Multi-Variable Optimization to Enhance the Performance of a Cantilevered Piezoelectric Harvester

Abstract

Over the past decade, various researchers have shown the practicality of extracting electrical energy from vibrating structures. The primary objective of this research was to optimize the parameters of a basic cantilever harvester to maximize the energy production of electricity from surrounding mechanical energy. A distributed parameter model and its modal solution were utilized to determine the design variables via a parametric investigation. It is important to ensure sufficient power generation when there is a wide range of frequencies being excited, without any specific frequency being emphasized. The cost function for the optimization problem was defined as the average power within a specific frequency range, considering geometric and physical restrictions. It was discovered that different parameters’ values can be utilized to generate varying maximum power levels for different frequency ranges. Furthermore, a comparative analysis was performed between the finite element method (FEM) using ANSYS and the current harvesting model. The comparison demonstrated a high level of concordance, enabling the utilization of the finite element method (FEM) for subsequent investigation.

1. Introduction

Over the past years, there has been a substantial increase in the implementation of digital processes and wireless communication networks across diverse sectors. Nevertheless, some systems, particularly those situated in distant or inaccessible areas, necessitate continuous functioning, hence resulting in expensive and disruptive battery replacements. Wireless sensors, with power needs ranging from 10 to 100 microwatts, provide a potential answer to this problem by utilizing energy from the surrounding environment. This feature allows for uninterrupted operation without the necessity of regular battery changes, hence improving the dependability and effectiveness of these systems.

A highly efficient approach to execute energy harvesting in such applications involves utilizing piezoelectric materials, which can convert mechanical vibrations from the surroundings into electrical energy. Vibrations of a low level commonly occur in various settings, such as household appliances, power plants, bridges, aircraft, automobiles, and even the human body. These vibrations offer an easily accessible energy source that can be utilized to power low-energy gadgets. An energy harvesting design that is very efficient has been identified as a cantilever construction with piezoelectric material attached either to both the upper and lower surfaces or to a single side. The vibration of the cantilever causes it to flex, resulting in the generation of stress in the piezoelectric layers, which in turn produces an electrical charge.

Roundy et al. [1] conducted a study where they examined and contrasted various potential sources of ambient power with fixed energy sources. They explored the potential for converting environmental vibrations into electrical energy. Roundy and Wright [2] employed a two-layer bending element with a cantilever model that has a single degree of freedom to construct and enhance a piezoelectric generator.

Erturk and Inman [3] presented a precise solution for a piezoelectric cantilever energy harvester. The harvester is excited by simple movements, specifically transverse and limited rotational translations. The solution is based on assumptions derived from the Euler–Bernoulli beam theory. Erturk and Inman [4] conducted an experimental test of their model using the same prior assumptions. Sodano and Inman [5] developed an analytical model of a beam with a piezoelectric component by utilizing Hamilton’s theory to estimate the energy generated by the piezoelectric effect, based on the assumptions of the Euler–Bernoulli beam theory. Scruggs [6] introduced the concept of a feedback controller to enhance the power output of an energy harvesting system. The theory is utilized for vibratory systems that are activated in a random manner and exhibit a wide range of frequencies in their stationary response. Renno et al. [7] examined the impact of damping on power and discovered both quantitative and qualitative impacts. The study examined the consequences of electromechanical coupling and the influence of a circuit containing an inductor. In order to improve the results of the harmonic base excitation model for the transverse and longitudinal vibrations of the single degree of freedom (SDOF), Erturk and Inman [8] incorporated a correction factor into the SDOF model of the cantilevered beam. De Marqui Jr. et al. [9] proposed an finite element plate model with electromechanical coupling that can accurately forecast the power generated by a piezoelectric energy harvester.

Lee et al. [10] conducted a study that examined the performance of substituting the widely used silicon substrate with graphene, and the commonly used lead zirconate titanate (PZT) with ZnO. Graphene greatly improved performance when the thickness was 200 nm.

Xiong et al. [11] enhanced the energy conversion efficiency of a piezoelectric cantilever beam energy harvester by introducing a cavity in the metal substrate. The rectangular aperture resulted in a decrease in frequency, an increase in bandwidth, and an amplification in both voltage and power output. The study verified the model’s accuracy by employing COMSOL and MATLAB to analyze the eigenfrequency and frequency domain. Garg and Dwivedy [12] utilized both theoretical and experimental methods to examine the efficiency of a piezoelectric energy harvester under parametric excitation. Xu and Li [13] introduced a novel stochastic averaging method to analyze the behavior of a bistable energy harvester under random excitation. Keshmiri et al. [14] introduced a novel nonlinear cone-shaped piezoelectric cantilever beam design that has a generation efficiency 19.76 times greater than that of the conventional uniform cantilever generator.

