

The device *(a)* is a Laser vibrometer (type: LSV250) connected to a computer in order to measure the displacement magnitude of the seismic mass. The acceleration of the vibration source is measured by an accelerometer *(e),* the measured signal is provided to the electronic

maximum) or from *Vcc* to 0 Volt (for negative speed maximum).

delivered by the previous stage change of state (rising or falling edge).



The Figure 10 shows the displacement gain achieved by the present approach as a function of the input vibration frequency compared to same system without the rebound mechanism. This gain is defined as the ratio between the relative displacement at off resonance obtained by using the present approach over the displacement obtained without activating the rebound when the maximum of speed is occurred. As expected by theory, the gain depends effectively on the input frequency, the gain is more important for frequencies much higher than the first resonant frequency because the amplitude of the relative displacement is close to the physical maximum since the vibration frequency is close the resonant one. This figure shows a difference between the theoretical expectation and the experimental results in terms of displacement gain due to the fact that in the theoretical study the damping induced by the actuators themselves was not taken into account.

**Figure 10.** Experimental and theoretical results of the relative displacement gain achieved by the rebound technique

These results present a great advantage of the rebound technique for the increase of the VEH efficiency over a wide frequency band. The amplification of the seismic mass displacement allows more mechanical energy extraction when the resonant frequency is not equal to the input one. Nevertheless, the drawback that remains for the present approach is a large power consumption to actuate the piezoelectric actuators. The energy required for each rebound is estimated at 200µJ. Further works are under investigation in order to reduce as low as possible this consumption.

### *3.1.7. Conclusions*

250 Small-Scale Energy Harvesting

rebound technique

Seismic mass displacement gain

Components Definition

the actuators themselves was not taken into account.

Theory Experiment

**Table 2.** Structure components definition

Tip mass

Acceleration sensor

Cantilever made of stainless *k1*

Piezoelectric actuators APA 400M

The Figure 10 shows the displacement gain achieved by the present approach as a function of the input vibration frequency compared to same system without the rebound mechanism. This gain is defined as the ratio between the relative displacement at off resonance obtained by using the present approach over the displacement obtained without activating the rebound when the maximum of speed is occurred. As expected by theory, the gain depends effectively on the input frequency, the gain is more important for frequencies much higher than the first resonant frequency because the amplitude of the relative displacement is close to the physical maximum since the vibration frequency is close the resonant one. This figure shows a difference between the theoretical expectation and the experimental results in terms of displacement gain due to the fact that in the theoretical study the damping induced by

**Figure 10.** Experimental and theoretical results of the relative displacement gain achieved by the

These results present a great advantage of the rebound technique for the increase of the VEH efficiency over a wide frequency band. The amplification of the seismic mass displacement allows more mechanical energy extraction when the resonant frequency is not equal to the input one. Nevertheless, the drawback that remains for the present approach is

<sup>45</sup> <sup>50</sup> <sup>55</sup> <sup>60</sup> <sup>65</sup> <sup>70</sup> <sup>75</sup> <sup>80</sup> <sup>85</sup> <sup>90</sup> <sup>0</sup>

Vibration frequency (Hz)

The casing/support connected to the vibration source

In this part of the chapter, a new approach for amplifying the movement of the seismic mass at off resonance have been shown by theory and experiments. A relative displacement seven times higher than the one achieved with a single resonator at off resonant frequency (90Hz) was shown. This best gain occurs at twice the natural frequency of the structure. This relative displacement gain corresponds to an output electrical power gain equal to 49, which represents a good prospect in the field of energy harvesting. This technique ensures a dynamic amplification of the relative displacement in real time, with a high efficiency without the need of a control loop, a simple measure of the sign of the acceleration is sufficient to control the whole system. Nevertheless, the electrical consumption of the actuator applying the rebound is still too large to make the system completely autonomous. This first demonstrator validate the principle with an actuator over-sized, a significant reduce of its consumption promises a good perspective in this rebound technique.

If this rebound mechanism can be applied for a large number of vibration types (up to random vibrations), it is nevertheless interesting to inspect if other techniques, with narrower applications but easier to use, can be used to enlarge the frequency response. The next part presents two ways to follow a main vibration frequency that moves during the time.

### **3.2. Active tuning of the resonant frequency:**

In the present section, two approaches for a dynamic tuning of the resonant frequency are given. These techniques could be applicable where the main vibration frequency is susceptible to change over time. It could be used for vibrations spread over a wide bandwidth as well, except that in this case the system will track only one main frequency. For both techniques given in what follows, the idea is to make a tuning of the resonant frequency by changing the stiffness of a piezoelectric material.

The most used structure shape for piezoelectric transduction in case of harvesting mechanical vibration is presented by the figure below:

**Figure 11.** Piezoelectric cantilever shape

The structure is composed of three main components:


All these components are bounded together as shown in the previous figure.

