1 Introduction

Advances in low power Very Large Scale Integration (VLSI) design, CMOS Complementary Metal Oxide Semiconductor (CMOS) fabrication and MEMS Micro-Electro-Mechanical Systems (MEMS) technology have considerably reduced power requirements of devices. Energy harvesting is the process by which ambient energy is captured and converted into electricity for enabling self-powered devices. In the coming years, energy harvesting can totally eliminated the need for the battery as an energy source (Priya and Inman 2009). Recently, MEMS technology has been extensively used to realise micro energy harvesters to meet the low power requirement of microelectronics and wireless sensors used in Internet of Things (IoT) application (Iannacci 2019). Replacing batteries in a wide wireless sensor network (WSN) can be tricky and expensive. Vibration Energy Harvester (VEH) has become increasingly popular as a self-sufficient and sustainable power solution for microsystem like WSN that is battery free (Roundy et al. 2004). A VEH uses ambient vibration to produce electrical energy using piezoelectric, electromagnetic and electrostatic transduction mechanisms (Priya and Inman 2009). Due to higher power density, ease of miniaturization, easy manufacturing and implementation, Piezoelectric Energy harvester (PEH) has been the focus of most researchers (Liu et al. 2018). PEHs are usually designed as unimorph or a bimorph cantilever beam and tip mass to reduce resonant frequency. They employ rectangular cantilever beams with a substrate and piezoelectric layer/layers where piezoelectric elements experience non-uniform strain that varies linearly along the length with maximum strain at the fixed end of cantilever beam (Yang et al. 2018). Overstrain and fatigue around the fixed end become concerns that can affect the lifetime of the PEH and have led the researchers to look for different geometries with more uniform strain distribution (Baker et al. 2005; Muthalif and Nordin 2015). To this end, tapered cantilevers with a trapezoidal profile are an optimal shape to improve power output for low frequency operation (Zhang et al. 2017). However flexible cantilever beams still suffer from disadvantage of high resonance frequency, narrow bandwidth and low voltage and power output. Additionally, ambient vibration source is typically random, low and wideband (Jones et al. 2001) and therefore a mechanism is required to lower the operating frequency and widen the operational bandwidth. Indrajit and Biswas (2018) showed that a device must be long, narrow and thin to have low resonant frequency (Indrajit and Biswas 2018). The classical vibration theory states that “higher vibration modes will have strain nodes, where the cancellation of charge occurs due to dynamic strain field changing direction along cantilever length (Erturk and Inman 2008)”. The segmentation of electrode makes the piezoelectric harvester suitable for multi-mode frequency excitation (Erturk et al. 2009; Zizys et al. 2015). Alternatively, in order to resolve narrow bandwidth, multi DOF systems with multiple vibration modes have been used (Tang and Yang 2012; Nabavi 2019). The approach involves array of harvesters to realize multiple resonant structures thereby increasing the effective operational bandwidth. The major drawbacks of the method involves inefficient device volume since only a single harvester contributes to the device output at a particular frequency. Iannacci et al. (2015) proposed and experimentally verified a MEMS based VEH with Four-leaf Clover (FLC) design to achieve multi-modality for wide operating frequency (Iannacci et al. 2015, 2016). The design concept exploits the same idea of increasing mechanical DOFs that enable multiple resonant modes of vibration for a single mechanical resonator structure. Furthermore, the complex design of FLC is optimized for suitable selected modes in order to lie in a predefined frequency band to find a trade-off between geometrical complexity and performance (Sordo et al. 2016; Janusas et al. 2014) experimentally evaluated multi-mode electromagnetic PEH with two resonant modes by not covering the strain nodes of second vibration mode having lower resonant frequency to match the environmental vibrations (Janusas et al. 2014). A similar finding is reported by Krishnasamy and Lenka (2018) where segmentation of piezoelectric layer has lower resonant frequency and improved power compared to segmentation of electrodes in both first and third vibration modes (Krishnasamy and Lenka 2018). Nonlinearity in a system is of great importance while designing any type of VEH (Ramlan et al. 2009). One method to achieve non-linearity is by varying nonlinear stiffness or motion by coupling with magnet, popularly known as piezo-magneto-elastic harvester. Shahruz (2008) investigated a piezoelectric harvester with a magnetic proof mass on the tip of piezoelectric beam and magnets in its vicinity. He reported that when the magnets have appropriate strengths and is placed appropriately, the vibration of the beam-mass system can be amplified; thus increasing the harvested energy. The major drawbacks seen in implementation of nonlinear piezo-magneto-elastic harvesters are difficulty in micro manipulation and weight control of bulk magnet. Apart from narrow bandwidth and low power generation, fatigue becomes a concern for long-term vibration energy scavengers. In a PEH, 1600 microstrain for PZT unimorph film and 500 microstrain for bimorph is considered a strain safe limit to avoid fracture (Roundy et al. 2005).