Although there have been notable improvements in piezoelectric energy harvesting, the challenge of tuning these devices to function efficiently across a wide range of vibrational frequencies is still a significant problem. Most previous research has focused on particular frequency ranges or constrained operational conditions, thereby limiting the generalizability of their results to real-world settings where vibrational frequencies are frequently unpredictable and diverse. This work employs an established analytical solution to simulate the behavior of cantilevered piezoelectric harvesters. The accuracy of the solution is verified by comparing it with finite element modeling (FEM). After confirming its validity, the analytical solution is used to conduct a thorough analysis of many parameters and optimize the system. The parametric study methodically examines the impact of crucial parameters, including beam length, beam width, substructure thickness, piezoelectric layer thickness, and the piezoelectric constant, on the effectiveness of energy harvesting. Subsequently, an optimization problem is defined and resolved to maximize energy production by precisely adjusting these parameters. Our objective is to improve the adaptability and efficiency of cantilevered piezoelectric harvesters by including these methods. This would enable them to efficiently harness energy from various frequencies of environmental vibrations. This study not only fills the gaps left by prior research but also offers a strong framework for constructing sophisticated energy harvesting devices that are better suited to real-world, fluctuating situations.

2. Mathematical Modeling

2.1. Electromechanical Model of the Harvester

This study examined a harvester, including a piezoelectric layer positioned on top and attached to a substructure layer positioned at the bottom. The two layers were presumed to be perfectly adhered together at one end, with the other end free, forming a cantilever beam harvester, as depicted in Figure 1. To develop the analytical model, Euler–Bernoulli assumptions were utilized for this problem and it was assumed that the conducting electrodes completely covered the top surface. Furthermore, a load resistance was attached to the piezoelectric layer. Ultimately, a fundamental oscillatory motion is exerted on the harvester from its base.

The electric field, stress, and strain of the harvester were determined by applying the piezoelectric constitutive laws as specified in the IEEE Standard on Piezoelectricity [15]:

$$T_1^S=Y_S{S}_1^S$$

$$T_1^p=Y_p(S_1^p-d_{31}{E}_3)$$

(1)

$$S_1=s_{11}^E{T}_1+d_{31}{E}_3$$

(2)

where T and S are the stress and the strain, respectively, E₃ is the electric field, d₃₁ is the piezoelectric constant, $s_{11}^E$ is the elastic compliance at the constant electric field, and Y is the Young’s modulus. Subscript 1 refers to the direction of axial strain (x-direction), while subscript 3 is the direction of polarization (y-direction). Superscripts p and s stand for PZT and substructure, respectively.

The coupling term is derived from:

$$\vartheta =-{\displaystyle \frac{Y_p{d}_{31}b}{2h_p}}(h_c^2-h_b^2)$$

(3)

where h_b is the distance from the neutral axis to the bottom of the PZT layer, and h_c represents the distance from the neutral axis to the top of the PZT layer, where h_p denotes the PZT thickness and b is the beam width, as depicted in Figure 2.

$$\begin{array}{c}YI{\displaystyle \frac{{\partial }^4{w}_{rel}(x,t)}{\partial x^4}}+c_{s}I{\displaystyle \frac{{\partial }^5{w}_{rel}(x,t)}{\partial x^4\partial t}}+c_a{\displaystyle \frac{\partial w_{rel}(x,t)}{\partial t}}+m{\displaystyle \frac{{\partial }^2{w}_{rel}(x,t)}{\partial t^2}}\\ +\vartheta \upsilon (t)\times \left[{\displaystyle \frac{d\delta (x)}{dx}}-{\displaystyle \frac{d\delta (x-L)}{dx}}\right]=-m{\displaystyle \frac{{\partial }^2{w}_b(x,t)}{\partial t^2}}-c_a{\displaystyle \frac{\partial w_b(x,t)}{\partial t}}\end{array}$$

(4)

Here, c_a and c_s are viscous air damping and the equivalent coefficient of strain rate damping, respectively; I is the equivalent area moment of inertia of the composite cross-section; w_b and w_rel are base motion and the transverse deflection of the beam to its base, respectively. YI represents the bending stiffness, m is the mass per unit length of the beam, $v\left(t\right)$ is the voltage, and L denotes the beam length.

The equations of the current and voltage can be expressed as follows [3]:

$$i(t)=-\underset{x=0}{\overset{L}{\int }}{d}_{31}{Y}_p{h}_{pc}b{\displaystyle \frac{{\partial }^3{w}_{rel}(x,t)}{\partial x^2\partial t}}dx-{\displaystyle \frac{{\epsilon }_{33}^{S}bL}{h_p}}{\displaystyle \frac{d\upsilon (t)}{dt}}$$

(5)

$$\upsilon (t)=R_{l}i(t)=-R_l\left[{\int }_{x=0}^L{d}_{31}{Y}_p{h}_{pc}b{\displaystyle \frac{{\partial }^3{w}_{rel}(x,t)}{\partial x^2\partial t}}dx-{\displaystyle \frac{{\epsilon }_{33}^{S}bL}{h_p}}{\displaystyle \frac{d\upsilon (t)}{dt}}\right]$$

(6)

where ${\epsilon }_{33}^s$ and R_L represent the permittivity at constant strain and the load resistance, respectively.