One way to quantify the structure capability to change its resonant frequency is to estimate resonant frequency tuning ratio. For a cantilever based piezoelectric structure, the tuning ratio is given by the equation below:

$$\frac{f\_{\text{max}} - f\_{\text{min}}}{f\_{\text{min}}} = \sqrt{\frac{I\_b + 2\mathbf{x}\_1 \mathbf{x}\_0 I\_p}{I\_b + 2\mathbf{x}\_0 I\_p}} \frac{I\_b \frac{L^3}{3} - 2\mathbf{x}\_0 I\_p \left(L^2 L\_p + \frac{L\_p^3 - L^3}{3} - L L\_p^2\right)}{I\_b \frac{L^3}{3} - 2\mathbf{x}\_1 \mathbf{x}\_0 I\_p \left(L^2 L\_p + \frac{L\_p^3 - L^3}{3} - L L\_p^2\right)}} - 1} \tag{7}$$

*Ib* and *Ip* represent the moment of inertia of the substrate and the piezoelectric part, respectively.

*L* and *Lp* represent the length of the beam and the piezoelectric layers, respectively.

$$\mathbf{x}\_0 = \frac{Y\_{p\text{ min}}}{Y\_b}, \quad \mathbf{x}\_1 = \frac{Y\_{p\text{ max}}}{Y\_{p\text{ min}}}$$

*Yp* min and *Yp*max: the minimal and maximal value of the piezoelectric Young's modulus.

*Yb*: the Young's modulus of the substrate

Equation (7) shows that the tuning ratio depends on the sizes of the structure, on mechanical material properties (*x0)* and electromechanical properties (*x1)*. The choice of the piezoelectric material is based on its Young's modulus sensibility to the external conditions, a high sensibility will allow more change of the piezoelectric Young's modulus, and then a high tuning ratio.

#### *3.2.1. Application of a DC electric field*

#### *3.2.1.1. Introduction*

Mechanical and electromechanical properties of piezoelectric materials depend on external constraints. Among these properties, there is the effect of an applied electric field on the stiffness of the piezoelectric material. Thus, using a good piezoelectric material, in terms of stiffness variation under a static electric field, a high resonant frequency tuning ratio could be achieved. Adjusting the level of the applied electric field will adjust the resonant frequency and hence a controllable resonant frequency VEH could be achieved.

#### *3.2.1.2. Theory of the approach*

252 Small-Scale Energy Harvesting

The structure is composed of three main components:

extracted energy into electrical one.

the relative displacement.

ratio is given by the equation below:

*Yb*: the Young's modulus of the substrate

*3.2.1. Application of a DC electric field* 

*3.2.1.1. Introduction* 

respectively.

max min 1 0

*I xxI f f*

2

*b p*

 *The substrate*: the substrate is usually added to piezoelectric harvester for two aims: enhancing the effective mechanical quality factor of the structure, removing the stress neutral line of the whole structure outside the symmetry axes of the piezoelectric part in order not to reduce the output generated power by electrical charges compensation. *The piezoelectric part*: stressed under mechanical stress, it converts the mechanical

*The seismic mass*: used to adjust the resonant frequency and enhance the amplitude of

One way to quantify the structure capability to change its resonant frequency is to estimate resonant frequency tuning ratio. For a cantilever based piezoelectric structure, the tuning

3 3 <sup>3</sup> min <sup>0</sup> 2 2

*f I xI <sup>L</sup> L L*

*L* and *Lp* represent the length of the beam and the piezoelectric layers, respectively.

0 1

*Yp* min and *Yp*max: the minimal and maximal value of the piezoelectric Young's modulus.

*x x Y Y*

2

*Ib* and *Ip* represent the moment of inertia of the substrate and the piezoelectric part,

*b p p*

3 3 3

*L L L I xI LL LL*

*b pp p*

*b pp p*

*I xxI LL LL*

0

2 2 3 3

1 0

min max

, *p p b p*

*Y Y*

Equation (7) shows that the tuning ratio depends on the sizes of the structure, on mechanical material properties (*x0)* and electromechanical properties (*x1)*. The choice of the piezoelectric material is based on its Young's modulus sensibility to the external conditions, a high sensibility will allow more change of the piezoelectric Young's modulus, and then a high tuning ratio.

Mechanical and electromechanical properties of piezoelectric materials depend on external constraints. Among these properties, there is the effect of an applied electric field on the stiffness of the piezoelectric material. Thus, using a good piezoelectric material, in terms of

3 3

min

2 2

*p*

1

(7)

All these components are bounded together as shown in the previous figure.

