Dynamic Analysis and Energy Harvesting Potential of Slitted Cantilever Beam Fitted with Piezoelectric Transducer

Abstract

This research investigates how slitted beams fitted with piezoelectric transducers (PZTs) behave when generating energy from environmental vibrations. By changing slit widths, slit lengths, and tip masses, we investigated the frequency response of these beams using analytical methods and finite element analysis (FEA). The obtained results demonstrate that resonance peaks are brought closer together, and coupling between vibrational modes is enhanced by larger slit spacing, whether or not anti-resonance dips are present, though the magnitudes of these peaks are affected by the width of the slits. The possibility of energy harvesting can be enhanced when resonance peaks are shifted and amplified by heavier tip masses. The FEA results support the analytical model, showing other characteristics such as sharp dips or anti-resonance troughs. This work provides valuable information for future design improvements by highlighting the significance of optimizing slit parameters and tip masses to enhance the efficiency of piezoelectric energy harvesters. This study demonstrates the benefits of using slitted beams with piezoelectric energy harvesting and provides recommendations for their efficient design.

1. Introduction

Harvesting the energy of structural vibrations is becoming increasingly significant as a viable approach to providing an autonomous and independent power source for numerous low-power electronic devices. This process involves converting mechanical energy from ambient vibrations into electrical energy using various transduction processes, such as piezoelectric, electromagnetic, and electrostatic processes. Using different transduction techniques including piezoelectric, electromagnetic, and electrostatic techniques, the mechanical energy from ambient vibrations is transformed into electrical energy in this process. The increasing need for long-term, trouble-free power solutions for things like wireless sensors, wearable electronics, and Internet of Things (IoT) systems, where it can be difficult or impossible to replace or recharge batteries, motivates this strategy. Energy harvesting, which makes use of ambient vibrations in the environment, provides a sustainable and reliable power source, lessening the need for traditional power plants and paving the way for more sustainable, environmentally conscious infrastructure. Therefore, it a great option for situations that need continuous, long-term operation, applications in hostile conditions, or in isolated areas.

Piezoelectric materials produce an electrical charge when subject to mechanical pressure. This characteristic renders them well suited for capturing energy from surrounding vibrations. Scientists have focused their efforts on enhancing the effectiveness and energy generation capability of piezoelectric energy harvesters by making improvements in materials and developing creative structural designs [1,2,3,4,5]. In their study, Wang and Song [6] employed zinc oxide nanowires to fabricate piezoelectric nanogenerators with a high energy output. Their findings highlight the potential of nanostructured materials to enhance energy conversion efficiency, mostly attributed to their large surface area-to-volume ratio. To increase the power output, He et al. [7] investigated the use of cantilever-based piezoelectric harvesters with tip mass. It was found that the output power and resonance frequency range of these devices could be increased by adjusting the tip mass, dimensions, and position. In actual situations where vibration frequencies may vary, such a modification can increase the device’s efficiency.

Altering the form of the cantilever beam is an alternative method used to improve the efficiency of piezoelectric energy harvesters. Various geometries, such as trapezoidal, triangular, or serpentine shapes, offer more flexibility, can reduce resonance frequencies, and enhance strain distribution [8,9,10,11]. In their study, Zhang et al. [12] investigated how different beam forms impact the efficiency of piezoelectric energy harvesters. The researchers found that a trapezoidal beam design resulted in greater power output compared to a rectangular beam, mostly because of improved stress distribution over the length of the beam. Chimeh et al. [13] present a low-frequency piezoelectric MEMS harvester with a serpentine structure, optimized using a neural network and genetic algorithm. The design achieves a 121.7 Hz resonant frequency and 0.73 μW power output, demonstrating efficient performance in a compact device.

The integration of piezoelectric materials into various structural systems has been a significant area of research. D’Ambrogio et al. [14] developed a flexible piezoelectric composite by aligning PZT particles within a PDMS matrix using dielectrophoresis. The structured material shows enhanced dielectric and piezoelectric properties, making it suitable for sensors and energy harvesting applications across various surfaces. Iqbal et al. [15] developed a hybrid energy harvester combining piezoelectric and electromagnetic mechanisms within an insole design. The device efficiently captures low-frequency biomechanical energy from walking, optimizing power generation for wearable electronics. Toledo et al. [16] introduced two piezoelectric microactuator devices that were fine-tuned for in-plane displacement using finite element technique simulations. Their research focused on optimizing actuation efficiency by conducting a comparative analysis of low-stiffness and high-stiffness actuators. They showed that the low-stiffness actuator outperformed the high-stiffness actuator because of its lower stiffness coefficient and greater responsiveness to applied voltage. In a similar vein, Díaz-Molina et al. [17] investigated the efficiency of in-plane piezoelectric unimorph microactuators comprising several material combinations, including AlN, PVDF, and polyimide. According to their research, soft material combinations were best suited for high-displacement applications since they provided the highest displacement per volt, particularly when thinner substrates were utilized. Piezoelectric energy harvesting devices were thoroughly analyzed by Stamatellou [18], who highlighted the possibility of nonlinear dynamics and material improvement to improve energy capture efficiency in low-frequency situations.

