## Abstract

This work presents a novel design based on weakly-coupled beam-based structural elements with engineered nonlinearities that hosts a piezoelectric thin film for energy harvesting. The H-shaped structure are finite elements simulated to understand their frequency response, which includes local and global mode interactions. The inherently nonlinear structures are then designed to enable the activation of the 1:1, 1:2, and 1:3 internal resonances that broadens the frequency response and increases its amplitude. The H-shaped beam-based structures are 3D printed and their dynamic behavior is experimentally investigated using LDV to get insights on how to integrate them with piezoelectric elements. The experimental power measured from individual patches of an H-shaped piezoelectric loaded structure is 5.26 µW, which is relatively less than the power of a piezoelectric loaded doubly-clamped beam (9.18 µW). This discrepancy primarily arises from differences in size between the two batches. However, by integrating the voltage outputs from both patches on the H-shaped coupled beam structure, a synergistic effect is achieved, yielding a combined power of 14.31 µW. This concentration is observed across a broader frequency spectrum, specifically ranging from 180 to 200 Hz, as compared to the single peak exhibited by the doubly-clamped beam. These findings highlight the exceptional potential of the H-shaped coupled-beam structure and emphasize their remarkable efficiency of the piezoelectric energy harvesting.

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## 1 Introduction

The significance of Energy Harvesting (EH) technology has been magnified by the growing demand for alternative energy sources and robust self-powered systems. EH from a system’s surrounding environment has emerged as a highly promising technology that can play a vital role in addressing the challenge of the global energy crisis due to escalating energy consumption, environmental degradation, and mounting pressure on the ecosystem [1, 2]. Furthermore, EH has the potential to enhance the functionality of wireless sensor networks and enable the realization of self-powered sensors. Energy Harvesters (EHs) have attracted significant attention as promising energy sources for supplying power to sensors in remote or hard-to-reach locations. Their potential can be attributed to their capability of converting normally wasted kinetic energy into usable electrical energy to power sensors located in remote areas, inaccessible locations, or harsh environments.

Vibrations carry inherent power that have become a popular area of interest in the field of energy harvesting. Ambient vibrations, which have a wide range of frequency spectrum and low amplitudes, hold significant promise as an energy source. Researchers have been exploring various methods to capture this energy for use in different applications. Some recent studies highlighted the use of topological metamaterial beam vibrations [3] to develop energy harvesters for self-powered sensors in heavy railways [4], optimize electrostatic vibration energy harvesters based on MEMS electret devices [5], and integrate inertial pendulums for zero-energy sensor applications in freight trains [6]. Upon careful analysis of the literature, it is apparent that vibration energy harvesting is a highly promising and versatile source of energy. This emphasizes the criticality of acknowledging the potential of vibrations as energy sources that can be efficiently harnessed. Air Conditioning (AC) units are widely available sources of wasted vibration energy. Hence, developing high-performance EHs that utilize the vibration of AC units is desired to power indoor air quality sensors and structural health monitoring active indicators. Utilizing AC vibrations to generate electric power is central for this paper.

Vibration Energy Harvesters (VEHs) are devices designed to convert ambient kinetic energy, specifically vibrations, into electrical energy. The potential for VEHs to power electronic devices in remote and harsh environments has garnered significant attention in recent years. VEHs are particularly promising devices due to their ability to harness energy from various vibrating sources, including human and machine motions [7,8,9,10], as well as environmental vibrations [11, 12]. VEHs offer a highly advantageous solution for powering low-power electronic devices, such as sensors, wireless communication modules, and remote controls. The development of VEHs for the purpose of empowering the next generation of sensors presents a remarkable opportunity that should not be overlooked [13]. Different mechanisms have been explored to realize VEHs, such as electromagnetic [14,15,16], electrostatic [17,18,19], magnetostrictive [20,21,22,23], triboelectric [24,25,26] and piezoelectric material systems [27,28,29].