The dependence that exists between the stiffness of the piezoelectric material and the strength of the applied electric field could be noted from the complete equation of piezoelectricity given by the following expression:

$$\varepsilon\_{lj} = \mathbf{s}\_{ljlm}^E \sigma\_{lm} + d\_{ljn} E\_n + \frac{1}{2} \tau\_{ljlmpq}^E \sigma\_{lm} \sigma\_{pq} + \frac{1}{2} a\_{ljnr} E\_n E\_r + \kappa\_{ljlmn} \sigma\_{lm} E\_n \tag{8}$$

This equation features two different parts, the first one relies the deflection to the stress by a constant parameter (*sE*), while the second one shows a non constant coefficient of proportionality between the deflection and the mechanical stress, this non constant parameters depends on the applied electric field (*En*). This effect reflects the non linear behavior of piezoelectricity when it is subjected to a DC electric field. This effect varies from one type of piezoelectric material to another; it depends also on how the electric field is applied on the material in terms of strength and direction.

Equation (8) is the general piezoelectric equation, but considering our cantilever design, some assumptions can be taken into account: the mechanical behavior of the cantilever is elastic, only one component of the electric field vector of the stress and strain tensors are taken into account. Hence, this leads to the following simplified equation relating the stress to the strain and the applied electric field:

$$
\sigma\_x = s\_{11}\sigma\_x + d\_{31}E\_3 + \frac{1}{2}a\_{113}E\_3^2 + \kappa\_{113}\sigma\_x E\_3 \tag{9}
$$

This leads to the following expression of the mechanical stress:

$$
\sigma\_x = \frac{1}{s\_{11} + \kappa\_{113}E\_3} \varepsilon\_x - \frac{d\_{31}E\_3 + \frac{1}{2}a\_{113}E\_3^2}{s\_{11} + \kappa\_{113}E\_3} \tag{10}
$$

The expression relaying the stress to the strain shows clearly the dependence that exists between the applied electric field and the stiffness of the piezoelectric material. However, it is difficult to determine the parameters appearing in this model, in most cases they are not provided by suppliers. Another simple equivalent model has been proposed by Thornburgh et al [20], reflecting the same effect by using a simple relation. This model is based on the linear constitutive equations of the piezoelectric material, we keep only the first and the second terms of the equation (9). Except that the effect of the DC electric field is reflected on the piezoelectric coefficient (*d31)*. The value of this one depends on the level of the applied electric field. The equation showing this dependence is given by (11), where the *d31* is the piezoelectric strain coefficient at *0kV/cm* and *q31* is called piezoelasticity coefficient:

$$d\_{31}^{\*} = d\_{31} + q\_{31} \varepsilon\_{x} \tag{11}$$

Using this relation, the stress function of the strain can be expressed as:

$$
\sigma\_x = \overset{\circ}{Y}\_p \varepsilon\_x - \beta \tag{12}
$$

�� � : is the expression of the piezoelectric effective Young's modulus as a function of the applied electric field and is expressed as follows:

$$\text{With } Y\_p' = \frac{1 - q\_{31}E\_3}{s\_{11}} \text{ and } \beta = d\_{13}E\_3.$$

For a bimorph cantilever shape, the resonant frequency is then expressed as follows:

$$f\_r \frac{1}{2\pi} = \sqrt{\frac{w}{4(M + 0.24M\_b)L^3} \left( Y\_p (1 - q\_{31}E\_3) \left( 6t\_s^2 t\_p + 12t\_s t\_p^2 + 8t\_p^3 \right) + Y\_s t\_s^3 \right)}\tag{13}$$

The optimization of the tuning ratio involves in first step the choice of a piezoelectric material. This material should allow a high resonant frequency shift under a low applied DC electric field without compromising the efficiency of the mechanical-to-electrical conversion. Finding a piezoelectric material having a high electric coupling can induce a material with a low quality factor reducing the harvesting power, it is then important to find a material with a good compromise, we introduce then a new figure of merit for choosing the piezoelectric material taking into account the following parameters:

*The electromechanical coupling coefficient k31:* 

It is necessary to choose a material with a high electromechanical coupling coefficient, this will improve the efficiency of the electromechanical conversion.

*The maximum electrical field supported by the piezoelectric material (Emax and Emin):* 

As it can be seen from the expression of the resonant frequency (13), the highest the supported electric field is, the highest and the tuning ratio is. It is then important to take into account the limits of the applied electric field imposed by the piezoelectric material.

*The coefficient of piezo-elasticity:* 

The effect of the applied electric field on the stiffness of the piezoelectric material is described by the coefficient of the piezo-elasticity *q31* as shown by equation (11). This means that a material with a high coefficient of piezoelasticity will provide a high resonant frequency tuning ratio.