To overcome the drawbacks of conventional linear PEHs, we propose a novel concept combining trapezoidal shaped PEH with segmented piezoelectric bimorph layers at the strain nodes of higher vibration modes of the PEH. The trapezoidal shape of the harvester will have more uniform strain distribution, which in turn results in efficient use of device structure for piezoelectric voltage generation while reducing the resonant operating frequency. Whereas, segmentation of bimorph piezoelectric layers further reduce the frequency and increase the generated output voltage by evading the cancellation of charge that occurs at strain nodes of higher vibration modes. The analytical model of the proposed design concept is also developed. The comparative analysis is performed between conventional rectangular PEH, non-segmented trapezoidal PEH and the proposed segmented trapezoidal PEH.

2 Model development and simulation

Figure 1 shows a schematic of continuous trapezoidal PEH. It has proof mass attached on beam’s free end to adjust resonant frequency. PZT-5H is chosen as piezoelectric material because of its high volume of piezoelectric coefficient and dielectric constant. Top and bottom of the steel substrate are attached with PZT layers. Thin electrodes are affixed on both side of PZT layers, which can be neglected for analytical models. The varying beam’s width over the length L is given as,

$$ b(x) = ~b_{0} - {\text{ }}cx,{\text{ }}0{\text{ }} \le x \le L $$
(1)

where, \( c = (b_{0} - b_{L} /L) \)) is tapering parameter.

Fig. 1
figure 1

Trapezoidal PEH with continuous bimorph piezoelectric PZT layers

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Equation of Motion (EoM) of a bimorph piezoelectric layer beam connected in parallel is given by Euler–Bernoulli assumptions (Erturk and Inman 2009),

$$\text{E}\text{I}\left(\text{x}\right)\frac{{\partial }^{4}\text{w}(\text{x},\text{t})}{\partial {\text{x}}^{4}}+{\upvartheta }\left(\text{x}\right)\text{v}\left(\text{t}\right)\left[\frac{d{\updelta }\left(\text{x}\right)}{dx}-\frac{d{\updelta }(\text{x}-\text{L})}{dx}\right]+\text{m}\left(\text{x}\right)\frac{{\partial }^{2}\text{w}(\text{x},\text{t})}{\partial {\text{t}}^{2}}=-\left[{\text{M}}_{\text{t}}{\updelta }\left(\text{x}-\text{L}\right)+\text{m}\left(\text{x}\right)\right]\frac{{\partial }^{2}{\text{w}}_{\text{b}}(\text{x},\text{t})}{\partial {\text{t}}^{2}}$$
(2)

where, \({w}_{b}\) is the excitation displacement of base in z direction, w is beam transverse deformation, \({\updelta }\left(\text{x}\right)\) represents a Dirac delta function, \(\text{v}\left(\text{t}\right)\) is the alternating piezoelectric voltage and \({\text{M}}_{\text{t}}\) is mass of the tip. Rest of terms change along length as,

Bending stiffness,

$$\text{E}\text{I}\left(\text{x}\right)=\text{b}\left(\text{x}\right)\left[{\text{E}}_{\text{s}}\frac{{\text{h}}_{\text{s}}^{3}}{12}+\frac{2{\text{E}}_{\text{p}}}{3}\left({\left(\frac{{\text{h}}_{\text{s}}}{2}+{\text{h}}_{\text{p}}\right)}^{3}-\frac{{\text{h}}_{\text{s}}^{3}}{8}\right)\right]$$
(3)

