Simulation and Characterization of a Nonlinear Dual-Frequency Piezoelectric Energy Harvester ^†

Abstract

In this paper, we present a concept, simulation and characterization results of a dual-frequency piezoelectric energy harvester with magnetic frequency tuning capabilities. We demonstrate that the frequency-agile multi-mode capability enables the device to harvest on a wider range of operating frequencies than classical vibration harvesters.

1. Introduction

Vibration based energy harvesting is particularly well suited for industrial applications, where wireless sensor networks may be implemented in inaccessible environments. The growing interest in such systems motivated the development of self-adaptive harvesters to act as a sustainable power source. However, most of the vibration based energy harvesters are spring-mass-damper systems, which generate maximum power only when their resonance frequency coincides with the ambient vibration frequency. As a consequence, further research work was invested in the development of new resonator designs and in elaborating different strategies for performance optimization of these devices; e.g., by considering multi-resonant structures as in Wu et al. [1]. Alternatively, bandwidth broadening can be achieved by tuning the operating frequency. For example, Eichhorn et al. [2] proposed a vibration energy harvesting with a learning algorithm for a resonance frequency adaptation. A method to simplify modelling and simulation of the nonlinear piezoelectric energy harvester using nonlinear springs has been proposed in [3]. D. Hoffmann et al. [4] developed an autonomous energy harvesting system, which is able to adapt its operating frequency to the dominant ambient vibration frequency. In [5] we presented a full experimental characterization of a single- and dual-frequency resonator with the potential use as an energy harvester. Recently, in [6], we studied the behavior of a dual-frequency piezoelectric energy harvester incorporating permanent magnets for bi-directional frequency tuning.

In this paper, we study the behavior of a dual-frequency piezoelectric energy harvester. The geometry consists of a so-called folded beam resonator with two fundamental mode shapes appearing at two close frequencies. It incorporates neodymium permanent magnets for bi-directional frequency tuning of both resonances. Thanks to a three-dimensional magnetostatic model of the magnet configurations, we were able to determine the force vector acting on the movable magnet on the beam segments as a function of beam displacement and distance to the fixed magnet. We defined a two-dimensional fit function and identified respective coefficients by numerical optimization. This analytical force-gap-displacement relationship will be implemented as nonlinear forces in a numerical model for transient simulation.

2. Simulation of Dual-Frequency Piezoelectric Energy Harvester

In order to estimate the power output of the proposed energy harvester, we implemented a model of the structure in the finite element (F.E) simulation tool ANSYS Multiphysics. First, we considered the mechanical resonator, thoroughly studied in [5], with two identical masses m= 7.6 g. By performing a modal analysis, we defined the mode shapes and their eigenfrequencies. The two lowest resonance frequencies obtained from the simulation are f_1,sim = 63.182 Hz and f_2,sim = 77.457 Hz, which match the experimental values of f_1,exp = 62.630 Hz and f_2,exp = 76.072 Hz. Furthermore, we considered piezoelectric films into our 3D model, in order to estimate the voltage and the power output. The patch dimensions are 60 × 10 × 0.2 mm³ on the outer beams and 48 × 18 × 0.2 mm³ for the inner beam as depicted in Figure 1. The structure is under a harmonic base acceleration of 0.5 g. The material properties of the piezoelectric ceramic originate from PIC-255, supplied by PI Ceramics [6].

These results demonstrate that the considered resonator amplifies significantly the displacement at the two first fundamental modes, leading to a significant mechanical strain distribution in the beam segments. Hence, the piezoelectric thin films polarize and generate surface charges. Subsequently, they show the possibility of harvesting energy from the first two resonance frequencies and illustrate that the deformation appears either in the inner or in the outer beam.

3. Simulation of the Magnetic Forces

The simulation of the bi-directional frequency tuning effect in different configurations and orientations requires a full characterization and the implementation of the magnetic forces into our model. Therefore, we considered a parametrized magnetostatic simulation of a pair of neodymium permanent magnets with N42 magnetization and with the geometry shown in Figure 2.