*Dielectric losses coefficient:* 

The limiting factor of the present approach is the dielectric losses of the material. A material with a high dielectric losses coefficient presents more leakage current. This will induce higher power consumption for the management electronic of the system. It is then important to choose a material with low losses.

The figure of merit taking into account the different constraints above, can be expressed as:

$$\mathcal{A}\_{p-1} = \frac{k\_{31}^2 \left(E\_{\text{max}} - E\_{\text{min}}\right)}{\tan\left(\delta\right)} q\_{31} \tag{14}$$

After an overview of the most used piezoelectric material, it was found that the best material for this application is the PZN-PT, allowing the best compromise between the resonant frequency tuning and the electromechanical conversion. Despite the best performance of this material, it remains actually quite expensive compared to others.

#### *3.2.1.3. The experimental validation of the approach:*

*The manufactured device* 

254 Small-Scale Energy Harvesting

� <sup>=</sup> ������� ���

> �� 1

�� �

With ��

\*

 *x px Y* 

For a bimorph cantilever shape, the resonant frequency is then expressed as follows:

4�� � ���4����� ����1�����������

Using this relation, the stress function of the strain can be expressed as:

applied electric field and is expressed as follows:

and �=�����

material taking into account the following parameters:

will improve the efficiency of the electromechanical conversion.

*The electromechanical coupling coefficient k31:* 

*The coefficient of piezo-elasticity:* 

frequency tuning ratio.

*Dielectric losses coefficient:* 

to choose a material with low losses.

�� <sup>=</sup> � �

31 31 31 *<sup>x</sup> ddq*

'

: is the expression of the piezoelectric effective Young's modulus as a function of the

The optimization of the tuning ratio involves in first step the choice of a piezoelectric material. This material should allow a high resonant frequency shift under a low applied DC electric field without compromising the efficiency of the mechanical-to-electrical conversion. Finding a piezoelectric material having a high electric coupling can induce a material with a low quality factor reducing the harvesting power, it is then important to find a material with a good compromise, we introduce then a new figure of merit for choosing the piezoelectric

It is necessary to choose a material with a high electromechanical coupling coefficient, this

As it can be seen from the expression of the resonant frequency (13), the highest the supported electric field is, the highest and the tuning ratio is. It is then important to take into

The effect of the applied electric field on the stiffness of the piezoelectric material is described by the coefficient of the piezo-elasticity *q31* as shown by equation (11). This means that a material with a high coefficient of piezoelasticity will provide a high resonant

The limiting factor of the present approach is the dielectric losses of the material. A material with a high dielectric losses coefficient presents more leakage current. This will induce higher power consumption for the management electronic of the system. It is then important

*The maximum electrical field supported by the piezoelectric material (Emax and Emin):* 

account the limits of the applied electric field imposed by the piezoelectric material.

��� � 1�����

(11)

(12)

� � ���

�������

�� (13)

The Figure 12 is a picture of the fabricated structure, it consists on a bimorph cantilever shape. The structure sizes and the main electromechanical properties are presented in the table below:

**Figure 12.** Picture of the fabricated structure


**Table 3.** Characteristics of the fabricated structure

*The experimental results:* 

The Figure 13 presents the relative displacement amplitude as a function of the input frequency for different applied DC voltage on the piezoelectric layers. It presents also the structure resonant frequency as a function of the applied DC voltage. A tuning ratio of the resonant frequency up to 20% has been obtained by applying an electric field from -1 to +6kV/cm. Theory shows that a tuning ratio of 26% should be reached in these conditions, this discrepancy is due to the errors introduced during the fabrication process. The bounding of the different layers has been done using an adhesive. The given theoretical model does not take into account the parameters of this adhesive, which explain for the main part the difference between the experimental and theoretical results.

**Figure 13.** The effect of the applied DC electric field on the resonant frequency

The most important challenge while designing a VEH able to adjust its resonance frequency automatically, is the power balance between the converter output power and the power required to drive the frequency tuning. As explained before, most of developed wideband VEH have a negative power balance. In the next section, a new low power consumption electronic is proposed, this electronic is under development within the *CEA-Leti*, it allows a dynamic tuning of the resonant frequency by tracking the maximum output power point.

#### *3.2.1.4. The resonant frequency tuning electrical circuit*

The drive electrical circuit is composed of two principal parts, the power circuit and the control circuit. The first one allows the flow of the power between an electrical energy storage element and the piezoelectric material in order to apply a DC electric field, while the second one controls the level of the applied electric field according to the shift that exists between the vibration frequency and the resonant one.