Piezoelectric Vibration Energy Harvesters (PVEHs) are considered a highly viable option for VEHs compared with other vibration energy harvesting methods [30,31,32,33,34,35,36]. PVEHs are renowned for their impressive power density, scalability, and compact design. Therefore, they are considered to be excellent candidates for onboard power sources [37, 38]. Piezoelectric materials have the unique ability to generate an electric voltage when exposed to mechanical stress or strain. Thus, they are ideal materials for energy harvesting from vibrating objects. Yu et al*.* [39] presented a bird-shaped broadband energy harvester in an effort to mimic a bird in flight and achieve a wide operating bandwidth. Xie et al*.* [40] introduced a multimodal E-shaped energy harvester with a built-in bi-stability and internal resonance that was able to harvest energy from both the upward and downward movements of a central mass. These novel designs have the potential to improve the energy harvesting efficiency of piezoelectric devices, making them suitable for various energy harvesting systems. As a result, PVEHs have been utilized in various applications such as structural health monitoring [41, 42], and wearable devices [43, 44].

The advent of Nonlinear Vibration Energy Harvesters (NVEHs) represents a promising technology that has the potential to enhance the performance of PVEHs. This innovative products opened the possibility of capturing broader range of ambient vibrations, including those that are typically difficult to capture with traditional Linear Vibration Energy Harvesters (LVEHs) due to their low amplitude and high frequency. As a result, NVEHs are poised to become critical enablers of more efficient and sustainable energy harvesting technologies for various applications. The literature on NVEHs highlights various techniques that utilize oscillators and resonators for power generation. Nonlinear techniques, such as Duffing, bi-stability, parametric, and stochastic, have been explored to improve the power density of harvesters [45]. The latter review emphasizes the need for dependable and productive energy harvesting methods to promote sustainable power generation. Recent research focused on nonlinear techniques to broaden operational bandwidth, with new designs and configurations proposed for energy harvesters, such as inter-well modulation [46], bi-directional schemes [47], tri-directional harvesters [48], energy harvesters with ball tip masses [49], and piezoelectric arrays with motion limiters [50]. Nonlinear two-degree-of-freedom systems [51] and multi-stable piezo-magneto-elastic harvester arrays [52, 53] are other nonlinear systems that have been introduced to improve performance.

Coupled structures have favorable characteristics that can significantly improve the operational bandwidth of energy harvesters, and recent research has demonstrated innovative techniques to achieve this. For example, Upadrashta and Yang [53] proposed a nonlinear spring-mass-damper system that utilizes a series of piezoelectric energy harvesters connected in parallel and coupled by a magnetic force, resulting in an increased output voltage and a broader frequency range. The experimentally tested harvester array produced a power density of 1.2 μW/cm^{3} over a frequency range of 10–90 Hz. Similarly, Shim et al*.* [54] developed a nonlinear broadband piezoelectric energy harvester that employed a coupled beam array to broaden the bandwidth of the energy harvester using a combination of nonlinear coupling and mechanical resonance. Numerical simulations predicted that the energy harvester could generate a peak power density of 0.30 mW/cm^{3} over a frequency range of 25–45 Hz. These designs offered superior performance over broad frequency ranges and hold significant potential for improving energy harvesting in various applications.

Internal resonance is the energy transfer among the different modes of vibration when two or more modes have commensurate ratios such as 1:2 or 1:3. Also, the structure should have strong nonlinearities to trigger the coupling and experience low damping conditions. This phenomenon has been explored for wide range of applications such as improving the sensitivity of resonator-based mass sensors [55], stabilizing the frequency of resonators for timing applications [56], signal filtering applications [57]. Also, in maximizing the generated power by increasing the bandwidth of the vibration-based energy harvester [58,59,60,61,62,63]. In [64], the authors developed a generalized topology for designing resonators with multiple commensurate ratios. They concluded that the relative angle among the connected beam plays a critical role to determine the frequency characteristics. Reference [63] demonstrated a novel design based on a tunable auxiliary oscillator coupled to the main nonlinear oscillator. The frequencies of the system can be easily adjusted to achieve the commensurate condition necessary to activate the 2:1 internal resonance. Using this technique, they achieved a 130% increase in the bandwidth. Following the same approach, the authors in [59] demonstrated 97% improvement in the bandwidth compared to that of the linear system. In the literature, different structure such as the L-shape [60] and portal frame [58] have been employed to demonstrate the influence of internal resonance on improving the resonator bandwidth.