The magnet’s dimensions are 10 × 10 × 5 mm³. For simplification purposes, we ignore the rotation of the magnet as the beam deflects and consider only vertical displacement. The simulation considered different orientations (attractive and repulsive) and different configurations (axial and vertical). However, and due to the approach similarities between the different configurations, we consider only attractive magnets in the vertical configuration. The simulation has been performed for different gap values and for different vertical displacements in the interval from −6 to 6 mm as depicted in Figure 2. In order to be able to simulate the effect of the magnetic tuning of our dual-frequency resonator model, we consider spring elements with nonlinear (displacement-dependent) stiffness. Therefore, we defined a global two-variable fitting function (Figure 2). The fitting function shows an excellent match with the simulation results.

As a first step, the defined functions have been implemented into a 2D ANSYS model of a simple (80 × 10 × 1 mm³) stainless steel cantilever (same geometry as the simple beam resonator in [5]). The forces have been considered using the nonlinear spring element COMBIN39 [7]. The natural frequency of our simple resonator is f₀ = 56.296 Hz, which we use as a starting point for our transient simulation. By considering attractive forces in the vertical configuration, the overall stiffness of the structure will decrease. We run a transient simulation with a harmonic base excitation and linearly decreasing excitation frequency (sweep down ‘s.d.’) as shown in Figure 3.

The graph shown in Figure 3 represents the primary transient analysis results. They demonstrate the frequency tuning effect of the initial resonance of the resonator, showing that the magnetic forces implemented into our model led to a structure softening. Due to the considerable solution time of a full transient simulation, we considered an equivalent nonlinear lumped spring-mass-damper resonator in SIMPLORER. We parametrized the excitation frequency such that it sweeps down. The results in Figure 3 illustrate the same effect with the ones from the full model and both data show a nice match in terms of frequency with the experimental data set in [5].

4. Frequency Tunable Dual-Frequency Resonator

We consider the geometry depicted in details in Figure 4. It consists of two identical 80 mm long arms (referred to as outer beams 1 and 2), stretching from the base and mechanically connected via a common end to a 60 mm long inner beam, which extends in turn towards the base.

The resonator is a stainless steel structure, incorporating a pair of external neodymium permanent magnets, exerting forces on another permanent magnet, which is mounted on the flexible beam. The net magnetic force superimposes with the restoring force of the deflected beam. Thus, an effective spring constant arises, leading to the desired frequency shift. The structure is excited with a harmonic base acceleration with a magnitude of a = 0.2 g generated by a vibration test equipment.

The experimental results in Figure 4 show the effect of gap variation on resonance frequency of the dual-frequency resonator and illustrate a maximum bidirectional frequency shift of up to 12%. In [5] we demonstrated the independency of tuning the two frequencies, which opens up the opportunity to reduce the frequency gap between the two modes and enables the possibility for a frequency overlap. However, we reported a considerable drop in displacement amplitudes while tuning, after exceeding certain gap thresholds, which constitutes a limitation of such a strategy.

5. Conclusions

In this work, we simulated the behavior of the dual frequency mechanical resonator incorporating piezoelectric thin films for energy harvesting purposes. We demonstrated the dual-frequency feature and that energy can be harvested at two different frequencies. A three-dimensional magneto-static simulation of the magnetic forces between the two permanent magnets has been performed to define a general force-gap-displacement relationship, which enables us to run gap dependent simulations with control features and to study the behavior of the targeted self-adaptive piezoelectric energy harvester in more detail. We also implemented the already defined functions into a 2D simple resonator model and presented our primary transient analysis results. The simulation results demonstrated that the implemented forces trigger a structure softening leading to a decrease of the natural frequency and showed the possibility to consider a reduced model to make the simulation more efficient. Finally, we experimentally investigated magnetic frequency tuning of a dual-frequency resonator. It has been shown that both operating frequencies can be tuned by up to 12%. As the tuning of one mode does not affect the other, the approach can also be used for reducing the frequency gap between the two modes.