2.2. Base Motion

In this section, the solution steps of the electromechanically coupled equations of the harvester are described. The relative motion of the beam is governed by

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

(7)

where η_r(t) is modal coordinate of a clamped-free beam for the rth mode, while ${\varphi }_r$ is the orthonormal mode shapes of the free vibration of a cantilever beam which can be governed using the following equation [3]:

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

(8)

Here, λ_r is associated with the nth natural frequencies and is determined using the following characteristic equation:

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

(9)

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

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

(10)

The undamped modal frequency is defined as:

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

(11)

and the modal coupling term

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

(12)

The mechanical damping ratio can be assumed.

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

(13)

The circuit time constant is

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

(14)

2.3. Frequency Response Function (FRF)

In this section, all vibration modes are neglected, except the transverse vibration mode. It is also assumed that the voltage modulus is the ratio of the voltage output to the base acceleration, which can be obtained from [3]:

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

(15)

where ω is the frequency of the base excitation.

Using Ohm’s law, the current modulus FRF is:

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

(16)

Here,

$${\gamma }_r^w=\underset{x=0}{\overset{L}{\int }}{\varphi }_r(x)dx$$

(17)

To derive the power modulus, multiply the current FRF with the voltage FRF. The power modulus represents the power output relative to the square of the base acceleration.

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

(18)

Here, Y₀ is the amplitude of the base motion.

3. Parametric Study

Using the distributed parameter model presented above, a unimorph harvester is presented, with the geometric, material, and electromechanical parameters given in

Table 1. Geometric, material, and electromechanical characteristics of the sample harvester.

Description	Symbol (Unit)	Value
Beam length	L (mm)	100
Beam width	b (mm)	20
Substructure thickness	hs (mm)	0.5
Piezoelectric thickness	hp (mm)	0.4
Substructure Young’s modulus	Ys (GPa)	100
PZT Young’s modulus	Yp (GPa)	66
substructure mass density	ρs (kg/m3)	7165
PZT mass density	ρp (kg/m3)	7800
Piezoelectric constant	d31 (pm/V)	−190

.

By utilizing the energy harvester with the specified parameters outlined in Table 1, it was determined that the first three modal frequencies fall within the range of 0 Hz to 1000 Hz. Hence, to gain a more comprehensive understanding of how various parameters impact the overall power output, particularly when altering the modal frequencies, a nominal frequency range of 0 to 1000 Hz is considered. This study examines four primary FRF functions: the voltage applied across a resistive load, the current through a resistive load, the generated electrical power, and the relative motion transfer from the base to the beam tip. The voltage and current frequency response functions provide valuable insights into how changes in certain parameters may impact the overall output power. Alternatively, studying relative motion provides a more comprehensive understanding of how the harvester will mechanically react to parameter changes. It also elucidates the decrease or increase in power caused by mechanical factors, anti-resonance nodes, damping, stiffness, and various modes such as torsional and lateral.

Additionally, the impact of the load resistance is examined. This study explores two primary scenarios: a short circuit condition and a open circuit state. If the load resistance approaches zero (Rl → 0), the circuit is regarded as a short circuit with a short circuit natural frequency, ω_r^sc. Conversely, when the load resistance is extremely high (Rl → ∞), the circuit is regarded as an open circuit with an open circuit natural frequency, ωr^oc. The values within the range of ω_r^sc and ω_r^oc are regarded as moderate.

There are two types of damping believed to be present in the system: internal damping (strain rate) c_sI, which is directly related to stiffness, and external damping (air damping) c_a, which is directly related to mass. Experimental methods can be used to determine the values of damping. The values ζ₁ = 0.010 and ζ₂ = 0.013 were employed in this study [3]. By utilizing the specified harvester characteristics outlined in Table 1, we determined the values of the first and second modal frequencies, which were ω₁ = 47.8 Hz and ω₂ = 299.6 Hz, respectively. The damping constants in the governing equations were calculated as c_sI/YI = 1.2433 × 10⁻⁵ s per radian and c_a = 4.886 radians per second.

3.1. Frequency Response of the Voltage Output

The load resistance R_L significantly affects the dynamic behavior of the system. When the load resistance is close to zero (R_L → 0), the behavior of the circuit is predicted to be similar to a shorted circuit. Conversely, the system is anticipated to be in an opened circuit state when the load resistance (R_L) approaches infinity (R_L → ∞). The voltage frequency response function (FRF) is a measure of the ratio of the voltage output to the base acceleration, as described above in Equation (17). Figure 3 displays the voltage frequency response function (FRF) for five distinct load resistance values, ranging from 10² Ω to 10⁶ Ω. It is important to note that the voltage output is significantly influenced by the load resistance. The greatest voltage is attained at a frequency of (${\omega }_1^{sc}=$ 47.8 Hz) for the short circuit scenario (R_L = 10² Ω). However, when the circuit is open, (R_L = 10⁶ Ω), the resonance frequency is (${\omega }_1^{oc}=$ 48.8 Hz).