Mass per unit length,

$$m\left(x\right)=\text{b}\left(\text{x}\right)\left(2{\rho }_{p}{h}_{p}+{\rho }_{s}{h}_{s}\right)$$
(4)

Electromechanical coupling term,

$$\vartheta \left(x\right)=b\left(x\right)\frac{{E}_{p}{d}_{31}}{{h}_{p}}\left[\left(\frac{{h}_{s}^{3}}{4}\right)-{\left({h}_{p}+\frac{{h}_{s}}{2}\right)}^{2}\right]$$
(5)

where, the thickness of each PZT (\({\text{h}}_{\text{p}}\)), thickness of Steel substrate (\({\text{h}}_{\text{s}}\)), length of beam (L) are geometry parameters. \({\rho }_{p}\), \({\rho }_{s}\) are density of PZT and steel respectively while \({E}_{p}\) and\({E}_{s}\) are the Young’s modulus of PZT and steel material respectively. Table 1 shows the geometry and material properties of the structure.

Table 1 Geometric dimension and material properties
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Modal Analysis is important to find the mode shape of the vibrating beam. The transverse displacement of the beam is expressed using Galerkin procedure in the form (Erturk and Inman 2008),

$$\text{w}\left(\text{x},\text{t}\right)=\sum _{i=1}^{n}{\varPhi }_{i}\left(x\right){\eta }_{i}\left(t\right)$$
(6)

where,\({\varPhi }_{i}\left(x\right)\) is the mass normalized displacement eigenfunction and \({\eta }_{i}\left(x\right)\) is the coordinate of a cantilever beam of the ith mode. “The ith vibration mode has i − 1 strain nodes (Erturk and Inman 2008)” and the ith mode shape \({\varPhi }_{i}\left(x\right)\) is found by using above equations and the necessary boundary condition of a cantilever beam (fixed-free) as,

$${\varPhi }_{i}\left(x\right)={C}_{r}\left[\text{cos}\frac{{\beta }_{r}}{L}x-\text{cosh}\frac{{\beta }_{r}}{L}x+{\xi }_{r}(\text{sin}\frac{{\beta }_{r}}{L}x-\text{sinh}\frac{{\beta }_{r}}{L}x)\right]$$
(7)
$${\xi }_{r}=\frac{\text{sin}{\beta }_{r}-\text{sinh}{\beta }_{r}+{\beta }_{r}\frac{{M}_{t}}{m\left(x\right)L} (\text{cos}{\beta }_{r}-\text{cosh}{\beta }_{r}) }{\text{cos}{\beta }_{r}+\text{cosh}{\beta }_{r}+{\beta }_{r}\frac{{M}_{t}}{m\left(x\right)L}\left(\text{sin}{\beta }_{r}-\text{sinh}{\beta }_{r}\right)}$$
(8)

where, \({C}_{r}\)is modal amplitude constant found by normalizing \({\varPhi }_{i}\left(x\right)\) base on the orthogonality settings (Erturk and Inman 2009). And \({\beta }_{r}\) is eigenvalues of the system obtained from characteristic equation

$$1+\text{cos}\beta \text{cosh}\beta +\beta \frac{{M}_{t}}{m\left(x\right)L}(\text{cos}\beta \text{sinh}\beta -\text{sin}\beta \text{cosh}\beta )=0$$
(8)

Second derivative of the \({\varPhi }_{i}\left(x\right)\) in Eq. (7) i.e. \(\frac{{d}^{2}{\varPhi }_{i}\left(x\right)}{d{x}^{2}}\) gives the strain distribution. For the trapezoidal bimorph beam shown in Fig. 1, the strain distribution curve is plotted for four modes (Fig. 2) in MATLAB. For a third mode of vibration, the strain nodes are found to be located at 0.1341 L and 0.4972 L as observed in Fig. 2, where L is the length of the beam.