The aim of this study is to demonstrate the benefits of utilizing weakly coupled beams as opposed to a single beam element in energy harvesting applications. To achieve this, we employ a model comprising a set of coupled beams that are weakly connected near the fixed supports. The structure is designed using the Finite Element Method (FEM) to realize the activation of internal resonances that widen and intensify the frequency response. Our focus is directed towards examining internal resonance phenomena, which gets activated provided that the ratio among the natural frequencies is commensurate. Next, we produce 3D-printed samples with relevant dimensions. These samples are then tested experimentally to obtain their frequency responses. The results of this investigation are expected to contribute to the understanding of the behavior of weakly coupled structural components under sinusoidal base excitation which simulates the vibration induced by host rotating equipment’s, such as air conditioning units. It is expected to also provide insights into the potential applications of 3D printing technology in the field of structural dynamics. To test coupled beams as energy harvesters, piezoelectric elements are attached to the 3D printed samples to convert kinetic energy into electric energy and measure output power.

## 2 Design and experimental work

### 2.1 Numerically inspired design considerations

An H-shaped coupled-beam (H-structure), shown schematically in Fig. 1, is comprised of two doubly-clamped beams that are joined by a coupling beam, are numerically investigated. Employing Finite Element Analysis (FEA) using COMSOL MULTIPHYSICS 5.6, a comprehensive investigation is conducted on the H-structure to ensure the activation of internal resonance by having commensurable or nearly commensurable ratios of their first four global frequencies. The process of parametric investigation involves studying two critical geometrical variables of the coupling beams, namely their length (L_{3}) and location (L_{1} = L_{2}), as presented in Fig. 1. The two primary beams are considered as a pair of identical beams, each is 100 mm in length, 10 mm in width, and 1 mm in thickness.

The effect of coupling length is studied within a range of 1 mm–50 mm, while the coupling location is defined by the distance of the midline of the coupling beam to the clamping end of the two primary beams, which took length values in the range from 6 to 55 mm. The first four global mode shapes of the designed H-structure are shown in Fig. 2**.**

Figure 3 illustrates the variation of the first four global frequencies of the H-structure under various coupling lengths and locations. Notably, the first and second global frequencies exhibit a smooth, gradual transition as coupling parameters vary. However, when the coupling is located at 40 mm or greater, the third and fourth global frequencies display abrupt, non-smooth shifts. This phenomenon highlights the substantial influence of the coupling of beams on the overall structural global frequencies.

The structural global frequency variation serves as a notable indicator of its energy harvesting potential, particularly evident when observing 1:1, 1:2, and 1:3 ratios between the global frequencies, signifying internal resonance. The analysis aims to discern the predominant frequencies linked with the primary beams. Our investigation reveals that an infinite array of parameter values can be utilized to attain the necessary commensurable or nearly commensurable ratios between global frequencies, as depicted in Fig. 4.

An important objective is to activate internal resonances of the H-structure. Thus, weakly coupled cases are identified by structural minute cross-sectional area and proximity to the clamping end, which is usually under 15% of a main beam’s length [65]. Therefore, we have deliberately opted for a coupling location at 12 mm, corresponding to 12% of a primary beam’s length. With this choice, the commensurate ratio is realized and a parametric study is done using the weakly coupled location (L_{2} = 12 mm) for different coupling length (L_{3}) that was varied from 1 to 50 mm.

As shown in Fig. 5a, it is evident that for short coupling length, the primary trend leads to an augmented disparity between the first and second global frequencies, however, they converge with increasing the coupling length. On the other hand, as the length of the coupling beam is increased, the third and fourth global frequencies decrease. This decrease causes the two global frequencies to come closest to each when the coupling length is around 25 mm, then the global frequencies diverge from one another for the cases of higher coupling lengths, as demonstrated in Fig. 5b. If a coupling length of 20 mm is selected, then the associated global frequencies and their respective ratios detailed in Table 1. One can note that the desired commensurable or nearly commensurable ratios to be 1:1 and 2:1.

### 2.2 Device fabrication

The H-structure is prototyped out of PLA (Polylactic Acid), typically used as a 3D printing “ink”. The inherent nonlinear characteristics of the PLA material provided an excellent opportunity to trigger the quadratic nonlinearity and activate the internal resonance of the printed samples. Fabrication is done using the 3D-printing of the type Ender-3 V2 from Creality. To ensure that the samples are of the highest quality, optimal printing parameters are utilized, as outlined in Table 2.