Figure 4 illustrates the relationship between the voltage output and load resistance for excitations at frequencies corresponding to short and open circuits. When the load resistance is low (in a short circuit condition), the voltage output is greater than in the open circuit condition until the curves overlap at a load resistance of approximately 43,246 Ω. At the open circuit resonance frequency, the voltage output is higher than at the short circuit. At high values of load resistance, the voltage output becomes less responsive to changes in the load resistance, and the slope for both open and short circuits decreases, as depicted in Figure 4.

3.2. Frequency Response of the Current Output

The current was determined by dividing the voltage by the load resistance, as introduced in Equation (16), as well as the current FRF. Figure 5 shows the current output frequency response function (FRF) for different load resistance values, with a closer look at the first mode of the current FRF for five distinct load resistance values. The current exhibits an inverse relationship with the voltage, meaning that as the load resistance increases, the amplitude of the current drops in a consistent and continuous manner. Figure 6 illustrates the correlation between the current output and load resistance when the system is excited at the short circuit and open circuit resonance frequencies of the first mode. Furthermore, it is evident that the current remains nearly constant when the load resistance is low. Additionally, the short circuit resonance frequency exhibits a higher current output compared to the open circuit resonance frequency, until the two curves intersect at R_L = 4000 Ω. After this point, the current begins to decrease as the load resistance increases, with a steep slope. Following this crossing point, when the load resistance is increased, the current output at the resonance frequency of the open circuit becomes greater than that of the short circuit.

3.3. Frequency Response of Power Output

Figure 7 depicts the power output frequency response function (FRF) for five distinct load resistance values, with a magnified view of the first mode. In contrast to the voltage FRF and current FRF, the power FRF does not follow a monotonic pattern. When the load resistance is increased or decreased, there is a point of intersection between the R_l = 10³ Ω and R_l = 10⁴ Ω curves at a frequency of 193.7 Hz. This intersection occurs off resonance, indicating that the same amount of power can be obtained using either resistance value. The magnified display reveals the characteristics of both short circuit and open circuit behavior, demonstrating the presence of junctions among the FRF curves. The initial point of intersection occurs at a frequency of 48.19 Hz, where the resistance values of 10⁴ Ω and 10⁵ Ω intersect.

Based on previous findings, the voltage and current at the resonance frequency when the circuit is short-circuited are higher compared to the frequency when the circuit is open, until the load resistance reaches the value of R_l = 40.9 kΩ. After this point, the voltage and current at the open circuit frequency increase. The power behavior is anticipated to remain consistent, as power is determined by multiplying the voltage and current. Figure 8 demonstrates that the power output is greater at the short circuit resonance frequency until it reaches the load resistance value of 40.9 kΩ. Beyond this threshold, the power output is larger at the open circuit resonance frequency. Furthermore, both open circuit and short circuit graphs exhibit maximum values and yield the same power output.

3.4. Beam Frequency Response

The relative tip motion frequency response function (FRF) is the quotient of the amplitude of vibration displacement at the tip of the beam and the amplitude of the base displacement. It is also referred to as the relative motion transmissibility. In this study, the motion at the tip of the beam is of utmost significance, as it is at this location where the transverse displacement reaches its highest point during vibration modes. The mechanical frequency response function (FRF) of the relative tip motion can be mathematically represented as

$${\displaystyle \frac{w_{rel}(L,t)}{Y_0{e}^{j\omega t}}}={\displaystyle \sum _{r=1}^{\infty }\left[{\gamma }_r^w-{\chi }_r\left(\frac{{\displaystyle \sum _{r=1}^{\infty }\frac{j\omega {\varphi }_r{\gamma }_r^w}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }}}{{\displaystyle \sum _{r=1}^{\infty }\frac{j\omega {\varphi }_r{\chi }_r}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }+\frac{1+j\omega {\tau }_c}{{\tau }_c}}}\right)\right]}\times {\displaystyle \frac{m{\omega }^2{\phi }_r(L)}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }}$$

(19)

Figure 9 shows the relative tip motion frequency response function (FRF) for both the uncoupled system and the coupled system. The linked system was tested with five different load resistance values, focusing on enlarged mode 1. The figure demonstrates that the relative tip motion is not affected by changes in the load resistance in the low load resistance zone. An anti-resonance occurs at a frequency of 538.6 Hz (for open circuit) and 536.7 Hz (for short circuit). Anti-resonance refers to a state in which the impedance of an electric, acoustic, or dynamic system decreases significantly, reaching a value close to infinity. The amplitude of anti-resonance is observed to be minimal, but it would be limitless if there were no mechanical damping or losses.

Upon examining the magnified view of mode 1, it can be observed that the first natural frequency shifts from 47.8 Hz to 48.8 Hz. This alteration in frequency is responsible for the reduction in vibration amplitude at 47.8 Hz, rather than being caused by damping. The load resistance is rising because of the electromechanical effect, causing a decrease in the vibration amplitude at the resonance frequency when the circuit is short-circuited.