Fig. 2
figure 2

Strain distribution curve of trapezoidal PEH for 1st, 2nd, 3rd and 4th vibration mode

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Based on the strain distribution curve, the continuous trapezoidal beam of length L is segmented at 13.41 mm and 49.72 mm along the length to design segmented trapezoidal PEH for 3rd vibration mode. The final segmented trapezoidal model is shown in Fig. 3 that will work in 1st, 2nd and 3rd vibration modes.

Fig. 3
figure 3

Proposed Segmented Trapezoidal PEH for 3rd mode of vibration

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Figure 4 shows the cross section of the bimorph cantilever beam with a substrate layer (hs) sandwiched between two piezoelectric layers (hp). The discrete PZT layers created after segmentation are connected in parallel to a common load resistance R for maximum voltage as shown in Fig. 5.

Fig. 4
figure 4

Piezoelectric Beam cross section neglecting the thin electrode

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Fig. 5
figure 5

Side view of beams with discrete piezoelectric segments of proposed PEH showing electrical connection

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The equivalent circuit is represented in Fig. 6. Where, \({C}_{1}\)\({C}_{2}\) and \({C}_{3}\) are equivalent capacitance of symmetric PZT segments. The Kirchhoff’s current equation gives the voltage-strain relationship (Krishnasamy and Lenka 2018),

$$\sum _{n=1}^{3}{C}_{n}\left(x\right)\frac{dv\left(t\right)}{dt}+\frac{v\left(t\right)}{R}-{I}_{p}\left(t\right)=0$$
(10)
$$ C_{1} \left( x \right) = \frac{{\varepsilon _{{33}} L_{1} \mathop \smallint \nolimits^{} b\left( x \right)}}{{h_{p} }},0 < x < L_{1} $$
(11)
$$ C_{2} \left( x \right) = \frac{{\varepsilon _{{33}} \left( {L_{3} - L_{2} } \right)\mathop \smallint \nolimits^{} b\left( x \right)}}{{h_{p} }},L_{2} < x < L_{3} $$
(12)
$$ C_{3} \left( x \right) = \frac{{\varepsilon _{{33}} \left( {L - L_{4} } \right)\mathop \smallint \nolimits^{} b\left( x \right)}}{{h_{p} }},L_{4} < x < L $$
(13)
$${I}_{p}\left(t\right)=\sum _{n=1}^{3}\left({\int }_{{L}_{i-1}}^{{L}_{i}}\left(b\left(x\right){{E}_{p}d}_{31}{h}_{pc}\frac{{\partial }^{3}w\left(x,t\right)}{\partial {x}^{2}\partial t}+{C}_{n}\left(x\right)\frac{dv\left(t\right)}{dt}\right)dx\right)$$
(14)
$${h}_{pc}=\frac{{h}_{p}+{h}_{s} }{2}$$
(15)

where i = 1, 3, 5 is segmented lengths of the beam. \({\text{L}}_{5}\)= L is the total length and n is the number of PZT segments, n = 1(0< x <\({L}_{1}\)), n = 2(\({L}_{2}\) < x < \({L}_{3})\), n = 3 (\({L}_{4}\) < x< L) after segmentation. The constants \({d}_{31}\) and \({\epsilon }_{33}\) gives the piezoelectric constant and permittivity of piezoelectric material respectively. The measured piezoelectric voltage is \(v\left(t\right)\) = \({I}_{p}\left(t\right)\times\)R across the load resistance. P = \(\frac{{v\left(t\right)}^{2}}{R}\) gives the generated output power.

Fig. 6
figure 6

Equivalent circuit of piezoelectric segments connected in parallel across a load resistance R

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The model is created using Finite Element Method (FEM) software COMSOL MULTIPHYSICS 5.5. The software calculates the generated piezoelectric output voltage and power generated using piezoelectric effect transduction in the piezoelectric harvester. The cantilever is constrained to be fixed at one end and the other end is free to vibrate. In terms of electrical boundary conditions, the inner faces of piezoelectric layer is grounded, while the top input acceleration of 0.5 g (g = 9.8 \(m/{s}^{2}\)) is applied to induce mechanical strain upon the device. The structure is meshed before performing any simulations analysis. The meshing has resulted in 26,256 elements for a total number of degrees of freedom of 169,463 for a segmented trapezoidal PEH. Eigen frequency study is done to find the Eigen frequencies and strain distribution of different vibration modes of the harvester. Frequency domain analysis is executed to study the performance across a frequency sweep around the resonant Eigen frequencies.