Electromagnetic energy harvesting is efficient for extracting energy from low-frequency vibrations and utilizes the movement between a magnet and a coil to generate an electric current, according to Faraday’s Law of Electromagnetic Induction [19,20,21,22]. El-Hami et al. [23] created a prototype device that captures electromagnetic energy. The device consists of a vibrating beam with a permanent magnet. Their findings demonstrated that these systems could provide a significant amount of power with which to operate low-power sensors and wireless devices. This highlights the significance of adjusting the damping factor to achieve the highest possible energy conversion efficiency. Liu et al. [24] conducted a comprehensive review of inerter-based vibration isolation systems, highlighting their applications across various fields such as automotive, civil, and ocean engineering. They particularly focused on the advantages of using inerters for both vibration suppression and energy harvesting, providing insight into the mechanical network’s analogy to electrical systems and demonstrating the potential of inerters to improve system performance and reduce complexity.

Electrostatic energy harvesting utilizes the movement between conducting surfaces and changes in capacitance to produce electrical energy. This technology’s compatibility with microfabrication processes makes it extremely appropriate for MEMS applications [25,26,27]. Meninger et al. [28] introduced an electrostatic converter that transforms vibrations into electricity utilizing capacitors with adjustable capacitance. Their research demonstrated the practical application of these converters in settings with low frequency. It also explored the possibility of using other dielectric materials to improve the efficiency of these energy harvesters.

Hybrid energy harvesters utilize multiple transduction processes to enhance energy conversion efficiency and expand the range of frequencies at which they may operate. These technologies are especially beneficial in circumstances where the attributes of surrounding vibrations exhibit substantial variations [29,30,31,32]. Muthalif et al. [33] created a hybrid energy harvester that combines piezoelectric and electromagnetic transduction methods. Their empirical findings demonstrated that the hybrid system had superior power generation capabilities when compared to individual harvesters, particularly in situations characterized by a diverse array of vibration frequencies. Lagomarsini et al. [34] introduced a hybrid harvester that combines piezoelectric and electrostatic processes. Their research revealed that the hybrid method has the potential to greatly improve energy conversion efficiency and deliver a more consistent power output in different vibratory conditions.

Numerical simulations and finite element method (FEM) evaluations are essential tools for optimizing the design and enhancing the performance of energy harvesters. These techniques enable researchers to simulate intricate shapes, qualities of materials, and limits on the behaviour of energy harvesters and to forecast their performance in different scenarios [35,36,37,38,39]. Pan et al. [40] examined the efficiency of a piezoelectric energy harvester featuring a beam with a zigzag shape. Their research showed that the zigzag arrangement effectively amplified the voltage output and expanded the operational frequency range. Li et al. [41] utilized numerical simulations to enhance the performance of a hybrid energy harvester that integrates piezoelectric and electromagnetic principles. Their findings demonstrated that hybrid designs can exhibit superior efficiency and more consistent energy output in comparison to single-mode harvesters. Karadag et al. [42] conducted a study that utilized the FEM to develop and enhance the design of a piezoelectric energy harvester with a curved cantilever beam. Their research revealed that the curved form had the potential to promote strain distribution and optimize energy conversion efficiency. In addition, they investigated the impact of several piezoelectric materials on the performance of the harvester. Madinei et al. [43] employed numerical simulations to examine the efficacy of a hybrid energy harvester that utilizes both piezoelectric and electrostatic transduction mechanisms. Their research demonstrated that the hybrid harvester was able to sustain optimal performance throughout a broad spectrum of vibration frequencies and environmental conditions.

Current research has primarily concentrated on novel designs and materials for devices that capture and convert energy. Bagheri et al. [44] introduced novel methodologies to enhance the efficiency and utilization of piezoelectric energy harvesters. The efficacy of these methodologies is improved through the integration of artificial intelligence. In order to enhance energy harvesting from low-frequency vibration sources, Kulah and Najafi [45] conducted a thorough examination of frequency up-conversion methods and provided valuable insights by discussing strategies that are based on impact, extraction, and snap-through.