Figure 10 illustrates how changing the load resistance affects the relative tip motion when the base is excited at the resonance frequencies of the first vibration mode, under both short circuit and open circuit conditions. The movement at the end of the beam is crucial in determining the capacity of the harvester. When the load resistance is raised to 10⁶ Ω, the amplitude of vibration at this frequency begins to rise. As a result, the vibration amplitude of frequency does not consistently exhibit a monotonic pattern when the load resistance is increased or decreased. At the resonance frequency when the circuit is open, the amplitude of vibration initially decreases gradually and then rapidly increases as the load resistance increases.

From Figure 8 and Figure 10, it is evident that at the short circuit frequency, the power output steadily increases as the load resistance increases until it achieves its maximum value. Simultaneously, the vibration amplitude is reduced. Continuing to raise the load resistance further reduces the power output and increases the amplitude of vibration. At the resonance frequency when the circuit is open, raising the load resistance results in an increase in power output and a little decrease in vibration amplitude. However, beyond this point, the vibration amplitude grows rapidly as the load resistance continues to increase.

3.5. Average Power Effect

In this section, a parametric study is presented to analyze the effect of changing different parameters on the power output of the harvester and examine the sensitivity of these parameters on the generated power.

Some harvesters exhibit a significant power output within a certain frequency range, namely at their resonance frequency, but experience a sharp decrease in power outside this range. The primary goal is to achieve the highest average power across a wider range of frequencies. The average power within the required frequency range [ω₁:ω₂] is defined as

$$P_{avg}={\displaystyle \frac{{\displaystyle {\int }_{{\omega }_1}^{{\omega }_2}}(Power FRF)d\omega }{{\omega }_2-{\omega }_1}}$$

(20)

By substituting Equation (18) into Equation (20), we obtain

$$P_{avg}={\displaystyle \frac{{\displaystyle {\int }_{{\omega }_1}^{{\omega }_2}}\left(\frac{{\left(\sum _{r=1}^{\infty }\frac{-jm\omega {\varphi }_r{Y}_r^w}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }\right)}^2}{R_l{\left(\sum _{r=1}^{\infty }\frac{j\omega {\varphi }_r{X}_r}{{\omega }_r^2-{\omega }^2+j2{\zeta }_r{\omega }_r\omega }+\frac{1+j\omega T_c}{T_c}\right)}^2}\right)d\omega }{{\omega }_2-{\omega }_1}}$$

(21)

The specified range in this case study spans from 0 to 1000 Hz. This range was selected to ensure that the first three modal frequencies were covered. Alternatively, the frequency range for the harvester can be specified based on its characteristics and the existing ambient vibration.

Figure 11 displays the mean power frequency response of commonly utilized piezoelectric materials. Furthermore, this figure shows an amplified representation of the initial modal frequency within the resonance zone. According to the figure, PZT-507 material generates the highest power output among the eight materials, while PZT-807 produces the lowest power.

Table 2. Different piezoelectric material properties.

	PZT-5H1	PZT-507	PZT-806	PZT-401	PZT-807	PZT-5A	PZT-5A1	PZT-5A2
ρ (kg/m3)	7500	7800	7500	7600	7600	7800	7750	7750
d31 (nm/V)	−0.274	−0.360	−0.123	−0.132	−0.097	−0.19	−0.157	−0.171
E (GPa)	72	72	72	72	72	66	72	72

presents some of the piezoelectric parameters that can significantly impact the power output of an energy harvester, including the constant d₃₁, density ρ, and modulus of elasticity E. The modal frequencies of all materials exhibit minimal deviation either to the right or to the left. This alteration in natural frequency mostly arises from variations in density and modulus of elasticity among the piezoelectric materials. The PZT-5A material exhibits the largest density and lowest modulus of elasticity, which results in the anticipation of the lowest natural frequencies.

In this analysis, we examined the impact of different parameters on the average power output. We utilized Equation (21) and considered frequencies ranging from 0 to 1000 Hz.

Figure 12 illustrates how changing the magnitude of the piezoelectric constant affects the average power produced by the harvester. Figure 12 shows that as the absolute value of the piezoelectric constant increases, the average power rises significantly. Therefore, selecting a material with a greater piezoelectric constant can effectively raise the power output of the harvester without altering its major features.

The piezoelectric constant directly influences the modal coupling of the harvester, in conjunction with the modal coupling term (χ_r). Moreover, the piezoelectric constant directly affects the mass normalized term (φ_r), meaning that an increase in the piezoelectric constant will result in a corresponding rise in the mass normalized term.