Two types of samples are created: a doubly-clamped beam and an H-structure. The doubly-clamped beam is 100 mm long, 10 mm wide, and 1 mm thick, as shown in Fig. 6a. On the other hand, the H-structure has a coupling beam of 20 mm in length located at 12 mm with the same width and thickness as the primary beams, as depicted in Fig. 6b.

### 2.3 Parameter extraction

The material properties of 3D-printed objects can be greatly influenced by multiple factors, including printing technology, parameters, material composition, and post-processing techniques. As such, it is essential to fully characterize the material to predict structural dynamic characteristics accurately. This ensures that the resulting models provide reliable representations of the actual behavior of the structure with desired performing capabilities.

#### 2.3.1 Material density

Material’s density is measured for multiple samples, and the results show good agreement with the standard PLA material density, with only a 4.9% percentage error, as shown in Table 3.

#### 2.3.2 Flexural rigidity

The fabrication process can generate internal axial force, which has a significant impact on natural frequencies of the doubly-clamped beam, unlike cantilever beam. To investigate this effect, the undamped eigenvalue problem is solved, considering the axial force term in the governing equation and corresponding boundary condition for doubly-clamped beam. We present the detailed procedure in the appendix (see Appendix [A]). Modulus of elasticity can be found using the following fundamental formula:

where f_{1} is the first frequency, \({\upomega }_{\text{non}1}\) is the nondimensional natural frequency corresponding to the induced axial load (N), \(\text{E}\) is the modulus of elasticity, I = 1/12 × b × h^{3} is moment of inertia for the beam cross section, ρ is the material’s density, A = b × h is the cross-sectional area, and l is the beam’s length.

By utilizing the information provided in Table 4 and substituting back the numerical value in Eq. 1 results in a Modulus of elasticity of 2.9409 GPa which is comparable to the typical value ranging from 2.7 to 3.2 GPa [66].

#### 2.3.3 Damping coefficient

The damping phenomenon in materials arises from the material’s internal friction, which impedes the amount of energy that can be absorbed and dissipated during vibration. PLA is typically recognized for having low damping coefficient when compared to other materials. Consequently, Appendix B provides comprehensive insights into the utilization of two widely employed methodologies, namely the half-power method and the decay method, for the precise determination of damping coefficient. The experiments show a good agreement between the two methods where the values of 0.01696 and 0.01610 are determined using the half-power bandwidth method and the decay method, respectively.

### 2.4 Experimental setup

Figure 7 shows the full experimental setup for characterizing the structures under base excitations simulate the vibrations induced by the AC units. An integrated power amplifier Mini shaker (Modal Shop model # 2007E01) to electrodynamically actuate the structure over wide range of frequencies and amplitudes. The frequency and level of excitation are controlled using a Keysight 33500B Waveform Generator. The response of the structure is then monitored by a Laser Doppler Vibrometer (OMETRON VH-1000D type 8338), and data from the LDV is acquired using a Keysight EXTR054A Mixed Signal Oscilloscope.

## 3 Results and discussion

### 3.1 Dynamic response

#### 3.1.1 Doubly-clamped beam

In this section, we examine the dynamic response of the doubly-clamped beam under base excitation. Through this examination, one can gain insights into the H-structure’s linear and nonlinear behaviors when subjected to varying levels of excitation. To accomplish this, we conducted an experiment, in which we subjected a doubly-clamped to different levels of excitation while sweeping through a range of frequencies around its first natural frequency that is obtained numerically. Figure 8 displays the time history and frequency response curves for both forward and backward frequency sweeps at three different levels of excitation, namely 0.75 g, 1 g, and 1.25 g. The obtained results show a linear response at excitation levels of 0.75g (Fig. 8a, b) and 1 g (Fig. 8c, d). However, at an excitation level of 1.25 g, a hardening effect was observed in the frequency response curve, as shown in (Fig. 8e, f).

The hardening effect observed in the frequency response curves in Figs. 8e, f indicate a nonlinear behavior of the doubly-clamped beam at 1.25 g. This effect is characterized by an increase in the beam’s natural frequency as the amplitude of the excitation increases. This behavior is usually observed in systems with cubic nonlinearities. In the case of the doubly-clamped beam, the hardening effect is due to the geometric nonlinearities introduced by the large deflections that occur at high excitation levels. These large deflections cause changes in the stiffness of the beam that leads to hardening effect.