This study concentrates on the dynamic behaviour of slitted beams with piezoelectric transducers (PZTs), specifically PZT-5A, to explore their potential for energy harvesting applications. The presence of slits in the beam is anticipated to impact the coupling between various vibration modes, thereby influencing the beam’s resonance and anti-resonance properties. Precise measurements and material attributes are used in the simulation of the slitted beams, including a material substructure with notable mechanical strength and damping capabilities. By investigating various combinations of slit lengths and tip masses, we show how such structural alterations impact the frequency response and energy-harvesting efficiency of the beams.

An analytical model and finite element analysis (FEA) are used to examine the beams’ frequency response functions (FRFs). The latter enables us to validate the analytical model and offer insights into the structural dynamics and optimization of slitted beam designs for enhanced energy harvesting performance.

By shedding light on how structural alterations, particularly slit arrangements and tip-mass modifications, can be employed to optimize energy harvesting in practical applications, this study fills a significant knowledge gap in the field of energy harvesting research. The results of this study are intended to direct the creation of more effective piezoelectric harvesters, enabling a wider range of uses, such as in settings with fluctuating vibration frequencies.

2. Electromechanical Model

In this study, we investigated the mathematical model of a slitted cantilever beam energy harvester with a PZT-5 layer on top, as shown in Figure 1. The beam is modelled using Euler–Bernoulli beam theory, considering tip masses on both free ends and incorporating damping effects. It is assumed that plane sections remain plane and perpendicular to the neutral axis after deformation. The figure is for illustrative purposes, but it is important to note that the PZT material is brittle and requires substantial support from the substrate and intermediate layers to prevent cracking during testing, as seen in commercially available designs.

For a cantilever beam with a piezoelectric layer, the strain $S$ due to bending can be expressed as [46]

$$S\left(x,z,t\right)=-z{\displaystyle \frac{{𝜕}^{2}w\left(x,t\right)}{𝜕x^2}}$$

(1)

where z is the distance from the neutral axis of the beam, and w(x,t) is the transverse displacement of the beam at a distance x from the fixed end and at time t. This displacement w(x,t) represents the deflection of the beam in the direction perpendicular to its length due to bending.

The piezoelectric constitutive equations relating the mechanical stress T and electric displacement D to the mechanical strain S and electric field E are [47]

$$\left\{\begin{array}{c}T=c^{E}S-eE\\ D=eS+{\epsilon }^{S}E\end{array}\right.$$

(2)

where $c^E$ is the stiffness at a constant electric field, $e$ is the piezoelectric stress constant, and ${\epsilon }^S$ is the permittivity at constant strain.

The electric field in the piezoelectric layer is related to the voltage $V\left(t\right)$ across the thickness h_p of the piezoelectric layer as follows [48]:

$$E_3={\displaystyle \frac{V\left(t\right)}{h_p}}$$

(3)

The bending moment $M_p\left(x,t\right)$ due to the piezoelectric effect in a slitted beam can be derived by integrating the stress over the piezoelectric layer:

$$M_p\left(x,t\right)={\int }_{{-h}_p/2}^{h_p/2}T\left(x,z,t\right){\mathrm{b}}_{\mathrm{p}}dz$$

(4)

where ${\mathrm{b}}_{\mathrm{p}}$ is the width of the piezoelectric layer, while $h_p$ is the thickness of the piezoelectric layer.

Substituting for $T(x,z,t)$ and integrating, we obtain

$$Mp(x,t)={\int }_{{-h}_p/2}^{h_p/2}\left[c^E\left(-z{\displaystyle \frac{{𝜕}^{2}w(x,t)}{𝜕x^2}}\right)-e{E}_3\right]b_{p}dz$$

(5)

Considering the piezoelectric coupling term θ [49],

$$\theta =-e{\int }_{{-h}_p/2}^{h_p/2}{b}_{p}dz=-d_{31}{E}_p{b}_p{h}_p$$

(6)

where $d_{31}$ is the piezoelectric strain constant, and $E_p$ is Young’s modulus of the piezoelectric material.

For a slitted beam, the coupling term must consider the reduced cross-sectional area due to the slit(s). The effective width $b_{eff}$ can be approximated by subtracting the width of the slits from the total width:

$$b_{eff}=b_p-n{w}_{slit}$$

(7)

where n is the number of slits, and $w_{slit}$ is the width of each slit.

Thus, the modified coupling term θ_eff is

$${\theta }_{eff}={-d}_{31}{E}_p{b}_{eff}{h}_p{\displaystyle \frac{{𝜕}^{2}w\left(x,t\right)}{𝜕x^2}}={-d}_{31}{E}_p\left(b_p-n{w}_{slit}\right)h_p$$

(8)

The introduction of slits in a cantilever beam alters the stiffness and mode shapes, leading to mode coupling, which results in two resonance peaks and anti-resonance. The slit changes the effective stiffness, splitting a single resonance into two distinct peaks.