The block diagram of the control circuit is given by Figure 14 below. The aim of this electronic is to determine how much is the resonant frequency lower (or higher) than the vibration one. First of all, this shift is determined by measuring the phase difference between the acceleration of the vibration source and the piezoelectric voltage. At resonance the phase shift between these two signals is equal to a quarter of a period, when the resonant frequency is higher than the vibration one, this phase shift is higher than this quarter of a period and inversely when the resonant frequency is lower than the vibration one. Thus, after measuring this phase shift, two cases may occur: (i) the phase shift is lower than a quarter of a period, this means that the voltage already applied on the piezoelectric material is higher than the voltage that should be applied across the material (*fr*<*fvib*). (ii) the phase shift is higher than a quarter of a period, this means that the voltage already applied on the piezoelectric material is lower than the voltage which should be applied across the material (*fr*>*fvib*). In the first case, energy is transferred from the electrical energy storage element into the piezoelectric material, the switch *k1* is closed first, the close time of both switches should correspond exactly to the energy expected to be injected into the piezoelectric capacitance to reach the right DC voltage across the piezoelectric material. After switching on *k1* during the right time, the expected energy is stored into the magnetic core, it is then switched off and *k2* is switched on, the stored energy in the magnetic core is then injected in the piezoelectric capacitance and the voltage applied across the piezoelectric material attends its intended value. The process for the second case is the same as the first one, except in this case the energy transfer is made in the other direction in order to decrease the voltage across the piezoelectric material by closing first *k2* and then *k1*. The energy is then restored to the electrical energy storage. The power used to maintain the right voltage across the piezoelectric material is just the losses that occur during the power transfer in the fly-back converter.

**Figure 14.** Resonant frequency adaptation circuit

#### *3.2.1.5. The power balance*

256 Small-Scale Energy Harvesting

Tip mass displacement (µm)

resonant frequency up to 20% has been obtained by applying an electric field from -1 to +6kV/cm. Theory shows that a tuning ratio of 26% should be reached in these conditions, this discrepancy is due to the errors introduced during the fabrication process. The bounding of the different layers has been done using an adhesive. The given theoretical model does not take into account the parameters of this adhesive, which explain for the

main part the difference between the experimental and theoretical results.


**Figure 13.** The effect of the applied DC electric field on the resonant frequency

*3.2.1.4. The resonant frequency tuning electrical circuit* 

<sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>400</sup> <sup>0</sup>

Input frequency (Hz)

between the vibration frequency and the resonant one.

The most important challenge while designing a VEH able to adjust its resonance frequency automatically, is the power balance between the converter output power and the power required to drive the frequency tuning. As explained before, most of developed wideband VEH have a negative power balance. In the next section, a new low power consumption electronic is proposed, this electronic is under development within the *CEA-Leti*, it allows a dynamic tuning of the resonant frequency by tracking the maximum output power point.

260

270

280

Resonant frequency (Hz)

290

300


Applied electric field (kV/cm)

The drive electrical circuit is composed of two principal parts, the power circuit and the control circuit. The first one allows the flow of the power between an electrical energy storage element and the piezoelectric material in order to apply a DC electric field, while the second one controls the level of the applied electric field according to the shift that exists

The block diagram of the control circuit is given by Figure 14 below. The aim of this electronic is to determine how much is the resonant frequency lower (or higher) than the vibration one. First of all, this shift is determined by measuring the phase difference between the acceleration of the vibration source and the piezoelectric voltage. At resonance the phase shift between these two signals is equal to a quarter of a period, when the resonant frequency is higher than the vibration one, this phase shift is higher than this quarter of a period and inversely when the resonant frequency is lower than the vibration one. Thus, after measuring this phase shift, two cases may occur: (i) the phase shift is lower than a quarter of a period, this means that the voltage already applied on the piezoelectric material is higher than the The Figure 15 below shows the power balance of the whole VEH including the resonant frequency tuning system. It is assumed that the resonant frequency is adjusted by step of 0.1Hz and it can be changed up to 10 times a second. The accelerometer consumption is not taken into account.

The mechanical, dielectric and electrical losses are investable, these losses are common for all VEH systems. The most important point shown in this power balance is the positive net output power. The whole system is self sufficient including the power required by the electronic for racking the vibration frequency and also the power needed to apply the right DC voltage across the piezoelectric material.

**Figure 15.** Complete power balance of the whole system

This result is very promising for the next generation of piezoelectric VEH, it shows that the real time resonant frequency tuning can be energetically positive. Nevertheless, the actual piezoelectric material which is in good agreement with this technique is still expensive (few 100€ per cm3) and their quality factor is quite limited (< 100).

In the next part of this section, another solution for a dynamic resonant frequency adjustment is given. This solution is quite similar to the previous one, it consists to adjust the electrical load connected to the piezoelectric material in order to adjust the electrical stiffness of the piezoelectric material.