Figure 7 shows the FEM simulated model of the rectangular PEH (100 mm \(\times 20 \text{m}\text{m}\)), trapezoidal PEH and segmented PEH in their first mode of vibration having geometry and material properties as listed in table I. It can be seen from Fig. 7a that stress distribution of rectangular harvester is not uniform and is concentrated at fixed end of the cantilever beam. Figure 7b shows that stress distribution is improved a little in trapezoidal harvester compared to rectangular harvester while Fig. 7c shows stress distribution of segmented trapezoidal harvester is almost uniform along the entire length of the harvester.

Fig. 7
figure 7

FEM model of a rectangular PEH b trapezoidal PEH c segmented trapezoidal PEH in their fundamental mode of vibration with frequency 64 Hz, 60 Hz and 58.379 Hz respectively. The colour legend shows the von mises stress (N/\({\text{m}}^{2}\)) distribution

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3 Result and discussion

In order to study the effect of introducing segmentation in an optimal trapezoidal shaped PEH, we used the FEM model with optimal geometry parameters to find the necessary output behaviour in the three vibration modes of the proposed harvester. The Fig. 8 compares the strain distribution of rectangular PEH, a trapezoidal PEH and the segmented trapezoid PEH in a graph. It can be observed that segmented trapezoid has more uniform strain along the length followed by continuous trapezoid and then rectangular harvester for same mechanical input. The undercut in strain distribution of segmented trapezoidal PEH as seen in Fig. 8 is due to discontinuity at strain nodes. The uniform strain in trapezoidal shaped cantilever beam will reduce the fatigue concern due to concentration of strain around the fixed end of a rectangular cantilever beam.

Fig. 8
figure 8

Comparison of strain distribution of rectangular, trapezoidal, segmented trapezoidal piezoelectric energy harvester (PEH) along the length

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The optimal electrical connection is found by finding the optimal load resistor connected across the output terminal as shown in Fig. 5. Figure 9a is the voltage vs. load resistance graph for a rectangular PEH that shows the voltage is maximum and gets saturated in 19.4 V for optimum load of 50 kΩ. Figure 9b shows that the optimal resistor of trapezoidal PEH is 26 kΩ at 20.4 V while the optimal load of segmented trapezoidal PEH is 21 kΩ at 21.3 V as shown in Fig. 9c. The reduced load resistor will increase the generated output power. Figure 10 shows the comparison of frequency response curve of rectangular PEH, trapezoidal PEH and segmented trapezoidal PEH for optimum load of 50 kΩ, 26 kΩ and 21 kΩ respectively at their respective resonant frequencies. The generated voltage is maximum at first fundamental resonant frequency in case of all the harvesters. The generated voltage is highest for segmented Trapezoidal PEH followed by trapezoidal PEH and rectangular PEH for all the three mode frequencies. There is resonance peak at three frequencies indicating multi frequency operation with increased power generation at higher vibration modes of the different harvesters. It is also observed that the operating resonant frequencies of the trapezoidal PEH is reduced compared to rectangular harvester, which is further reduced after segmentation of the piezoelectric PZT layers at strain nodes. Figure 11 shows the power generated in the proposed segmented trapezoidal PEH structure at optimum load of 21 kΩ. The generated power is 21.6 mW, 0.23 mW and 0.15 mW at three resonant frequencies 58 Hz, 400 Hz and 1189 Hz respectively. The generated outputs of the rectangular and trapezoidal piezoelectric energy harvester are tabulated in Table 2.