The transverse displacement $w(x,t)$ of the beam can be expressed as a superposition of mode shapes:

$$w\left(x,t\right)={\sum }_{r=1}^2{\varphi }_r\left(x\right){\eta }_r\left(t\right)$$

(9)

where ${\varphi }_r\left(x\right)$ represents the mode shapes, and ${\eta }_r\left(t\right)$ denotes the modal coordinates.

The frequency response function (FRF) is

$$H\left(\omega \right)={\sum }_{r=1}^2{\displaystyle \frac{{\chi }_r{\varphi }_r\left(x\right)}{{\omega }_r^2-{\omega }^2+j2{\varsigma }_r{\omega }_r\omega }}$$

(10)

The anti-resonance occurs when the contributions from the two modes cancel out:

$${\sum }_{r=1}^2{\displaystyle \frac{{\chi }_r{\varphi }_r\left(x\right)}{{\omega }_a^2-{\omega }_r^2}}=0$$

(11)

At this anti-resonance frequency ${\omega }_a$, destructive interference causes a dip in the response. This dynamic behaviour is essential for optimizing energy harvesting in slitted beams.

In combining the above equations, the electromechanical model for the slitted beam piezoelectric energy harvester, including the coupling term, is

$$\begin{gathered} YI{\displaystyle \frac{{𝜕}^4{w}_{rel}(x,t)}{𝜕x^4}}+c_{s}I{\displaystyle \frac{{𝜕}^5{w}_{rel}(x,t)}{𝜕x^4𝜕t}}+c_a{\displaystyle \frac{𝜕w_{rel}(x,t)}{𝜕t}}+m{\displaystyle \frac{{𝜕}^2{w}_{rel}(x,t)}{𝜕t^2}}+{\theta }_{eff}{\displaystyle \frac{𝜕V\left(t\right)}{𝜕t}}\delta \left(x\right)= \\ -m{\displaystyle \frac{𝜕w_b\left(x,t\right)}{𝜕t}}\delta (x)-c_a{\displaystyle \frac{𝜕w_b\left(x,t\right)}{𝜕t}}-{\displaystyle \frac{{𝜕}^{2}w\left(x,t\right)}{𝜕x^2}} \end{gathered}$$

(12)

where YI is the bending stiffness of the beam, C_sI is the internal strain rate, C_a is the external air damping coefficient, m is the mass per unit length, $\delta \left(x\right)$ is the Dirac delta function representing the location of the piezoelectric layer, and $w_b$ is the motion of the beam carrying the piezoelectric layer.

The voltage across the piezoelectric layer can be derived from the piezoelectric constitutive equations and the electromechanical coupling. For a piezoelectric cantilever beam with a tip mass, the voltage V(t) generated due to mechanical strain can be expressed as [48]

$$V\left(t\right)={-R}_L{\int }_0^L{d}_{31}{Y}_p{h}_p{b}_p{\displaystyle \frac{{𝜕}^3{w}_{rel}(x,t)}{𝜕x^t𝜕t}}dx-{\displaystyle \frac{1}{{ϵ}_{33}^s}}{\int }_0^L{\displaystyle \frac{𝜕v\left(t\right)}{𝜕t}}{b}_p{h}_{p}dx$$

(13)

where $R_L$ is the load resistance, $d_{31}$ is the piezoelectric strain constant, and ${ϵ}_{33}^s$ is the permittivity at constant strain.

The current $i\left(t\right)$ through the load resistance due to the voltage generated can be derived using Ohm’s law and the relationship between the voltage and the charge generated in the piezoelectric layer and is given by

$$i\left(t\right)={\displaystyle \frac{v\left(t\right)}{R_L}}$$

(14)

The relative motion of the beam $w_{rel}(x,t)$ can be described as a series of mode shapes and corresponding modal coordinates. This approach uses the orthonormal mode shapes of the beam, which are solutions to the homogeneous part of the differential equation representing the beam’s movement, and is given by

$$w_{rel}(x,t)={\sum }_{r=1}^{\infty }{\varphi }_r\left(x\right){\eta }_r\left(t\right)$$

(15)

where η_r(t) is the modal coordinate of a clamped-free beam for the rth mode, while ${\varphi }_r$ is the orthonormal mode shape of the free vibration of a cantilever beam, which is governed by the following equation [50].