### *3.2.2. Adaptation of the electrical load [21]*

This subpart presents a declination of the previous solution for dynamic resonant frequency tuning. This one is based on the dependence that exists between the mechanical stiffness of a piezoelectric material and the electrical conditions to which it is subjected. In fact, the deflection of a piezoelectric bulk material under an applied mechanical force is higher when it is placed in short circuit (zero electric field) than in open circuit (no charge displacement), this means that the stiffness is lower in case of short circuit conditions than in open circuit conditions. The relation between the stiffness at short and open circuit is given by (14).

$$\mathbf{s}^{D} = \mathbf{s}^{E} - d^{2} \;/\; \mathbf{s}^{T} \Leftrightarrow \mathbf{s}^{E} = \frac{\mathbf{s}^{D}}{1 - k^{2}} \tag{15}$$

 with


One way to obtain a variable stiffness between these two limits (*sD* and *sE*) is to connect the piezoelectric material with an adjustable non-dissipative electrical load. However, this electrical load should not affect the quality factor of the structure. Thus, connecting a variable capacitor seems to be a good compromise, able to change the resonant frequency with less power losses. The value of the capacitance set the effective dielectric permittivity of the piezoelectric material and then the piezoelectric material stiffness as it is shown by the equation (16), where *Yp*: is the effective Young's modulus for the piezoelectric, *Cp* is the piezoelectric capacitance, *Csh* the capacitance connected with the piezoelectric material, and A is the effective section of the piezoelectric material. Hence, by adjusting the value of the connected capacitance (*Csh*), it is possible to adjust the value of the piezoelectric stiffness and then the resonant frequency.

$$Y\_p = \left(\mathbf{s}^{\varepsilon} - \frac{d\_{33}^2 A}{t\_p \left(\mathbf{C}\_p + \mathbf{C}\_{sh}\right)}\right)^{-1} \tag{16}$$

In case of a cantilever shape like in the Figure 11, the resonant frequency of the harvester becomes:

$$f = \frac{1}{2\pi} \sqrt{\frac{3\left(s^E - \frac{d\_{31}^2 A}{t\_p \left(\mathbb{C}\_p + \mathbb{C}\_s\right)}\right)^{-1} I\_p + Y\_b I\_b}{L^3 M}}\tag{17}$$

With *M*: the effective mass, *tp* : the thickness of the piezoelectric layers ans *L* : represents the beam length.

#### *3.2.2.1. Choice of the piezoelectric material*

As for the previous technique, to choose the suitable material we define a new figure of merit. This one is based on a compromise between the following parameters:

*High electromechanical coupling coefficient kij:* 

For the present approach, the electromechanical coupling has a significant effect on the resonant frequency tuning ratio as shown by the equation (14). It is better to choose a material with a high electromechanical coupling.

*Low dielectric losses:* 

258 Small-Scale Energy Harvesting

**Figure 15.** Complete power balance of the whole system

stiffness of the piezoelectric material.

with



*3.2.2. Adaptation of the electrical load [21]* 

100€ per cm3) and their quality factor is quite limited (< 100).

This result is very promising for the next generation of piezoelectric VEH, it shows that the real time resonant frequency tuning can be energetically positive. Nevertheless, the actual piezoelectric material which is in good agreement with this technique is still expensive (few

In the next part of this section, another solution for a dynamic resonant frequency adjustment is given. This solution is quite similar to the previous one, it consists to adjust the electrical load connected to the piezoelectric material in order to adjust the electrical

This subpart presents a declination of the previous solution for dynamic resonant frequency tuning. This one is based on the dependence that exists between the mechanical stiffness of a piezoelectric material and the electrical conditions to which it is subjected. In fact, the deflection of a piezoelectric bulk material under an applied mechanical force is higher when it is placed in short circuit (zero electric field) than in open circuit (no charge displacement), this means that the stiffness is lower in case of short circuit conditions than in open circuit conditions. The relation between the stiffness at short and open circuit is given by (14).

2


*DE T E <sup>s</sup> s sd s*

 

One way to obtain a variable stiffness between these two limits (*sD* and *sE*) is to connect the piezoelectric material with an adjustable non-dissipative electrical load. However, this electrical load should not affect the quality factor of the structure. Thus, connecting a variable capacitor seems to be a good compromise, able to change the resonant frequency

<sup>2</sup> / 1

*D*

*k*

(15)

In order to reduce as much as possible the power losses, it is necessary to choose a material with a low dielectric losses. But, unlike the previous approach, the coefficient of dielectric losses will have no effect on the power consumption of the power management electronic.

*The effective permittivity:* 

For the present approach, it is better to have a material with a high electrical capacitance, which means high dielectric permittivity because it minimizes the effect of the parasitic capacitances on the adjustment of the resonant frequency. If the parasitic capacitance is at the same order as the piezoelectric one, the variation of the shunt capacitance will have a minor effect on the tuning ratio of resonant frequency.