#### 3.1.2 H-structure

We examine the dynamic response of the H-structure under base excitation. By carrying out this examination, one can gain insights into their resonance characteristics when subjected to base excitation. To accomplish this, we performed an experiment where the H-structure is subjected to a base excitation of 1.25 g, while sweeping through a frequency range near its first two global natural frequencies. Response is measured using a Laser Doppler vibrometer (LDV) at a distance of L/4 from the free end of the primary beams, as depicted in Fig. 9. The LDV used relies on a single point measurement and can measure up to 500 mm/s. The frequency sweeps were conducted carefully to ensure a steady state response is achieved before recording the measurements. Then, the recorded data is post processed to generate the fast Fourier transform which reveals the frequencies affecting the response.

Figure 10a shows the time history and frequency response of the H-structure under a base excitation of 1.25 g measured on the first beam. One can observe the first and second global frequencies of the structure are within the 130–230 Hz frequency range. The first global frequency occurs at a frequency of approximately 179 Hz, while the second global frequency lies at 196 Hz, which corresponds to energy transfer from the second beam to the first beam. Furthermore, the figure reveals the presence of an internal resonance effect occurring at ratios of 2:1 (values 350 Hz and 390 Hz) and 3:1 (values 483 Hz and 590 Hz). One should note that the low amplitude of these higher modes can be attribute to the fact that the laser measurement points, Fig. 9, are in close proximity to the nodal line of these modes, Fig. 2, these measurement points was carefully selected due the range limitation of the laser doppler vibrometer. These outcomes suggest that the frequency at which a beam vibrates has a significant impact on its resonant behavior. Internal resonances occur when the ratio between the frequencies of different vibrational modes has a nearly commensurable ratio. In addition to the forward sweep, experiments were also conducted for the backward sweep as well. The outcomes in the form of time history and frequency response curves are shown in Fig. 10b. When we measured the second beam response, as shown in Fig. 10c, we observed a clear distinction between the two beams. In particular, the peak detected at the second frequency is higher than the peak observed at the first frequency. This finding strongly supports our hypothesis that the first frequency corresponds to the first mode shape of the first beam. Additionally, the significant magnitude of the peak at the second frequency suggests that the first mode shape of the second beam has been excited. Tests are also carried out using a backward frequency sweep, for which time history and frequency response curves are depicted in Fig. 10d. It is observed that indicate that the two beams have different vibrational modes and can provide valuable insights for analyzing their behavior under different conditions. Also, the activation of the 2:1 and 3:1 internal resonance was clear.

### 3.2 Energy harvesting application

This section includes experimental investigations conducted to assess the feasibility of energy harvesting using both doubly-clamped beams and H-structure. The primary objective is to evaluate the practicality of Piezoelectric Energy Harvesting (PEH) and explore the potential of vibration energy harvesting under various conditions. These experiments aim to deepen our understanding of the H-structure’s behavior and identify methods for enhancing its energy harvesting efficiency.

We utilized an advanced Macro Fiber Composite™ (MFC) manufactured by SMART MATERIAL [67], detailed in Table 5, which incorporates a piezoelectric patch (PZT) as an active component for harnessing electrical energy generated through structural deformation. In literature, the integration of the piezoelectric film can be realized through various schemes, such as complete coverage across the entire surface of doubly clamped beam [54, 65, 68]. One should note that even though the position of the MFC on the doubly clamped beam is not optimized and might result in charge cancellation, the generated output voltage, using 1g base excitation, is slightly higher than the one generated by the H-structure, as shown in Fig. 16b and Fig. 17b. In the context of the H-structure, our simulations have revealed that the optimal placement for a smaller patch, particularly is at one-fourth distance from the edge, as shown in Fig. 11.

In our experimental configuration, shown in Fig. 11, the structures were mounted on a shaker to simulate vibrations generated by a vibrating equipment, such as an air conditioning unit. We used the M8507-P1 model (85 × 7 mm) for the doubly-clamped beam setup Fig. 11a and two identical M2807-P1 models (25 × 7 mm) to capture energy from the H-structure, shown in Fig. 11b.