Figure 13 illustrates the impact of altering the beam length of the harvester on the mean power production. It indicates that the average power increases in direct proportion to the beam length until it reaches an ideal point at around 210 mm, after which it begins to drop. It is a well-established fact that when the length of the beam increases, the modal frequencies move towards the lower frequencies in the frequency response figures. When the beam length is reduced within a defined frequency range, some modal frequencies might not be captured unless the frequency range is extended. For example, with a beam length of 50 mm and a frequency range from 0 to 1000 Hz, only the first modal frequency will be detected. However, with a beam length of 100 mm and the frequency range still set from 0 to 1000 Hz, three resonance frequencies will fall within this range. Hence, the two jumps depicted in the illustration can be elucidated by this principle: when the length of the beam surpasses a specific threshold, the third modal frequency becomes encompassed within the power function, resulting in a significant amplification of power. Similarly, the second jump occurs when the beam length reaches a certain value and the second modal frequency falls within the frequency range. The beam length has an impact on the circuit time constant, damping ratio, modal coupling term, and mass normalized term. In general, the length of the beam directly influences the power output of the harvester.

Figure 14 illustrates the impact of altering the thickness of the piezoelectric layer on the average output power within the specified frequency range. The figure demonstrates a significant decrease in average power output within the range of layer thicknesses from 0.1 mm to 0.6 mm. However, the decline in power output is less pronounced for piezoelectric thicknesses ranging from 0.6 mm to 2 mm. The change in slopes occurs because the moment of inertia of the beam increases proportionally with thickness. It is well known that as the moment of inertia increases, the modal frequency also increases. This implies that fewer modal frequencies will be captured within the given frequency range.

The thickness of the piezoelectric material directly affects several parameters, such as the modal coupling term, damping ratio, circuit time constant, beam mass, bending stiffness, mass normalized term, resonance, coupling term, and viscous air damping coefficient.

In general, when the thickness of the piezoelectric material increases, the power generated by the harvester decreases.

Examining the beam width is crucial in this study as it is one of the most significant aspects. Figure 15 demonstrates a significant rise in the amplitude of output power as the beam width increases. Additionally, the beam width influences several factors within the circuit, such as the time constant, damping ratio, modal coupling term, mass normalization, beam mass, bending stiffness of the composite cross-section, and the coupling term.

The thickness of the substructure is a crucial geometric factor that influences multiple aspects, such as the damping ratio, modal coupling term, mass normalization, beam mass, bending stiffness of the composite cross-section, coupling term, resonance, and the viscous air damping coefficient. We evaluated six distinct values for the thickness of the substructure. It was discovered that increasing the thickness of the substructure improves both the power output amplitude and the resonance frequencies. Figure 16 depicts the impact of substructure thickness on the mean power generated within the frequency range of 0 to 1000 Hz, while keeping all other factors constant, as specified by Equation (21). According to Figure 16, the average power generated by the harvester increases as the substructure thickness increases.

4. Optimization

The findings from the previous section provided a very good understanding of how the parameters influence the average power output and how sensitive the average power output is to each parameter. The length and width of the beam, the thickness of the substructure, the thickness of the piezoelectric layer, and the piezoelectric constant are the most sensitive parameters on the output power based on the conclusion from the previous section. In this section, each of the previously studied parameters is optimized and all other variables are kept constant for different frequency ranges. The purpose of the single variable optimization is to ensure that the numerical algorithm converges to the correct optimum values for each case.

Table 3. Single parameter optimization using the nominal values of the harvester.

Frequency Range (Hz)	Optimum Value of Beam Width (mm)	Optimum Value of Beam Length (mm)	Optimum Value of Substructure Thickness (mm)	Optimum Value of Piezoelectric Thickness (mm)	Optimum Value of Piezoelectric Constant (nm/V)	Optimum Value of Load Resistance (Ω)
Subjected	5 < b < 100	10 < L < 250	0.1 < hs < 1	0.1 < hp < 1	−0.4 < d31 < −0.1	10 < R < 1010
300 to 600	100	161.5	1	1	0.4	5.1186 × 103
400 to 700	100	83.5	1	1	0.4	3.9845 × 103
500 to 800	100	74.5	1	1	0.4	3.2806 × 103
700 to 1000	100	104.9	1	1	0.4	2.4403 × 103
0 to 200	100	160.3	1	0.4845	0.4	4.0946 × 104
0 to 1000	100	211.4	1	0.4585	0.4	4.0963 × 104

shows the optimal values of different parameters using the nominal values of the harvester as presented in Table 3.

As expected, the values of beam width and substructure thickness converged to the upper bound, which is consistent with the findings of the parametric study. The limits can be adjusted based on the range of the design variables used. The wider and thicker the beam, the higher the average power produced, regardless of the frequency range. On the other hand, from the optimization program, the maximum average power is produced at a beam width of 211.4 mm, which means there is an optimum value of the beam length. The bounds can be changed according to the design variable range that was issued. As shown in the previous table, at each frequency range, there is an optimum value of the beam length, which was expected from the results of the parametric study presented earlier. Therefore, if a maximum power needs to be produced from the energy harvester, one should first determine the frequency range in this environment and choose the beam length accordingly.