$${\varphi }_r\left(x\right)=\sqrt{{\displaystyle \frac{1}{mL}}}\left[\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}{\displaystyle \frac{{\lambda }_r}{L}}x-\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{{\lambda }_r}{L}}x-{\sigma }_r\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}{\displaystyle \frac{{\lambda }_r}{L}}x-\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{{\lambda }_r}{L}}x\right)\right]$$

(16)

where λ_r is related to the nth natural frequency and is obtained utilizing the following characteristic equation:

$$1+\mathrm{c}\mathrm{o}\mathrm{s}\lambda \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\lambda =0$$

(17)

where ${\sigma }_r$ is a dimensionless frequency number that can be expressed as

$${\sigma }_r={\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}{\lambda }_r-\mathrm{s}\mathrm{i}\mathrm{n}{\lambda }_r}{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}{\lambda }_r+\mathrm{c}\mathrm{o}\mathrm{s}{\lambda }_r}}$$

(18)

The undamped modal frequency can be defined as

$${\omega }_r={\lambda }_r^2\sqrt{{\displaystyle \frac{YI}{m{L}^4}}}$$

(19)

with the modal coupling term

$${\chi }_r=\vartheta {\displaystyle \frac{d{\varphi }_r\left(x\right)}{dx}}{|}_{x=L}$$

(20)

The mechanical damping ratio can be assumed to be [49]

$${\varsigma }_r={\displaystyle \frac{c_{s}I{\omega }_r}{2YI}}+{\displaystyle \frac{c_a}{2m{\omega }_r}}$$

(21)

The circuit time constant is [51]

$$T_c={\displaystyle \frac{R_l{\epsilon }_{33}^{S}bL}{h_p}}$$

(22)

Considering only the transverse vibration mode and assuming the voltage modulus is the ratio of the voltage output to the base acceleration, we obtain

$${\displaystyle \frac{\upsilon \left(t\right)}{-{\omega }^2{Y}_o{e}^{j\omega t}}}={\displaystyle \frac{{\sum }_{r=1}^{\infty }\frac{-jm\omega {\varphi }_r{\gamma }_r^w}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }}{{\sum }_{r=1}^{\infty }\frac{j\omega {\varphi }_r{\chi }_r}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }+\frac{1+j\omega {\tau }_c}{{\tau }_c}}}$$

(23)

where

$${\gamma }_r^w={\int }_{x=0}^L{\varphi }_r\left(x\right)dx$$

(24)

To obtain the power modulus, defined as the power output divided by the square of the base acceleration, we multiply the FRFs of the current and voltage.

$${\displaystyle \frac{P\left(t\right)}{{\left({\omega }^2{Y}_o{e}^{j\omega t}\right)}^2}}={\displaystyle \frac{{\left({\sum }_{r=1}^{\infty }\frac{-jm\omega {\varphi }_r{\gamma }_r^w}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }\right)}^2}{R_I{\left({\sum }_{r=1}^{\infty }\frac{j\omega {\varphi }_r{\chi }_r}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }+\frac{1+j\omega {\tau }_c}{{\tau }_c}\right)}^2}}$$

(25)

3. Results

3.1. Finite Element Analysis (FEA)

FEA was used to validate the analytical model and help provide a thorough understanding of the dynamic behaviour of slitted beams fitted with PZT. The FEA simulations were conducted using ANSYS software version 14.5, concentrating on beams with dimensions and material properties to match those used for the analytical model.

The model was constructed using the following dimensions: overall beam length of 142.0 mm, width 28.0 mm, with slit width 2.0 mm, and the slit commencing at a distance 50.0 mm from the fixed end (slit length 92.0 mm from the free end). The beams were composed of a 0.70 mm thick substructure topped with a 0.66 mm thick PZT-5A layer. To ensure that the stress and strain distributions, especially at the slitted regions and the free end of the beam, were accurately represented, the model used a fine mesh.

The fixed end of the beam was fully constrained, while the free end was able to vibrate freely. An ACEL command was used to apply base vibrations to the beam from the beam base, with an acceleration magnitude of −1 m/s² in the y-direction to simulate real-world vibrations.

With 500 substeps, a harmonic analysis covering a frequency range of 0 to 50 Hz was carried out to guarantee a thorough capture of the slitted beams’ resonance and anti-resonance behaviour. The electrical circuit’s load resistance was adjusted to 10 kΩ.

Figure 2 shows FEA results for slitted beams under various loading conditions and vibrational modes, generated using ANSYS software. The figure illustrates different deformation patterns and mode shapes, with higher deformations observed at the free end of the beam, both slitted and without a slit. These results confirm the analytical model and provide insights into the structural dynamics and optimization of slitted beams.

