*2.3. System-Level Simulation*

The system-level model consists of three parts as depicted in Figure 2. The multiphysics part comprises the ROM, which describes the mechanical and piezoelectric behavior of the harvester. The magnetic forces have been derived from magneto-static simulations [33] and implemented in the model as force functions 1 and 2. The magnetic force will alter the effective stiffness of the resonator. The damping ratio of the mechanical resonator depends on the structure's stiffness. Hence, it also varies while tuning. We address this by adjusting the damping ratio accordingly. The electrical part encompasses the rectification circuitry. A tuning control algorithm, based on maximum amplitude tracking, is included as well.

**Figure 2.** System-level model implemented in ANSYS twin builder, including a reduced order model of the harvester together with tuning actuation, the conditioning circuitry, and the tuning control algorithm.

#### 2.3.1. Mechanical Resonator Reduced Order Model

This subsection presents the validation of the reduced order model by comparison between results of a FE model, the reduced order model and experimental data. The MIMO system of the resonator is illustrated in Figure 3. Both models use three inputs: Base excitation "dis", (displacement amplitude of the ambient vibration), force on outer and inner beam "f\_outer" and "f\_inner", respectively. The tip displacement of the outer and inner beam "dis\_outer" and "dis\_inner" are the two outputs. The displacement amplitudes of the outer and inner beam of the FE model, respectively the reduced order model, are shown in Figure 3 for a displacement amplitude of 10 μm at the clamped part.

**Figure 3.** (**a**) The multiple-input multiple-output (MIMO) system of the mechanical resonator together with (**b**) the resonator reduced model validation through a comparison with the full finite element (FE) model. Both models are subjected to an excitation amplitude of 10 μm.

The obtained results from the reduced order model matched well with the FE model. The two fundamental modes appear at 63.1 and 77.5 Hz. Furthermore, the aforementioned frequencies have been compared to the experimental results obtained in [33], which are 62.6 and 76.1 Hz.

In [32,34] we presented a description of the magneto-static simulations, which enabled us to derive the magnetic forces involved in the frequency tuning. The same forces have been considered in the current system-level model. As shown in Figure 4 the magnetic forces yield a bidirectional frequency shift by up to 18%.

**Figure 4.** (**a**) Variation of the displacement amplitude of outer and (**b**) inner beam during frequency tuning. An 18% of bidirectional frequency shift can be achieved. The data indicates that frequency tuning does not affect the other resonance frequency.

The frequency tuning system-level simulations showed a relative error of approximately 3.1% and 1.8% for the first, respectively the second mode tuning (see Figure 5) when compared to experimental data. Yet, tuning towards smaller frequencies reveals an increasing discrepancy. We attribute this to some limitations of the underlying magnetostatic model, which neglects the rotation and lateral displacement of the magne<sup>t</sup> as the resonator undergoes deflection. Furthermore, as displacement amplitude increases while tuning towards smaller frequencies such deviations show a more pronounced effect.

**Figure 5.** (**a**) Experimental validation of the bidirectional frequency tuning simulation of first and (**b**) second resonance frequency.

#### 2.3.2. Piezoelectric Energy Harvester Reduced Order Model

After the validation frequency-agile resonator model, a reduced order model of the piezoelectric energy harvester has been derived. The corresponding MIMO system is illustrated in Figure 6.

Compared to the mechanical resonator model, the piezoelectric energy harvester model has two additional outputs "vol\_outer", "vol\_inner", referring to the voltage levels at the piezoelectric patches on the outer and inner beams. The results obtained from the reduced order and the FE model fit well and thus support the applicability of this reduced order modeling approach.

**Figure 6.** (**a**) The MIMO system of the piezoelectric energy harvester and (**b**) the harmonic response of the reduced order and full FE model subjected to a 10 μm excitation amplitude.

#### 2.3.3. Electrical Simulation

The system-level model integrates the reduced order model with electrical circuitry. Here, we considered a diode bridge for full-wave rectification, a capacitor for filtering, a buck converter for voltage regulation, and a resistive load with optimum resistance. The circuitry is connected to the two electrical ports of the reduced order model.

Rectification and Filtering

As presented in Figure 7, after rectification and filtering the AC voltage output, a DC voltage output with small ripple voltage is obtained.

**Figure 7.** (**a**) Simulation results of the AC voltage output before rectification and (**b**) the filtered DC voltage output of the piezoelectric harvester subjected to 0.2 g base excitation.