The optimum value of the piezoelectric thickness of the energy harvester mainly has two different scenarios. First, the values of the piezoelectric thickness converge to the upper bound in the absence of the first natural frequency. The second scenario occurs when the first natural frequency is included in the frequency range and an optimal thickness is found. As anticipated, the piezoelectric value always converges to the upper bound of the piezoelectric constant, which agrees with the previous findings. Therefore, it is better to choose a piezoelectric material with high piezoelectric constant to achieve higher average power output.

Finally, at each frequency range, there is a different optimal load resistance as depicted in Table 3, which indicates that if the frequency range of the environment is known, it is better to change the load resistance according to this frequency range to harvest the maximum power.

Now, the optimization problem is defined as minimizing the negative average power output $-P_{avg}$ subject to the bounds on design variables and geometric constraints shown in

Table 4. Design variable bounds.

Variable	Lower Bound	Upper Bound
Length (L)	10.0 mm	250 mm
Width (b)	5.0 mm	100 mm
Substructure thickness (hs)	0.10 mm	1 mm
Piezoelectric layer thickness (hp)	0.10 mm	1 mm
Piezoelectric constant (d31)	−400 pm/V	−100 pm/V

and

Table 5. Geometric constraints.

Constrain	Inequality
Width constrain	$\mathrm{b}<{\displaystyle \frac{L}{10}}$
Substructure thickness constrain	${\mathrm{h}}_{\mathrm{s}}<{\displaystyle \frac{L}{50}}$
Piezoelectric layer thickness constrain	${\mathrm{h}}_{\mathrm{p}}<{\displaystyle \frac{L}{50}}$

.

The boundaries can be adjusted based on the range of design variables that is utilized. In addition, the geometric restrictions are incorporated to ensure the validity of the beam assumptions. For instance, the thickness or width of the beam must not surpass its length. Hence, these limitations are incorporated into the formulation to ensure a rational geometry that can be effectively represented as a beam.

In order to solve the optimization problem, the MATLAB R2021a optimization toolbox function (fmincon) was adopted using a standard sequential quadratic programming (SQP) method. Different initial conditions were used in this multi-variable optimization to make sure that the problem is stable and always converges to the same optimum values.

Table 6. Harvester optimal values using multi-variable optimization.

Frequency Range (Hz)	L (mm)	b (mm)	hS (mm)	hP (mm)	d31 (pm/v)	R (KΩ)
300 → 600	242.40	24.20	1	1	−400	10.0
400 → 700	209.60	21.0	1	1	−400	10.0
500 → 800	50.0	5.0	1	1	−400	99.330
700 → 1000	50.0	5.0	1	1	−400	52.890
0 → 200	250.0	25.0	1	1	−400	35.440
0 → 1000	250.0	25.0	1	1	−400	23.750

presents the optimal values for various frequency ranges. For the 300 to 600 Hz frequency range, convergence was achieved after 24 iterations, yielding an optimal beam length of 242.4 mm and beam width of 24.2 mm. Within this range, the lower bounds were active for the piezoelectric constant and load resistance, while the upper bounds were active for the piezoelectric thickness and substructure thickness.

Within the 400 to 700 Hz range, the solution reached convergence after 33 iterations, resulting in an optimal beam length of 209.6 mm and beam width of 21 mm. As with the previous range, the lower bounds were active for the piezoelectric constant and load resistance, while the upper bounds were active for the piezoelectric thickness and substructure thickness.

In the 500 to 800 Hz range, convergence was achieved after 20 iterations, yielding optimal values of 50 mm for the beam length and 99.33 KΩ for the load resistance. In this range, the lower bounds were active for the piezoelectric constant and beam width, while the upper bounds were active for the piezoelectric thickness and substructure thickness.

In the 700 to 1000 Hz range, the solution converged after 16 iterations, yielding optimal values of 50 mm for the beam length and 52.89 KΩ for the load resistance. In this range, the lower bounds were active for the piezoelectric constant and beam width, while the upper bounds were active for the piezoelectric thickness and substructure thickness.

In the 0 to 200 Hz range, convergence was achieved after 13 iterations, with an optimal load resistance of 35.44 KΩ. In this case, the lower bound was active for the piezoelectric constant, while the upper bounds were active for the beam length, piezoelectric thickness, and substructure thickness.

Finally, in the 0 to 1000 Hz range, the solution converged after 12 iterations, with an optimal load resistance of 23.75 KΩ. In this frequency range, the lower bound was active for the piezoelectric constant, while the upper bounds were active for the beam length, piezoelectric thickness, and substructure thickness.

The main reason behind dividing the frequency into different ranges is to obtain a better understanding of the effect of each parameter on every modal frequency; for example, if the first modal frequency falls in the range between 0 and 200 Hz, while the second modal frequency falls in the range of 300 to 600 Hz, and the optimization program includes the range of 0 to 600 Hz, the result will always converge to the benefit of the first modal frequency because it will always be dominant. Hence, the frequency was divided into different ranges to provide insight into what will change in other ranges that include lower modal frequencies or off-resonance ranges.