The frequency response functions of slitted beams with one tip mass of 5 g and conventional beams with PZT layer and a tip mass of 5 g, as determined using ANSYS simulation, are shown in Figure 3. The regular beam shows a robust vibrational mode (resonance peak) close to 30 Hz, while the slitted beam shows resonance peaks at roughly 27 Hz and 43 Hz, and a sharp dip or anti-resonance between these peaks at about 31 Hz. The presence of the slit dramatically changes the dynamic behaviour of the beam, as demonstrated by this complex response in the slitted beam. The slit’s impact on energy dissipation at particular frequencies is demonstrated by the presence of the strong anti-resonance peak. Our findings confirm how significantly the vibrational properties and energy harvesting capability of beams with PZT can be affected by the inclusion of a slit and tip mass.

3.2. Comparison of FEA and Analytical Models

By comparing the predictions of our analytical model with the results of ANSYS FEA, we demonstrate our model’s accuracy and reliability. To compare the predictions of the analytical method with those of ANSYS and identify possible disparities, both techniques were used to determine the FRFs for the same slitted beams with a layer of PZT attached. With this dual approach, the dynamic behaviour of the beams in different configurations can be thoroughly examined, not only validating the analytical model but possibly offering valuable information for future piezoelectric energy harvester development.

A case study to investigate how various structural alterations affect the functionality of piezoelectric energy harvesters as predicted by our analytical model and ANSYS was carried out. By analyzing resonant frequencies, damping ratios, and total frequency response, we can determine how to best optimize the design for energy harvesting applications.

This work used PZT-5A, a piezoelectric material that possesses both high mechanical strength and a robust piezoelectric response. Young’s modulus for PZT-5A is 1.2035 × 10¹¹ Pa, giving it the rigidity required for the energy harvesting application, guaranteeing that the composite beam has sufficient stiffness to endure applied loads. Its density, which adds to the total mass of the beam structure, is 7750 kg/m³, which is a good match for the beam. PZT-5A’s damping coefficient, 1.2433 × 10⁻⁵, is an essential factor in determining how much energy is dissipated during dynamic loading and guarantees uniform energy dissipation properties across the beam. This case study calculated the FRFs of both regular and slitted beams using an analytical approach. Utilizing the given parameters and attributes, we could analytically predict the beam’s dynamic behaviour and resulting frequency response.

This study thoroughly examined how structural changes affect the performance of piezoelectric energy harvesters and offers insightful advice for improving their design and use in various applications.

Figure 4 shows the frequency response of two beam configurations, namely a regular beam and slitted beam, using analytical method. The x-axis shows the frequency, and the y-axis, the power magnitude. The natural frequency of the regular beam is roughly 28.9 Hz, and the slitted beam had a mass added at only one tip; there are two resonance peaks at 25.8 and 42.1 Hz, respectively, illustrating that the presence of a slit and tip mass cause the slitted beam to have two resonance frequencies and a sharp dip in response at about 35 Hz. The first peak shows the basic bending mode; the second peak comes from mode coupling brought about by the slits and tip mass design, therefore changing the beam’s stiffness. By greatly extending the working frequency range, these peaks help the energy harvester to gather energy from several vibrational sources. Especially in situations with different vibrational frequencies, the larger frequency response and mode coupling clearly offer a benefit by improving the energy collecting efficiency. The slit and tip mass provide complicated dynamic behaviour, which improves performance at some frequencies but causes sharp dips or anti-resonances at others. In order to optimize the design of piezoelectric energy harvesters for specific applications, these results imply that structural adjustments, such as the addition of a slit and tip mass, might substantially affect their frequency response and performance. Specifically, slitted beams present a viable way to enhance dynamic response and energy collection.

The beams’ resonance properties can be significantly changed by adjusting slit parameters such as slit width and length. To assist in maximizing the energy harvesting efficiency of slitted beams, this work explored the effect of slit characteristics on the frequency response of the beams.

Figure 5 compares the FRF of a slitted beam fitted with PZT using the analytical model and FEM. The analytical model shows two primary resonance peaks at 26 and 42 Hz and an anti-resonance drop at 35 Hz. With minor differences in the precise peak positions (28 and 44 Hz) and amplitudes, the FEM results show closely similar results. The first peak of both the analytical model and FEM nearly match, but the second peaks show small differences. In comparison to the analytical model, the FEM model’s anti-resonance dip is noticeably less, and the difference in frequency more pronounced. However, in general, the analytical model accurately depicts the fundamental features of the beam’s dynamic behaviour; nevertheless, additional refinement is required to better align the model with the FEM.

