## 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-to-reach 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 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

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

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.

### 2.1. Electromechanical EOMs

In this derivation, the states of the electromechanical system are the following:

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

where

where

where

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

where

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

(5) |

where

(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

(7) |

where the constant multiplying the

### 2.2. Modal decoupling

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.

where *i*^{th} transverse mode shape function, and *i*^{th} modal displacement. Given the configuration in Fig. 1 with a tip mass having a nontrivial mass

where

(10) |

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

(11) |

where

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

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

It remains to write the applied magnetic force

where *x*-coordinate of the tip mass centroid. The magnitude of this force

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

## 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.

where

In order to calculate the frequency transfer function, it is assumed that the electrical load can be represented by a resistor with value

Since the system of equations (12,19) is linear, the modal responses

where

In order to use Eq. (18), an estimate of the spectral density of the base acceleration

where the component amplitudes

(23) |

Consider first the case

(25) |

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

(26) |

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

Then, by mathematical induction,

Using this result in Eq. (23) gives

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

Eq. (30) indicates that the frequency transfer function

where

where

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

where

### 3.2. Nonlinear case, time domain

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.

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

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

Linearizing Eq. (36) about the point

Thus, the term in parentheses is the square of the effective natural frequency,

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,

again assuming that the effect of the electromechanical coupling is negligible. First, consider the case when the base is moving upward, i.e.

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:

If the times of the plucks are denoted

A similar formula can be derived for the pluck times when

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

where

where

(45) |

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

(47) |

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

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

## 4. Simulated response to sinusoidal base excitation

In this section, the response of the system to sinusoidal base excitation,

The first three natural frequencies of the beam are

The transfer functions for power harvested (normalized by

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

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).

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.

## 6. 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 non-linear device in lieu of simulation of the full dynamics equations.