#### *The coupling mode:*

As the vibrations are supposed to be straight and unidirectional, two modes could be used for ensuring an efficient electromechanical coupling, the longitudinal mode (polarization and mechanical stress axes are collinear), or the transverse mode where the polarization and mechanical stress axes are perpendicular. As the electromechanical coupling is higher in the longitudinal mode, this one enables a better tuning ratio.

Finally, by taking into account all these parameters and their effect on resonant frequency tuning, the figure of merit is:

$$FOM = \frac{k\_{3/}^2 \mathcal{L}\_{33}^T}{\tan \left( \delta \right)}, j = 1, 3 \tag{18}$$


**Table 4.** Comparison between the different piezoelectric materials

The table above shows the *FOM* of three different piezoelectric materials in two different modes:

It can be noted from this table that the PZN-PT in longitudinal coupling mode presents the best figure of merit and seems to be the suitable material for this method of resonant frequency tuning.

The next part presents the experimental results obtained with this technique on a structure prototype.

#### *3.2.2.2. The experimental validation:*

*The manufactured structure* 

The piezoelectric prototype developed by CEA-Leti is presented Figure 16. It consists on a bimorph piezoelectric cantilever, each piezoelectric layer is composed of a number of subparts mechanically bounded together in series. The polarization axis of each subpart is oriented on the direction of the resulted mechanical stress in order to work in the longitudinal mode. The electrodes of the subparts are connected in parallel, in order to obtain the highest equivalent capacitance.

This structure has been mounted on a shaker, the piezoelectric part has been coupled with a variable capacitance, and the obtained experimental results are presented in the next section.

*The experimental results* 

The first measurements show that the resonant frequency is equal to 208 Hz at short circuit condition and 294 Hz at open circuit condition, which represents **41% of tuning ratio**. The


**Table 5.** Characteristics of the fabricated structure

*The coupling mode:* 

tuning, the figure of merit is:

modes:

frequency tuning.

*3.2.2.2. The experimental validation:* 

obtain the highest equivalent capacitance.

*The manufactured structure* 

*The experimental results* 

prototype.

As the vibrations are supposed to be straight and unidirectional, two modes could be used for ensuring an efficient electromechanical coupling, the longitudinal mode (polarization and mechanical stress axes are collinear), or the transverse mode where the polarization and mechanical stress axes are perpendicular. As the electromechanical coupling is higher in the

Finally, by taking into account all these parameters and their effect on resonant frequency

2

tan

F.O.M(x1e3) PZT PMN-PT PZN-PT Transversal mode (31) 39.85 104.54 204.12 Longitudinal Mode (33) 136.96 447.7 618.52

The table above shows the *FOM* of three different piezoelectric materials in two different

It can be noted from this table that the PZN-PT in longitudinal coupling mode presents the best figure of merit and seems to be the suitable material for this method of resonant

The next part presents the experimental results obtained with this technique on a structure

The piezoelectric prototype developed by CEA-Leti is presented Figure 16. It consists on a bimorph piezoelectric cantilever, each piezoelectric layer is composed of a number of subparts mechanically bounded together in series. The polarization axis of each subpart is oriented on the direction of the resulted mechanical stress in order to work in the longitudinal mode. The electrodes of the subparts are connected in parallel, in order to

This structure has been mounted on a shaker, the piezoelectric part has been coupled with a variable capacitance, and the obtained experimental results are presented in the next section.

The first measurements show that the resonant frequency is equal to 208 Hz at short circuit condition and 294 Hz at open circuit condition, which represents **41% of tuning ratio**. The

*T j k FOM j* 

3 33 . , 1,3

(18)

longitudinal mode, this one enables a better tuning ratio.

**Table 4.** Comparison between the different piezoelectric materials

**Figure 16.** Picture of the fabricated structure

Figure 17 below presents the resonant frequency as a function of the shunt capacitance. The capacitance value is normalized to the blocked capacitance of the piezoelectric material (*Cs/Cp*). The blocked piezoelectric capacitance *Cp* is equal to 1nF.

**Figure 17.** Resonant frequency as a function of the shunt capacitance

A tuning ratio up to 31% is noted when the shunt capacitance varies between 0.07*Cp* and 10 *Cp*. The power extracted using an optimal load is equal to 320µW for 0.1g@250Hz.