3.3. Effect of Slit Parameters on Beam Dynamics

The arrangement of tip masses (m₁ and m₂) on the slitted beam and slit width and length are depicted schematically in Figure 6. To optimize the design of piezoelectric energy harvesters and enable more effective energy extraction from environmental vibrations, it is necessary to comprehend the effects of varying these parameters, (including whether to have a single distinct tip mass or two tip masses at the free ends of the beam) on the energy capture efficiency and resonance behaviour of the beam.

In this parametric study, the properties of the piezoelectric material, PZT-5A, included a Young’s modulus of 120.35 GPa, a density of 7750 kg/m³, a piezoelectric constant d₃₁ of −1.57 × 10⁻¹⁰ m/v, and elastic compliance at the constant electric field $s_{11}^E$ of 1.81 × 10⁻¹¹ [52]. These values are consistent with those reported in the specialized literature, confirming their suitability for energy harvesting applications [53]. The substructure is composed of aluminium with a Young’s modulus of 100 GPa and a density of 7165 kg/m³. The geometric dimensions of the beam and other parameters used in this study are provided in

Table 1. Baseline parameters used for the analysis of the slitted beam.

Property	Value
Beam Length	142 mm
Substructure Thickness hs	0.7 mm
Piezo Thickness	0.66 mm
Beam Width	28 mm
Slit Length	92 mm
Slit Width	2 mm
Load Resistance	1 × 104 Ω
Tip Mass (m1)	5 g
Tip Mass (m2)	0

.

3.4. Impact of Varying Single Tip Mass on Frequency Response of Slitted Beam

Optimizing the energy harvesting efficiency of slitted beams equipped with PZT requires, amongst other things, examining the impact of varying a single tip mass on the frequency response of the beams. Here, the mass on one free end is changed, while there is no tip mass on the other free end. Using this method, we can better understand how variations in tip mass affect resonance peaks and anti-resonance troughs, which in turn impact the dynamic response and energy capture capacities of the system. Designing piezoelectric energy harvesters that are more efficient requires an understanding of these effects. We analyzed the frequency response function under various single-tip mass situations to determine the best configurations for improved performance at higher frequencies.

Figure 7 depicts the FRF of a slitted beam with a PZT layer for three tip masses. Resonance peaks at around 27 Hz and 43 Hz are observed for all three configurations, with the magnitudes of these peaks increasing with increases in tip mass. This indicates that the addition of tip mass increases the beam’s potential to harvest energy. Also evident is anti-resonance between the main resonance peaks, which has a minimum value for the 5 g tip mass. In general, the inclusion of tip mass enhances the dynamic response, especially at higher frequencies, suggesting that optimizing tip mass can increase the performance of piezoelectric energy harvesters. Comprehending the effects of resonance and anti-resonance is essential for the design of energy harvesting devices that are highly efficient. Also, the value of the tip mass can make the harvester tuneable for different ambient vibrations.

3.5. Influence of Slit Width on Frequency Response of Slitted Beams

Optimizing the energy harvesting capabilities of slitted beams with PZT requires an understanding of how slit width affects the frequency response of the beams, the resonant frequencies, the resonance peak amplitudes, and the existence of anti-resonance dips, all of which are crucial elements in establishing the energy capture efficiency, as impacted by slit width. Our goal was to maximize the overall potential for energy harvesting by investigating dynamic response for slit widths of 2 mm, 4 mm, and 6 mm.

Figure 8 illustrates the dynamic behaviour of a slitted beam with varying slit widths (2 mm, 4 mm, and 6 mm). As the slit width increases, the resonance peaks shift to lower frequencies due to the reduced effective stiffness of the beam, making it more flexible. In mechanical systems, lower stiffness corresponds to lower natural frequencies, explaining the leftward shift of the peaks. Additionally, the distance between the two resonance peaks decreases as the slit width increases, which is attributed to enhanced mode coupling. A wider slit introduces more interaction between the beam’s vibrational modes, leading to closer resonance frequencies. This behaviour highlights how increasing slit width reduces stiffness and amplifies mode coupling, which are critical factors to consider in optimizing energy harvesting device designs using slitted beams.

Furthermore, the anti-resonance frequency, where the response dips significantly, also shifts to lower values as the slit width increases. This shift is again due to the reduction in stiffness and the changes in mode coupling. The anti-resonance dip becomes more pronounced as the slit width increases because the stronger coupling between vibrational modes leads to more destructive interference at certain frequencies, resulting in a deeper and more noticeable reduction in output at the anti-resonance point. This behaviour emphasizes the need to carefully design slit dimensions when tuning the dynamic response for specific applications, such as energy harvesting or vibration control.

