Energy harvesting is the process of converting ambient energy sources such as load, vibration, temperature, etc. into small electrical power. Piezoelectric materials are used for converting vibrations into usable electrical energy in many engineering applications, such as self-powered wireless sensors, radio transceivers, implanted biomedical devices, health monitoring, autonomous charging, automotive applications, etc. [1]. Their high-energy conversion efficiency and compatibility make them potential replacements or energy sources of batteries for small electronic devices. Cantilever beams are typically used for piezoelectric energy harvesting owing to their high average strain compared to other arrangements [2]. The strain profile of beams changes notably with geometry. Hence, the shape of the cantilever beam significantly affects the output power density. Although shape and geometry significantly influence the power output, a fundamental investigation of the power density for the same area or mass of the beam has not been adequately addressed well in the past.

Several studies on cantilever beams are available in the literature. These studies have mostly focused on enhancing the output power and working range. Baker et al. [3] analyzed various beam shapes with the goal of enhancing power output. Their findings showed that a cantilever beam with a trapezoidal footprint has the ability to generate 30% more power per unit volume compared to traditional rectangular beams. Zhang et al. [4] reported that the trapezoidal shape of a cantilever beam is more effective than that of a rectangular beam for piezoelectric energy harvesting. However, the experiment was performed for a small cantilever where the root width was greater than the beam length. In a computational study, Rosmi et al. [5] optimized the output power and demonstrated that modification of the micro-cantilever beam geometry can enhance the output power. A comprehensive study of rectangular and trapezoidal cantilever beams of the same volume, where the length of the trapezoid beam was increased to obtain the same volume, suggested an improved strain distribution and output power from the trapezoidal shape [6]. Lei et al. [7] examined different shapes of a cantilever beam and concluded that the truncated triangular beam provides a larger power output. Chen et al. [8] suggested a triangular cantilever beam over rectangular and trapezoidal cantilevers after analyzing these three geometries with uniform base widths and lengths. A similar investigation on microscale beams was conducted by Alameh et al. [9]. Patel et al. [10] examined the influence of the piezoelectric layer geometry over a rectangular beam and proposed a shorter and thinner piezoelectric layer over the host material to obtain a significant increase in energy storage. The effects of the length of the piezoelectric material segment on the resonance frequency, output power, and working range were reported by Salem et al. [11]. Pradeesh et al. [12] numerically investigated the effect of the position of the piezoelectric material and proof mass on the performance of a piezoelectric energy harvester. It was observed that when a piezoelectric material was placed at the fixed end, the energy harvester produced maximum power. The authors also investigated the effect of taper in thickness and width on cantilever beams and suggested that an inverted taper in thickness and width can produce more power than a typical rectangular cantilever [13]. According to Ibrahim et al. [14] using a taper in thick is more effective in power output than using a taper in width. Tang and Wang [15] investigated the impact of proof mass size on the performance of an energy harvester. Their reported that a small change in the proof mass geometry not only affects the resonant frequency, but also has a significant impact on the strain distribution along the beam and thus impacting the output power. Alameh et al. [16] analyzed the effects of proof mass on a piezoelectric vibration energy harvester. The authors suggested an optimized T-shaped harvester design to improve performance. Satyanarayana et al. [17] examined the change in output power from a piezoelectric cantilever energy harvester by changing the dimensions of the proof mass and the types of piezoelectric material. Maximum efficiency was obtained from PZT-5A material-based piezoelectric energy harvester with a proof mass dimension of 3.5 µm x 2 µm. Li et al. [18] examined a cantilever piezoelectric energy harvester with a curved L-shaped mass that improves power density and lowers the fundamental frequency compared to conventional cantilever harvesters. The curved L-shaped mass harvester was designed, fabricated, and tested. The result showed 20-31% lower fundamental frequency than a block-shaped mass harvester for the same power harvester volume. Palosaari et al. [19] presented the effect of the substrate layer thickness on the performance of a piezoelectric energy harvester with a bimorph arrangement where the resonant frequency was maintained uniform by changing proof mass, and a thinner substrate layer was found to be beneficial. Kim et al. [20] analyzed the effect of the thickness of the elastic layer on the output power for the same piezoelectric layer dimension and proof mass with a unimorph setting. The analysis showed an increase in output power with substrate layer thickness at the beginning and started to fall again after reaching a maximum. The result suggests that thickness of the substrate layer should be optimized for maximum power output. Sunithamani et al. [21] showed that an energy harvester with a disc-proof mass produces more power compared to a harvester with a ring-shaped proof mass. Matova et al. [22] investigated the effectiveness of tapered beams in MEMS piezoelectric energy harvesters and suggested that the tapering of short and wide beams does not affect the power output. The effect of the slope angle on a tapered cantilever beam was investigated by Simon et al. [23] and reported that a slope angle of 0.94 can improve the harvested power by a factor of 3.6 compared to a beam of uniform thickness. Reddy et al. [24] analytically and experimentally evaluated the effect of a rectangular cavity by varying the location of the cavity from the neutral axis of the beam on the output voltage and concluded that a beam with a cavity is capable of generating 75% higher voltage because the cavity is responsible for the shifting of the neutral axis of the beam, which in turn increases the strain and generated voltage. In a similar investigation of trapezoidal cavities, the authors concluded that the use of trapezoidal cavities further improves the output voltage over rectangular cavities [25]. Raju et al. [26] investigated the voltage and power generated by a cantilever structure with one to four rectangular cavities and it was found that the maximum voltage was produced with a single cavity section and two cavities. Usharani et al. [27]improved the frequency range of the piezoelectric beam energy harvester by using a double tapered cavity. Raju et al. [28] examined the effect of rectangular and trapezoidal cavities on tapered beams, and the result suggests that tapered beam in thickness and width with trapezoidal cavity provides the maximum power output among the analyzed beam configurations. Damya et al. [29] introduced a clamped-clamped beam and mass loading at the center type of energy harvester, which showed better performance than the conventional cantilever piezoelectric energy harvester. Lihua et al. [30] analyzed a cantilever piezoelectric energy harvester with an adjustable natural frequency by adding two boxes at two free edges along the fixed free direction and one rolling ball in each box. In 2009, Gao et al. [31] examined the effect of the ratio of the non-piezoelectric to piezoelectric length on the induced voltage and they concluded that the induced voltage per unit force increased with the length ratio, and the induced voltage per unit displacement was found to be the maximum when the ratio was unity. Wang et al. [32] discussed the dependence of the charge, voltage, and energy sensitivities on the elastic model ratio and thickness ratio of the elastic substrate and piezoelectric layer under different conditions in the case of a unimorph cantilever piezoelectric energy harvester. Zhou et al. [33] examined the performance of a piezoelectric simply supported beam energy harvester by changing the length of the piezoelectric material over the beam, and showed that energy harvesting performance can be improved by optimizing the length of the piezoelectric layer. In a more recent study, Izadgoshasb et al. [34] optimized the orientation of piezoelectric cantilever beam energy harvesting by taking vibrations from human motion. Gao et al. [35] investigated an energy harvesting technique using the piezoelectric cantilever with a cylindrical extension, where the effect of vibration was created by airflow.