We conducted comprehensive analyses to evaluate the electrical output characteristics of piezoelectric elements. Figures 12 and 13 illustrate the temporal variations in voltage for both the doubly clamped beam and H-structure configurations under the influence of a 1g base excitation. This graphical representation explains the dynamic voltage fluctuations inherent to each structural arrangement. Notably, the depicted alternating current (AC) profile highlights the dynamic nature of the electrical output from piezoelectric elements within these configurations. A rectifier circuit is utilized for converting the alternating current (AC) generated by each piezoelectric element to a direct current (DC).

Specifically, in the context of the H-shaped structure, it is noted that electric charge cancellation may arise due to changes in deformation sign. To address this potential issue, the voltage was aggregated post-rectification. A light-emitting diode (LED) was connected to the DC output voltage to represent a small power device, such as an indoor sensor. Thus, the constructed circuit would demonstrate the generation of useful electric output from PEH elements that could be used as a primary power source for regulating indoor climate conditions, as illustrated in Fig. 14.

For the full-wave rectifier configuration, a high-efficiency diode is selected, i.e., the 1N4148. this diode exhibits a moderate forward voltage drop as indicated in Table 6 Additionally, the use of electrolytic capacitors with suitable voltage and capacitance specifications plays a pivotal role in minimizing ripples and maintaining stability of the DC output, as outlined in Table 7.

In the context of a full-bridge wave rectifier circuit, precise load matching with the rectifier’s output is vital for optimal power conversion. Our experiments indicates that an approximately 70 kΩ load value was the most effective for both MFC patches, as shown in Fig. 15 and the summarized data in Table 8. This load optimization contributed to elevated efficiency and dependable voltage regulation within the rectification system. The harvested power is depicted in Fig. 15a as the total power, while the power density across the entire device is illustrated in Fig. 15b.

Thus, The approach taken for coming up with vibrational energy harvester is based on two pillars: (i) Analyzing LDV outcome to provide insights into structural dynamic behavior of the H-structure. (ii) Understanding mechanical to electric energy conversion of piezoelectric elements. Then, employing well-established normalized half-efficiency bandwidth (BW_{nhe}) to measure the effective bandwidth of piezoelectric energy harvesters [69]. The normalized half-efficiency bandwidth technique is widely utilized in analyzing frequency response curves, and identifying the frequency span where the harvester’s response drops to 0.707 of its maximum value. The calculation starts by examining the frequency response curve of the harvester, identifying its resonance frequency, and then determining the frequencies that define the bandwidth. This approach is crucial for applications related to structural dynamics in determining the effective bandwidth of piezoelectric energy harvesters as.

where the upper frequency ω_{Upper} and lower frequency ω_{Lower} are associated with 0.707 of the amplitude maximum value, with ω_{Center} denoting the frequency at which the peak occurs.

To illustrate further, LDV measurements are conducted on the doubly-clamped beams under the influence of a 1g base excitation. The frequency response, shown in Fig. 16a, provides a full understanding of the structural vibrational patterns. Figure 16b displays electrical measurements obtained from the piezoelectric material integrated into the same doubly-clamped beams, showcasing the electrical power generated from mechanical vibrations. Significantly, the effective bandwidth of the doubly-clamped beams is confined between 197 and 207 Hz (i.e., over 10 Hz bandwidth), and its central frequency is 201 Hz, resulting in a normalized half-efficiency bandwidth (BW_{nhe}) of approximately 0.05. The power reached 9.18 µW, which demonstrates the viability of our energy harvesting approach.

Turning our attention, now, to H-structure, the LDV measurements recorded on two distinct beams reveal distinct peaks corresponding to the first and second eigenfrequencies, as shown in Fig. 17a. This data illuminates the dynamic behavior of interconnected beams under external excitation. Figure 17b shows electrical measurements obtained from piezoelectric material integrated into the H-structure, demonstrating the effectiveness of the piezoelectric energy harvesting technique for this structural configuration. Even though the piezoelectric patches are significantly smaller compared to the doubly-clamped beam case, approximately 70% smaller, each patch exhibited a remarkable power output of 5.26 µW (equivalent to 50% of the output power observed in the doubly-clamped beam scenario) within the frequency range of 170–215 Hz. Additionally, the H-structure configuration demonstrates a broader spectrum of vibrational modes. Specifically, the first patch limits the effective bandwidth between 175 and 192 Hz, with a central frequency at 182 Hz, resulting in a normalized half-efficiency bandwidth (BW_{nhe}) of about 0.088. Likewise, the second patch constrains the effective bandwidth to the range from188 Hz to 207 Hz, with a central frequency at 198 Hz and a normalized half-efficiency bandwidth (BW_{nhe}) of approximately 0.096, thereby improving its electric energy harvesting capabilities.