Fig. 9
figure 9

Voltage vs. load resistance graph of a rectangular PEH b trapezoidal PEH c segmented trapezoidal PEH with 64 Hz, 60 Hz and 58 Hz respectively as input frequency in the fundamental mode (mode 1) of vibration

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Fig. 10
figure 10

Comparison of voltage generated at optimum load by rectangular PEH (50 kΩ), non-segmented trapezoidal PEH (26 kΩ) and segmented trapezoidal PEH (21 kΩ)

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Fig. 11
figure 11

Generated Output power of proposed segmented trapezoidal PEH for optimal load of 21 kΩ

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Table 2 Summary and comparison of results
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Table 2 summarises the output generated and makes a comparison of the rectangular PEH, trapezoidal PEH and segmented trapezoidal PEH in 1st, 2nd and 3rd vibration modes of the three harvester designs. There is increase in power generation by 91.66% at second vibration mode and improvement of 66.66% at third vibration mode after segmentation of bimorph piezoelectric PZT layers at strain nodes of trapezoidal PEH.

A comparative figure, area power density NPD (Normalize Power Density) with respect to both area and acceleration is calculated as in (16) to compare the performance of the improved structure with previous work as tabulated in Table 3. The proposed harvester has improved NPD compared to previous optimized structures of PEH.

Table 3 Comparison of Harvesters
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$$NPD = \frac{\text{P}\text{o}\text{w}\text{e}\text{r}}{\text{A}\text{r}\text{e}\text{a}\times {\text{a}\text{c}\text{c}}^{2}}$$
(16)

Before concluding this section, it is worth collecting some considerations around how the reported results could come to aid and support in other application fields of Microsystem (MEMS) technologies. In the first place, relying on a trapezoidal deformable beam to enhance the strain of the mass-spring system can improve the sensitivity of the miniaturised structure, if used for sensing purposes, e.g. inertial sensors or resonating sensor. Moreover, segmentation of electrodes might extend significantly the way mechanical deformations are transduced into electrical signals, therefore multiplying the sensing/actuating functions of the MEMS structure at stake. Finally, there exist aspects of reliability, as well, which are very important when dealing with Microsystem devices (Persano et al. 2016). To this end, the use of trapezoidal deformable structures can improve the mechanical reliability of MEMS-based devices.

4 Conclusion

The proposed PEH paves the way for a new and more practical design concept to improve the device performance with optimal geometry shape of the cantilever structure combined with segmentation concept. The trapezoidal shape offers a more uniform strain distribution to enable efficient use of the device volume while reducing the resonating frequency compared to conventional rectangular PEH. The segmentation of piezoelectric PZT layers at strain nodes of vibration mode avoids charge cancellation in the higher vibration modes thereby allowing it to operate in multi-mode with increased output generation. The harvester is designed for a 3rd by segmenting PZT at strain nodes, 13.41 mm and 49.72 mm along the length of the beam. Hence the proposed design has three operating resonant frequencies of 58 Hz, 400 Hz and 1200 Hz generating a power of 22.5 mW, 0.14 mW and 0.09 mW respectively at an optimum load resistance of 21 KΩ with 0.5 g input acceleration. Increase in power generation at 2nd vibration mode of 91.66% while an increase of 66.66% at third vibration mode is observed after segmentation of trapezoidal PEH. It is observed that segmentation of PZT layers further reduces the natural resonant frequency of the structure at higher vibration modes. It can be concluded that the proposed segmented trapezoidal design of PEH reduces the operating resonant frequency with performance improvement at all three modes of vibration compared to conventional rectangular PEH and trapezoidal PEH. Since the device is within the safe strain limit with more uniform distributed strain along the length, the structure is more reliable and will have increased lifetime. A multi frequency harvester can be obtained from a single simple resonator structure and avoid design complexity. The design can be used as low frequency and wide band vibration energy harvester in low powering WSNs, wearable and implantable devices. Piezoelectric sensors are one of the fastest growing technology in sensor market. The obtained results of the study can also be exploited to optimize and improve the performance of piezoelectric micro sensors and micro actuators. The future effort should be to achieve close range frequencies of multimodal energy harvester.