#### *3.2.2.3. Resonant frequency tuning drive electronic*

As mentioned before, the ultimate goal of the work is to develop an automatic system able to adjust in real time the harvester resonant frequency to the main frequency of the vibration source. The idea here is to couple the VEH with an adjustable capacitive load. The block diagram of the drive electronic is given by the following figure. The VEH is first connected to the adjustable capacitive load composed of two capacitances, *C1* and *C2*. It is supposed that the power conditioning circuit requires a very low voltage (≈3V) compared to the piezoelectric output voltage (>10V). Hence, the equivalent shunt capacitance that affects the piezoelectric stiffness is the sum of C1 with *C2*. Nevertheless, the capacitance *C2* enables a power transfer from the piezoelectric material to the power conditioning circuit and has more effect on the extracted electrical energy and then it enables an adjustment of the electrical damping.

**Figure 18.** Block diagram circuit for the resonant frequency tuning by adapting the electrical load

The objective of the drive electronic is to track the maximum output power flowing through the electrical load by adjusting the electrical damping (adjusting *C2*) and the resonant frequency (adjusting *C1* and *C2*). Hence, the reaction time of a change of *C2* is shorter than a change of *C1*, since the first has an effect of the transfer of energy while the second one on the resonant frequency. This electronic is composed of two loops, a slower one for adjusting the capacitance *C1* and a faster one to adjust the capacitance *C2*.

The first measurements show that the whole electronic consumes about 30µW which represents only 10% of the 300µW generated power.

### **4. Conclusions**

Through this chapter, it has been demonstrated that there is a real need to ensure a real time tracking of the vibration frequency. The issue of the adaptation between the input vibration characteristic and the mechanical characteristics of the VEH is capital for the development of robust and efficient VEH. Within this chapter, a special focus has been made on the solutions developed in *CEA-Leti* laboratory. Three solutions have been presented, the first one expects to amplify the relative displacement of any vibration type (random) occurring on a wide width frequency bandwidth, the second one consists to apply a DC electric field on the piezoelectric layer in order to adjust its stiffness and then the resonant frequency of the structure. The third solution consists to adjust the electrical load coupled with the piezoelectric material in order to adjust its stiffness and then the resonant frequency of the structure. Each solution has been validated experimentally, the first one enables an operation over one octave of frequency (50Hz to 100Hz), the second one a frequency tuning from (250Hz to 300Hz) and the last one from (210Hz to 280Hz). This chapter introduces also the drive electronic for each strategy, the drive electronic part is the most critical point for active techniques since it requires in most cases a huge amount of energy. First results obtained in CEA-Leti laboratory show that the resonant frequency tuning can significantly increase the net output power without consuming too much power to be managed (about 10% of the converter output power). Anyway, the successful of VEH systems is tightly related to the frequency bandwidth. So, the commercialization of such systems at large scale could not be done until overcoming efficiently the limits imposed by the bandwidth issue. Active techniques allowing positive power balance let us hope a large flexibility in the near future of VEH.

#### **Author details**

262 Small-Scale Energy Harvesting

electrical damping.

*C1*

Piezo layer

**4. Conclusions** 

*3.2.2.3. Resonant frequency tuning drive electronic* 

As mentioned before, the ultimate goal of the work is to develop an automatic system able to adjust in real time the harvester resonant frequency to the main frequency of the vibration source. The idea here is to couple the VEH with an adjustable capacitive load. The block diagram of the drive electronic is given by the following figure. The VEH is first connected to the adjustable capacitive load composed of two capacitances, *C1* and *C2*. It is supposed that the power conditioning circuit requires a very low voltage (≈3V) compared to the piezoelectric output voltage (>10V). Hence, the equivalent shunt capacitance that affects the piezoelectric stiffness is the sum of C1 with *C2*. Nevertheless, the capacitance *C2* enables a power transfer from the piezoelectric material to the power conditioning circuit and has more effect on the extracted electrical energy and then it enables an adjustment of the

**Figure 18.** Block diagram circuit for the resonant frequency tuning by adapting the electrical load

Set of the value of *C2* Set of the value of *C1*

*C2*

the capacitance *C1* and a faster one to adjust the capacitance *C2*.

represents only 10% of the 300µW generated power.

The objective of the drive electronic is to track the maximum output power flowing through the electrical load by adjusting the electrical damping (adjusting *C2*) and the resonant frequency (adjusting *C1* and *C2*). Hence, the reaction time of a change of *C2* is shorter than a change of *C1*, since the first has an effect of the transfer of energy while the second one on the resonant frequency. This electronic is composed of two loops, a slower one for adjusting

Power conditioning

> Maximum current tracking electronic

Signal conditioning

Electrical load

Load current

The first measurements show that the whole electronic consumes about 30µW which

Through this chapter, it has been demonstrated that there is a real need to ensure a real time tracking of the vibration frequency. The issue of the adaptation between the input vibration characteristic and the mechanical characteristics of the VEH is capital for the development of B. Ahmed Seddik, G. Despesse, S. Boisseau and E. Defay *LETI, CEA, Minatec Campus, Grenoble, France* 

#### **5. References**