It is essential to emphasize the cumulative approach of adding the voltage outputs from each distinct piezoelectric material patch to obtain the total output voltage for both patches, as illustrated in Fig. 17c. This allows to harness the full amount of energy generated by the H-structure, quantifying to an impressive 14.31 µW. Consequently, the bandwidth, ranging from 177 to 206 Hz, which corresponds to bandwidth of 29 Hz, with a central frequency of 191 Hz, yields a normalized half-efficiency bandwidth (BW_{nhe}) of around 0.152. This consolidated voltage output shows the effectiveness of amassing the overall efficiency and usability of the piezoelectric energy harvesting system, ensuring the capture and utilization of maximum electric energy from the beams’ mechanical vibrations.

In order to investigate the feasibility of utilizing captured piezoelectric converted energy, we utilize it to charge a battery. To accomplish this, we opt for a rechargeable lead-acid battery model (WP1224W) as the recipient of the piezoelectrically generated power. A careful monitoring of the electric charging process, commencing from almost a 1% state of charge (SOC), which is equivalent to the voltage of a 9.03 V battery. An hourly voltage measurement scheme is performed on the battery for 15 h. To assess power delivery, a 330-Ω load is introduced to measure both current and the corresponding power output. This rigorous approach allows for a comprehensive evaluation of the proposed energy harvesting system’s performance. Detailed data, including the measured battery voltage, state of charge, and current for the 330-Ω resistance, are presented in Table 9.

A time-based plot is established to illustrate the dynamic performance of our energy harvesting system’s power output. Figure 18 shows the evolving power delivery during the 15-h battery charging period.

In summary, our experimental analysis does demonstrate the effectiveness of our Piezoelectric Energy Harvesting (PEH) system. By harnessing piezoelectric energy, we’ve explored its potential across varying conditions. Notably, the H-structure’s configuration has exhibited an expanded frequency range for energy harvesting. This widens the scope for efficient power generation, marking a significant advancement in energy harvesting technology.

## 4 Conclusions

The study primarily focused on enhancing the frequency response of vibrating H-shaped coupled-beams structure under sinusoidal base excitation to act as a basis for piezoelectric energy harvesters, simulating AC units’ vibrations. Using insights from finite element simulations, an enhanced design of a weakly coupled beam-based structure is obtained. The coupled beams are capable of activating internal resonances at ratios of 1:1, 1:2, and 1:3, thereby expanding the effective frequency range for energy harvesting. The 3D printed structure from PLA-based material underwent rigorous experimental analysis to characterize their frequency response. By integrating piezoelectric elements into the constructed beam systems, vibrational kinetic energy is converted into electric energy, achieving a remarkable total power of 14.31 µW which is 1.6 times higher than that recorded by the doubly-clamped beam. Additionally, the proposed weakly coupled beam-based structure demonstrated a harvesting bandwidth of 29 Hz, three times wider than the doubly-clamped beam structure. Harnessing piezoelectric energy for battery charging purposes has yielded invaluable insights into its performance across diverse conditions. These achievements serve as a catalyst for further advancements in energy harvesting systems, offering the promise of sustainable power generation for low-power devices, such as IoT sensors.

## Data availability

No datasets were generated or analysed during the current study.

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## Acknowledgements

The authors acknowledge the support of the King Fahd University of Petroleum & Minerals through the Interdisciplinary Research Center for Communication Systems and Sensing under Grant #INCS2205.

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### Contributions

The authors confirm contribution to the paper as follows: study conception and design: H.D., N.J., M.H., data collection: H.D.; analysis and interpretation of results: N.J., M.H., H.D.; draft manuscript preparation: H.D. All authors reviewed the results and approved the final version of the manuscript.

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## Appendices

### Appendix A

By solving the following undamped eigenvalue i.e., governing equation Eq. A1 and the boundary condition Eq. A2, the variation in nondimensional natural frequency \({{\varvec{\upomega}}}_{\mathbf{i}}\) for the first six mode shapes of the doubly-clamped beam due to the presence of axial load ( N_{non} = α_{1} × N) where α_{1} = l2/(EI).

