**3. Experimental Investigation**

In order to characterize the presented harvester design, we integrate three macro fiber composite (MFC) piezoelectric patches supplied by SMART MATERIAL Corp. (M-8507-P2 on the outer beams and M-8514-P2 on the inner beam) with a resonator fabricated from steel, as depicted in Figure 14. The MFC patches are composed of piezoelectric rods embedded between layers of adhesive, interdigitated electrodes, and encapsulated with polyimide film. The patch dimensions are 60 × 7 × 0.18 mm<sup>3</sup> and 48 × 14 × 0.18 mm<sup>3</sup> for the outer and inner beam, respectively. The material properties of the MFC patches are given in Table 1.

**Figure 14.** Dual frequency piezoelectric energy harvester with macro fiber composite (MFC) patches on outer and inner beams. Permanent magnets are attached at the beam ends. The base excitation is applied to the clamped part on the left side.

**Table 1.** Material properties of the active area of the macro fiber composite (MFC) patches.


The harvester has been excited at the acceleration amplitudes of 0.5 and 1.0 g. The results presented on Figure 15 demonstrate the dual frequency feature of the harvester. The experimentally observed resonance frequencies *f* 1 *Exp* = 63.27 and *f* 2 *Exp* = 76.35 Hz match the simulation results *f* 1 *Sim* = 64.30 and *f* 2 *Sim* = 77.50 Hz. However, a lower voltage output has been observed. We attribute this to the adhesive tape which attaches the patches to the steel. In our assembly this degrades the strain transfer between the steel resonator and the piezoelectric layers when compared to solid bonding, e.g., using glue. Our simulations considered the adhesive tape as a material of high compliance (*E* = 450 kPa). The FE model implements a constant damping ratio, which yields correct amplitudes at the first mode and does not describe the damping at the second mode. A mode-specific or even frequency dependent damping ratio shall be applied instead. Furthermore, the slight frequency shift (up to 1.63%) between the model and the experiment results is caused by the additional mass of the solder paste used to electrically connect the inner patch. The patch attachment procedure and the limited reproducibility of the magnets positioning contribute in turn to such a frequency shift.

**Figure 15.** (**a**) Experimentally obtained voltage at the excitation levels of 0.5 and 1.0 g. Comparison of experimental data and (**b**) simulation results for an excitation level of 0.5 g.

We evaluated the harvester's power delivery using the power managemen<sup>t</sup> board 2151A provided by analog devices (see Figure 16), which also enables battery charging. The board integrates the LTC3331 chip, which provides a regulated voltage from various energy harvesting sources. The circuitry consists of an integrated low-loss full-wave bridge rectifier and a buck converter. The rechargeable coin-cell battery powers a buck-boost converter capable of providing voltages between 1.8 and 5.0 V. Depending on the available power from the harvester the board is either supplied by the harvester or the battery. An internal prioritizer switches between the power sources. If the harvesting source is available, the buck converter is active and the buck-boost is o ff and vice versa.

**Figure 16.** (**a**) Power managemen<sup>t</sup> boards 2151A from analog devices and (**b**) bq2557OEVM-206 from Texas Instruments used as power managemen<sup>t</sup> circuits.

The power managemen<sup>t</sup> board 2151A has been tested under different configurations as presented in Table 2.


**Table 2.** Harvester characterization at 0.5 g excitation level, using the 2151A power managemen<sup>t</sup> board.

Furthermore, we evaluated the efficiency of different power managemen<sup>t</sup> boards the 2151A and the bq25570EVM-206, designed for low power applications and providing only 1.8 V output voltage as depicted in Table 3. The efficiency in this case is nothing than the ratio between the output and the input power.

**Table 3.** Efficiency comparison of the power managemen<sup>t</sup> boards used as conditioning circuits for the designed harvester.


The experiments revealed that a maximum efficiency of approximately 50% can be reached using both boards with our harvester.

#### **4. Parametric Design Optimization**

One of the key features of the presented folded beam harvester design is the possibility to enhance the overall performance if the first two resonance frequencies appear closely spaced frequencies (co-resonance) and simultaneously provide the same power levels. This is a unique feature not provided by other multiresonant structures such as an array of two beams. All cantilevers of an array are subjected to the same base excitation, whereas in the case of the coupled resonator, the inner beam is subjected to the maximum tip displacement of the outer one, which is higher than the applied base excitation. This motivated us to investigate the possibility of optimizing the existing design.

The geometry of a vibration energy harvester determines its dynamic properties and thereby its operating frequency and the harvested power. Consequently, optimized dimensions yield higher power and better performance. For this purpose, the reference design was parameterized and optimized for an operating bandwidth centered at 75 Hz. The process of this optimization relied on FE models and is shown in Figure 17.

\* Non-dominated sorting genetic algorithm II.

\*\* Nonlinear programming by quadratic Lagrangian.

 **Figure 17.** Optimization process.

Firstly, the reference geometry was parameterized and subsequently optimized. Figure 18 presents the parameterized model. Table 4 gives the range of the seven geometry parameters. The size of the magnets and their positions was unchanged during the optimization process. A parameter range of ±50% has been chosen with respect to the reference design. The thickness *t* is a discrete parameter, because the device is fabricated from a metal sheet, which is available only at certain thickness values. The parameter *Li* has bounds chosen to enable efficient usage of space for all values of *Lo*. Constraining *Li* prevents the inner beam to overlap with the fixed support.

**Figure 18.** Parameterization of the reference geometry (light grey corresponds to steel, black to NdFeB, and yellow to PIC255).