3.6. Effect of Slit Distance on the Frequency Response of Slitted Beams

Optimizing the dynamic behaviour and energy harvesting efficiency of slitted beams using PZT requires examining the effect of slit length on the frequency response of the beams. The coupling effect between the resonance modes can be strongly influenced by the distance between the fixed end and the beginning of the slit. It is essential to comprehend how different slit distances impact the resonance peaks and coupling strength when building beams to optimize the amount of energy captured from ambient vibrations. We investigated the connection between slit geometry and dynamic response for three slit lengths: 67 mm, 92 mm, and 117 mm.

Figure 9 shows the results for three different slit lengths, namely, 67, 92, and 117 mm. Obviously, the frequencies of the first resonance peaks increase with a slit length of 117 mm and are well separated, and the anti-resonance dip becomes more pronounced. However, regarding the second resonance peaks, two observations can be made: First, the separation between the first and second peaks decreases with a decrease in slit length; secondly, the spread in the frequencies of the second peaks is much less than for the first peaks, again indicating stronger coupling, and the anti-resonance becomes less prominent. As would be expected for less slit length (67 mm), when the situation approaches that of the standard beam, the response approaches that of a standard beam without a slit. This trend suggests that the dynamic response and energy harvesting capability of the beam can be greatly affected by optimizing the slit distance.

The optimization of the dynamic response of the slitted cantilever beam through the meticulous modification of slit parameters greatly enhances its energy collecting capability. The inclusion of slits in the system results in mode coupling, therefore expanding the range of operating frequencies. This phenomenon enhances the efficiency of energy capture of the harvester from a broader range of ambient vibrations. Through the manipulation of the slit width, length, and distance from the fixed end, together with the adjustment of the tip mass, it is possible to shift and amplify the resonance peaks, thus resulting in increased energy production. The slitted beam is more effective in energy harvesting applications due to its improved design, which enables better adaptation to varied vibrational conditions. These structural changes increase the slitted beam’s total energy conversion efficiency, as confirmed by the results of FEA simulations and the analytical model. This suggests a viable way to improve the functionality of piezoelectric energy harvesters.

4. Conclusions

The goal of this study was to examine how different slit parameters affect the dynamics of slitted beams with PZT installed. The dynamic behaviour and energy harvesting potential of slitted beams using PZT was investigated in terms of the frequency response functions of such beams for different slit distances, slit widths, and tip masses, using both analytical techniques and finite element analysis.

Key findings of the study include the following:

• Impact of Slit Length: Resonance peaks are brought closer together by decreasing the slit length, which improves the coupling between vibrational modes. At larger slit lengths, the response becomes closer to that of a conventional beam without a slit, confirming that the best possible slit placement is essential to achieving maximum energy harvesting efficiency.
• Influence of Slit Width: Changes in the slit width have an impact on the resonance peaks’ amplitude and the existence of anti-resonance dips. While larger slits make anti-resonance dips more prominent, narrower slits show higher peak amplitudes, suggesting better potential for energy harvesting.
• Tip Mass Effects: Adding tip mass on one side of the beam shifts the resonance peaks and increases their amplitudes, strengthening the energy harvesting capability. Tip mass optimization can greatly enhance dynamic response and performance, as the anti-resonance effects seen between main resonance peaks become less pronounced with increasing tip mass.
• Validation using Finite Element Analysis: The main resonance frequencies are remarkably consistent between the analytical and FEA results. The FEA model provides a more comprehensive assessment of the beam’s dynamic behaviour, accounting for extra complexity, including secondary peaks and anti-resonance troughs.
• Future Work: Optimizing slit geometry further to improve the dynamic response and energy harvesting capability of the slitted beam could be the main emphasis of further work. Deeper understanding of the mode coupling effects and anti-resonance behaviours can come from parametric research altering the number of slits, their orientations, and their forms. Furthermore, key performance indices like energy efficiency and power density are crucial for optimizing slitted beams. Future work should focus on improving these through optimized slit geometries and materials. Furthermore, future work should investigate nonlinear dynamics and their effects on energy collecting efficiency, especially for uses in low-frequency situations. Furthermore, further investigations for self-tuning slitted beams that can adjust to different frequencies for more general and effective energy capture should consider the incorporation of smart materials, like shape memory alloys. Finally, the experimental validation of these cutting-edge designs in real-world environments would offer useful information for slitted beam energy harvesters’ scaling-up for pragmatic uses.

In summary, this study emphasizes how crucial it is to optimize tip masses and slit characteristics to improve the overall performance of piezoelectric energy harvesters. An all-encompassing strategy for comprehending and enhancing the design of slitted beams with PZT for effective energy harvesting could be provided through the combination of analytical and FEA techniques to further advance this potential technology.