The results reveal that the natural frequencies increase as the axial load increases, Fig.

19a. This is due to the axial loads generated during the fabrication process, which alter the stiffness of the structure, thereby affecting the natural frequency. Accordingly, the ratio between the successive non-dimensional natural frequencies is evaluated and presented in the Fig. 19b.

To evaluate the Youngs’ Modulus, a preliminary experiment is performed by exciting the doubly-clamped beam at a low-level base excitation using impulse excitation technique, the time response is shown in Fig.

20a. The latter enables the identification of the first natural frequencies by applying a FFT analysis for the time response, as depicted in Fig. 20b.

As illustrated in Fig. 20b, the first two frequencies recorded to be f_{1} = 166.734 Hz and f_{2} = 453.518 Hz, the ratio of these two frequencies found to be almost 2.72001. Subsequently, the ratio of these natural frequencies is compared to numerical outcomes in Fig. 19: (a) The variation for the non-dimensional Natural frequencies of the first six mode shapes, Fig. 19: (b) The The variation of the successive non-dimensional natural frequencies.

-b. This comparison facilitates the determination of the induced axial load. The latter value of the ratio corresponds to the nondimensional axial load of 2.232, as shown in Fig.

21a, b.

The axial load can be found to be 0.49281 N corresponding to the initial internal stress of 49.281 kPa. Regarding the H-shaped coupled-beams, we utilize an impulse test as a means of determining the frequencies of the structure through experimentation. Figure

22a displays the time response of the impulse test, while Fig. 22b depicts the frequency spectrum of the structure.

The found natural frequencies are then compared to the output of the parametric study to determine the nondimensional axial load. The parametric study was carried out using COMSOL Multiphysics 6.0 to ensure that the experimental and numerical results were consistent. The purpose of this study is to calculate the nondimensional axial load in the primary and coupling beams to be implemented in the numerical model. Accordingly, the primary beams experienced an estimated tension of 4.53 in the first beam and compression of − 15.59 in the second beam, while the coupling beam experienced a tension of 1.37. These nondimensional values corresponded to the induced axial loads during the 3D printing process, which were determined to be 0.86 N of tension, − 4.95 N of compression, and 0.35 N of tension.

### Appendix B

The half power method is a technique for calculating the damping coefficient of a damped harmonic motion system. The method entails measuring the system’s response power level at various frequencies and determining the half power frequency, which is the frequency at which the power level is half its maximum value. This value is then used to calculate the system’s damping coefficient using Eq. B1.

Where \({\text{f}}_{\text{upper}}, {\text{f}}_{\text{lower}},\text{ and }{\text{f}}_{\text{cent}}\) are the upper, lower, and central frequencies, and \(\upzeta \) is the damping ratio. To determine the damping coefficient using the half power method, a forward sweep was performed on a doubly-clamped beam around its first resonance frequency. The dynamic response of this sweep is shown in Fig.

23a, which displays the lower, center, and upper frequencies at 163.45 Hz, 169.105 Hz, and 166.734 Hz, respectively. These values can be substituted into Eq. B1 to get the damping ratio.

On the other hand, the decay method is a technique for determining the damping coefficient of a free-vibration system by measuring the logarithmic decrease of the amplitude over time. The damping coefficient can be computed using the amplitude decay curve’s logarithmic decrease following the Eq. B2 and Eq. B3:

where \(\updelta \) is logarithmic decrement, N is number of periods, \({\text{X}}_{1}\) is the amplitude of the first peak, and \({\text{X}}_{\text{N}+1}\) is the amplitude of N + 1 peak, and \(\upzeta \) is the damping ratio. Figure 23b displays the decay curve for the doubly-clamped beam’s free vibration, where the amplitudes of the 1st and 23rd peaks are recorded as 0.512077 and 0.0500612, respectively. These values can be used in Eq. B2 and Eq. B3 to determine the damping ratio.

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Daraghma, H., Jaber, N. & Hawwa, M. Exploiting internal resonance of 3D-printed weakly coupled structures for piezoelectric energy harvesting.
*Nonlinear Dyn* **112**, 16847–16869 (2024). https://doi.org/10.1007/s11071-024-09939-4

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DOI: https://doi.org/10.1007/s11071-024-09939-4