**Table 4.** Parameter ranges for the design optimization.

1 Discrete parameter since it is limited to commercial sheet metal; step size 0.5 mm.

A modal analysis and a harmonic analysis have to be performed to compute the objective values of the optimization. A modal analysis determines the eigenfrequencies for the first two modes, while a harmonic analysis computes the electrical power at these modes. The harmonic analysis implements a damping ratio of 0.8% which has been determined experimentally for a similar design. The base excitation acceleration amplitude was 0.01 g. The two outer piezoelectric elements were connected in parallel. The inner element was connected in series in order to obtain maximum power at an optimized load resistance. The results of these analyses yield the parameters and objectives presented in Equations (6)–(13) of which Equations (6)–(9) give the vibrational properties:

$$
\overline{f} = \frac{f\_1 + f\_2}{2} \tag{6}
$$

$$
abla f = \left| \text{75 Hz} - \overline{f} \right|\tag{7}$$

$$
\Delta f\_{rd} = \frac{f\_1 - f\_2}{\overline{f}} \tag{8}
$$

$$
\Deltaobj \, \Delta f\_{rel} = \left| 0.05 - \Delta f\_{rel} \right|. \tag{9}
$$

Here, *f*1 and *f*2 are the first two eigenfrequencies, *f* is their mean value and Δ*frel* is the relative operating frequency range. The two objectives *obj f* and Δ*frel* describe the intended operating frequency range. Upper bounds of 0.05 for *obj* Δ*frel* and 5 Hz for *obj f* control the convergence of the optimization algorithm.

Equations (10)–(13) are related to the electrical behavior:

$$V = -\mathbb{g}\_{31} \text{ t } \sigma\_1 \tag{10}$$

$$P = 2\pi V^2 \frac{\mathbb{C}\_{\mathcal{o}} \mathbb{C}\_{i}}{2\mathbb{C}\_{\mathcal{o}} + \mathbb{C}\_{i}} f \tag{11}$$

$$obj\,P\,ratio = \frac{\min(PD\_1, PD\_2)}{\max(PD\_1, PD\_2)}\tag{12}$$

$$
\rho\_{\rm obj} \,\overline{PD} = \frac{PD\_1 + PD\_2}{2},
\tag{13}
$$

where *V* is the approximated voltage of a piezoelectric patch, *g*31 is the piezoelectric voltage coe fficient for the 31 mode, *t* is the thickness of the piezoelectric patch, σ1 is the normal stress due to bending, *P* is the electrical power in an attached resistor of at optimum load value, *Co* and *Ci* are the capacitances of the inner and outer piezoelectric patches, *f* is the frequency and *PD*1 and *PD*2 are the power densities at the first and second eigenfrequency, respectively. The voltage was obtained analytically from the mechanical model in order to reduce the computational e ffort. This neglects the electromechanical back coupling. The power objectives *obj P ratio* and *obj PD* evaluate the frequency spacing of the two maxima and their amplitude ratio.

The optimization follows a two-step procedure: A global multi-objective optimization and subsequent local single-objective optimizations. Methods to decrease the computational e ffort such as a sensitivity analysis or a metamodel were omitted, as they were su ffering from insu fficient accuracy. The large design space and the nonlinear objective space require this two-step procedure where the first step searches for promising subspaces. A second, more refined step searches this subspace to find the final candidates. The multi-objective optimization employs the evolutionary algorithm NSGA-II, which iteratively evolves a set of start designs by selection, crossover, and mutation to satisfy the objectives of the optimization. The start population contained 3500 designs; each following generation comprised 100 designs. A crossover probability of 98% and a mutation probability of 1% defined the reproduction. The optimization converged for either 20 generations, a convergence stability of 2% or if 70% of the designs of a generation were Pareto-optimal. The Pareto set provides one start design per thickness for the single-objective optimizations. These start designs were selected to satisfy the two vibrational objectives to guarantee operation at resonance and broaden the harvesting. A subsequent single-objective optimization relied on the algorithm NLPQL for a gradient-based optimization. The local search deployed central di fferences and a finite di fference of 1%. The parameter ranges for each single-objective optimization were ± 10% of the start design. These local optimizations changed the definition of *obj* Δ *frel* to *obj* Δ *frel* = 0.01 − Δ *frel* . The optimization was considered completed if the change for the next iteration fell below 0.1% or if 20 iterations were reached. The local optimization comprised up to three single-objective optimizations.

Figure 19 presents the geometry of the reference design together with the individual optimized designs of all three thicknesses. The optimized design (*c*) can be compared to the reference design since both have the same thickness. Important di fferences are the length of the connection and the width of the outer beam, which results in an operating frequency close to 75 Hz.

**Figure 19.** (**a**) Reference geometry and (**b**) the three optimized designs with thicknesses of *t* = 0.5, (**c**) *t* = 1, and (**d**) *t* = 1.5 mm.

Figure 20 compares the power densities of the four designs in Figure 19. The power density of the reference design has two dominant peaks. However, the power drops beyond the bandwidth. In contrast, the designs (a) and (b) provide an operational frequency range with a power variation of only 3.5% centered at 75 Hz. Moreover, those designs also have higher peak power densities since their more thin steel structure is more compliant. Up to three local optimizations were performed for each design. Hence, a higher number of local optimizations will further improve the designs.

**Figure 20.** Power density of the reference design and the optimized designs. The co-resonance results in an extended operative bandwidth at comparable power levels.

In summary, an optimized geometry provides equal power at both resonance frequencies at even higher power density, as demonstrated with the 0.5 mm thick design.