From the above review, it has been found that a large volume of analysis has been performed to find novel aspects of piezoelectric energy harvesting and to maximize the power capacity. Some investigations have been carried out to analyze the effects of different cantilever beam geometries on energy harvesting parameters. However, most of the analysis about the influence of beam geometry has focused on rectangular, trapezoidal, and triangular cantilever beams. In this work, in addition to the above-mentioned beams, concave, convex, and V-cut cantilever beams were analyzed.

Furthermore, most of the investigations reported in the review conducted the analysis by coating the whole beam with the piezoelectric material [3,4,5,6,7,8,9,10]. However, in this study, only 1/3rd of the beam was coated with piezoelectric material in unimorph setting. It is known that the output from a cantilever energy harvester increases with an increased coating area of the beam. But it is also important to note that the cost of an energy harvester increases with the increased coating area, as piezoelectric materials (PZT-5A) are very expensive, which costs around $0.13/mm³ [36]. Taking this into account, when the cost limits the amount of piezoelectric material, should the harvester be designed in such a way that the base beam length is the same as the piezoelectric material or should a larger length of the base beam be used and a portion of it coated? The analysis for a fixed amount of piezoelectric material showed that when the base beam length was the same as the piezoelectric material, the power output was almost half compared to when the base beam length was three times the piezoelectric material length, and the piezoelectric coating is given to the fixed end of the beam. Therefore, simply by using a larger base beam and coating a portion of it with piezoelectric material, power output can be enhanced. To use a larger base beam, it is obvious that the cost of base beam material will increase. However, the fact is that aluminum, mostly used as the base beam, is around more than 50 times cheaper than piezoelectric materials (PZT-5A), with a cost of around $0.00015/mm³ [37]. Therefore, it is evident that attention should also be provided to beams with partial coating of piezoelectric materials.