Finite Element Model

This section describes the construction of a finite element model (FEM) using ANSYS software 14.5 and its comparison with the previous analytical model. Once the model is validated, it should be safe to create complex shapes for the harvester that provide a realistic result for future work. One of the main advantages of using ANSYS is that it has the ability to provide a coupled field analysis [11,16,17]. The coupled field analysis is carried out using the piezoelectric constitutive relationship between the electrical and structural fields.

The utilized energy harvester beam model comprises a piezoelectric layer on the upper side and a substructure layer on the lower side. The piezoelectric patch is coupled to a resistor, allowing for the measurement of voltage, current, and power across all nodes. The load resistance can be adjusted to any desired value, as shown in Figure 17.

The CIRCU94 element is assigned to the load resistance. The CIRCU94 model comprises two or three nodes that define the circuit component, with one or two degrees of freedom representing the circuit’s response. The substructure uses the SOLID45 element, with the material properties listed in

Table 7. Substructure material properties.

	ρ (kg/m3)	Y (GPa)	Poisson’s	Damping (kg/s)
Value	7165	100	0.3	$1.2433\times {10}^{-5}$

applied to it. SOLID45 is commonly used for the 3-D modeling of isotropic solid structures and is defined by eight nodes, each with three degrees of freedom: translation in the x, y, and z directions.

The SOLID5 element is classified as a piezoelectric patch in this investigation. SOLID5 can generate a piezoelectric field and exhibits coupling between different fields. It can also generate and manipulate three-dimensional magnetic, thermal, electric, and structural fields; however, the connection between these fields is limited. Each node of this element has a maximum of six degrees of freedom, and there are eight nodes in total. The PZT-5A2 material with Y-direction polarization is utilized in this case study. The modal analysis utilizes the block Lanczos mode extraction method to identify five modes and generate five extended mode forms. The Sparse solver with zero tolerance is utilized to obtain a complete harmonic solution approach. The substep count is 400, and the mesh size is 5 mm. The energy harvester has the following notional geometric dimensions: a substructure thickness of 0.5 mm, a beam width of 20 mm, and a beam length of 100 mm. The entire cantilever beam is fully covered from the top with a piezoelectric coating that has a thickness of 0.4 mm. The load resistance is 106 Ω. In order to capture a wider range of modal frequencies, the chosen frequency range is from 0 to 1000 Hz.

To validate the FEM solution with the analytical solution in Equation (4), the first modal frequencies were compared using both approaches. The results are presented in

Table 8. Frequency comparison in Hz between the analytical and FEM solutions.

Mode	Analytical	FEM	Error %
1	48.79	50.17	2.8%
2	301.50	308.62	2.35%
3	839.20	873.42	4.07%

, indicating a high level of agreement with an error of less than 5%.

Figure 18 presents a comparison between the FEM results using ANSYS and an existing harvesting model. The figure demonstrates a strong agreement between the FEM and analytical solutions. This successful validation of the FEM paves the way for further exploration of more complex energy harvester geometries.

5. Conclusions

This study introduced an optimization problem formulation based on a distributed parameter model of a cantilevered piezoelectric harvester, assumed to be excited transversely from the base. An analytical solution for the harvester beam was derived, and the design variables were identified through a parametric study. The optimization problem was then formulated and solved using the sequential quadratic programming (SQP) method, implemented through MATLAB’s optimization toolbox. Six design variables were selected to determine their optimal values across six different frequency ranges, demonstrating that each design variable has an optimal value that depends on the excitation frequency range. Six frequency ranges were selected to gain a better understanding of the changes in the ranges that include a lower modal frequency or off-resonance ranges.

Furthermore, a finite element model was validated against the analytical model, and a good agreement was found between them, introducing an advanced work with more complex geometric configurations.

The study revealed several critical insights:

• Optimization and Adaptability: The results underscore the importance of adaptability in the design of piezoelectric energy harvesters. By optimizing the design variables for specific frequency ranges, the harvester’s performance can be significantly enhanced, ensuring higher efficiency in various operational scenarios.
• Impact of Material Properties: The investigation into material properties highlighted that selecting materials with higher piezoelectric coefficients can lead to substantial improvements in power output. This suggests that material choice should be a key consideration in the design of future energy harvesting systems.
• Broader Application Potential: The methodology and findings of this study extend beyond the specific case of cantilevered piezoelectric harvesters. The optimization framework developed here can be adapted to other types of energy harvesters, potentially improving their performance under a variety of conditions.
• Future Directions: Expanding on this research, future studies could explore the optimization of energy harvesters with different geometric shapes to determine the most effective configurations. Additionally, investigating the integration and optimization of nonlinear effects, such as tailoring an optimal magnetic field, could further enhance energy harvesting efficiency. These efforts could lead to innovative designs that maximize energy capture under various operational conditions.
• Practical Implementation: Finally, the study’s validation of the analytical model through FEM and the identification of optimal design parameters pave the way for practical implementation. These findings could guide the development of commercially viable piezoelectric energy harvesters tailored for specific applications, such as in vibration-rich environments like industrial machinery or automotive systems.