This work performs a comprehensive study in which each beam shape is thoroughly investigated to obtain a geometric configuration for maximizing the power output from that beam type. Additionally, the analysis was conducted by maintaining a constant surface area and thickness for both the beam and piezoelectric material. The analysis, in which the surface area and thickness are held constant, has not been thoroughly investigated in previous researches. Finally, the power output from all beam types is compared to select an efficient design for maximizing the power output from a cantilever beam based energy harvester. Such an analysis has not been performed in previous studies.

Cantilever beam based energy harvesting is principally dependent on the structural resonance of the beam. The resonance frequency of a beam is heavily dependent on the mass distribution of the beam. Hence, when the geometric configuration of a beam is varied to increase the power output, the resonance frequency of the beam may shift. In this study, a relationship is presented to understand the dependence of the resonance frequency on the geometric variation of each beam shape.

The performance of the piezoelectric cantilever energy harvesters was numerically analyzed using the COMSOL Multiphysics software. Solid mechanics, electrostatic, and electric circuit modules were used. It has been determined that using tetrahedral elements is the most effective approach for this particular analysis. Initially, an eigenfrequency analysis was performed to obtain the natural frequency of the first mode (n=1). In this study, the entire analysis is focused on the first mode (mode-1), as it is correlated to the maximum power output. When mode-1 eigenfrequency is known, a frequency domain analysis was performed to investigate the power output variation around mode-1 natural frequency. A uniform beam of thickness 1 mm was maintained for all the beam types investigated in this study. An isotropic damping loss of 5% was considered for these structures [12]. In this analysis, the load on the energy harvester was kept constant at 10 kΩ and a proof mass of 0.17 gm was used. A 10 KΩ resistance was deliberately chosen, despite not being the optimal resistance for any of the beam shapes, to observe how a modification in shape influences power density when the resistance is arbitrary. This decision was made with the consideration that, in a practical application, the output from the harvested energy must be employed on a resistance associated with the application. In analyzing the beams, the geometry of each type of beam was varied systematically to obtain an efficient geometry for maximizing the power output. For a comparable argument between beams, some beam parameters (e.g., surface area, thickness, mass) were kept the same because the parameters can influence the power output. A similar approach was adopted for the PZT materials. A unimorph piezoelectric material (PZT-5A) of 0.5 mm thickness was placed at the fixed end of the beam as previous studies suggest that placing the PZT material close to the fixed end can ensure the highest electric potential [12]. It is known that the stress in a cantilever beam is usually concentrated near to the fixed end. Additionally, the power output from piezoelectric material depends on the stress that it experiences. One-third of the beam's surface area at the fixed end was coated with piezoelectric material in a unimorph configuration to address the high-stress concentration in this area. To make the comparison among the beams more convenient the output power is expressed in terms of power density, which is defined as the output power in μW per unit volume of piezoelectric material in mm³.

To validate the solution procedure, a 1 mm thick rectangular piezoelectric cantilever beam energy harvester with a length of 100 mm and width of 10 mm was used to validate the procedure of analysis with the established literature. A piezoelectric material with an equal length and width of 10 mm and thickness of 0.5 mm was attached at the fixed end of the cantilever. In this regard, the geometry and dimensions of the beam and piezoelectric material were taken as in the study by Pradesh et al. [12] for accurate predictions.

The model was simulated for output power by varying the frequency from 75 Hz to 110 Hz, and it has been obtained that the maximum output of 0.37mW occurs at 94.3 Hz, which is the resonance frequency of the model. This result appears to agree with the results of the work [12]. Hence, the chosen computational settings and parameters are found to be acceptable for fulfilling the objectives of this study.

Furthermore, a study was conducted to ensure the solution procedure was mesh-independent. The solution for different mesh element sizes was studied for a trapezoidal beam with an apex to base ratio of 0.5. The details of the mesh independence test are provided in Section 4.1.

In this study, five different types of cantilever beams were investigated with the aim to select an efficient geometry for producing the maximum power output from a low cost piezoelectric energy harvester. In performing the analysis, a few beams and PZT parameters (e.g., surface area, thickness, volume, and mass) were kept constant for a reasonable comparison. Table 1 below lists a comparison among the beams with respect to measured maximum power output and corresponding resonance frequency.

From the Table 1 it can be seen that the trapezoidal beam with narrower fixed end and wider free end produces the maximum power and in contrast the triangular beam produces the minimum. It can also be seen that there is a relationship between the maximum power output and the base width of the beam. In a general perspective, beams with a narrower base width, especially those featuring partial piezoelectric coating at the fixed end, tend to yield higher power output. This is because, by reducing base width, the area moment of inertia can be decreased, and the mass center of the beam can shift away from the fixed end. Consequently, the piezoelectric material attached to fixed end of the beam can experience a higher deflection and produces more power. In the case of the convex cantilever beam, the base width measured the minimum (2.53 mm), which was less than half compared to the trapezoidal beam. However, the power output of the trapezoidal beam is higher than that of the convex beam. Note that, the convex beam is a lengthwise symmetric beam. Hence, the mass center always remains at the midpoint of the beam. However, in the case of a trapezoidal beam, the base/fixed end width is the minimum, and beam width increases linearly throughout the length. Hence, the width at the free end is maximum. Because of the geometry, in a trapezoidal beam, the mass center stays closer to the free end, which warrants a higher deflection of the PZT material at the fixed end than in a convex beam.

Although the maximum power output is dependent on the amplitude of the PZT material, the resonance frequency is mainly dependent on the tip deflection of the beam. For a linearly varying geometry, the PZT deflection and beam tip deflection profiles remain proportional/consistent. However, in the case of nonlinear geometry variation (e.g., concave, convex, V-cut), the deflection of the PZT and beam tip are not proportional; however, they depend on the mass distribution of the beam. For example, in the case of a concave beam, the PZT deflection increases with an increase in the ratio, w. However, the beam tip deflection decreases at the same time. Hence, for a concave beam, the PZT and beam tip deflections are inversely proportional.

In a linearly varying configuration (e.g., trapezoidal and triangular), the power output and resonance frequency are inversely proportional. When the beam stiffness decreases, the resonance frequency also decreases. At the same time, owing to the decreased stiffness, the beam tends to bend more and produce more power. In the case of a nonlinear geometry, the power output and resonance frequency relationship are more complex and solely depends on the mass distribution. In the case of a concave beam, this relationship is linear, which is quite unique and can be useful for specific applications. In the case of both convex and V-cut beams, the power output and resonance frequency are neither proportional nor opposite. It is a combination of both and depends on the ratio.

While the trapezoidal beam stands out by generating maximum power output, it's crucial to note that the findings from studying alternative beam shapes hold their own significance. This is because cantilever energy harvesters tend to achieve peak output at their resonant frequencies. Such an analysis is valuable when choosing the most suitable beam for a specific application, where the precise frequency the beam will be exposed to plays a defining role.

In designing a piezoelectric energy harvester, cost is pivotal consideration. The investigation is conducted by considering the substantial expenses associated with piezoelectric materials. The aim of this research was to find a beam shape that could yield high output power density while utilizing a constrained amount of piezoelectric material. Hench, this is the reason why piezoelectric material coating is applied to just one third of the base beam’s surface. In this paper, a variety of cantilever beam shapes and geometries were investigated for maximizing the output power density. Each shape has been analyzed by varying the geometric parameters while keeping the surface area, volume, mass, and thickness of the beam and piezoelectric material constant. Previous studies have suggested that a trapezoidal beam with a narrower free end and wider fixed end width can produce a greater output. This effect hold true as long as piezoelectric material is applied along the entire length of the beam. However, to maintain the beam surface area constant for a trapezoidal shape, it is essential to adjust both the base width and free end width simultaneously. The current study has found a noteworthy outcome that a trapezoidal beam with a smaller base/fixed end width in comparison to the free end width can generate the maximum power when the modification is carried out simultaneously and coating is provided over one-third portion of the beam at the fixed end. This trapezoidal configuration surpasses the power output achieved by other beam shapes analyzed in this study Generally, beams with narrower base widths generate more power than those with wider bases. Within the trapezoidal beam, the power output capacity of the beam can be increased by increasing the apex to base width ratio. However, when the ratio is increased, it should be noted that the structural integrity of the structure is not compromised. This study also investigated the effect of beam shape/geometry on the resonance frequency of the beam. It has been found that for a linearly varying geometry, the resonance frequency and power output capacity of the beam are inversely proportional. For a nonlinear geometry, a different/variable relationship between the resonance frequency and the power output can be observed. The concave beam geometry exhibits a unique feature in which the power output increases with an increase in the resonance frequency.