1. Introduction
During the last few decades, linear and nonlinear energy harvesters have been a point of interest for the researchers and scientists for converting waste vibration energy into electrical energy [
1,
2,
3,
4]. In the most recent decade, a lot of improvement happened in nanotechnology which also affect the very-large-scale integration (VLSI) technology. Nanosensors devices or systems are being developed for various applications, most notably in healthcare and medical industries, military defense and environmental and agricultural industries. These systems can drive with very low power consumption. For this purpose, batteries are required to drive this type of device, which comes with a lot of drawbacks also. Batteries limit the size of Nanosensors devices or systems. Sometimes, the replacement of a battery can be costly, such as in peacemaker applications. However, there are a lot of energy sources, such as solar, thermal, vibration, or radiofrequency, liable to the working environment of the system. Apart from solar energy harvesting, energy extraction from vibration has gained much more attention from researchers, as it is widely available in the environment. Electromagnetic and piezoelectric technology are popular to extract energy for vibrating systems. Electromagnetic technology has a lot of drawbacks when scaling down in size due to increments in losses. For micro-electro-mechanical Systems (MEMS), piezoelectric energy harvesting technology provides more potential, as it is easy to integrate and highly sensitive to vibration.
Although, the concept of nonlinearity demands a complex understanding, it is advantageous in comparison with the linear harvester in terms of the bandwidth of excitation frequency. In the case of linear harvesters, the power reduces drastically if the excitation frequency is slightly away from its natural frequency [
5]; in contrast, with nonlinearity in the harvester, the effective harvesting frequency bandwidth can be increased significantly.
Researchers have explored various mechanisms to introduce the nonlinear dynamic phenomenon in energy harvesting applications. Mann and Sims [
6] used magnetic restoring force to levitate an oscillating center magnet for electromagnetic energy harvesting. Harmonic base excitation has been considered as input energy to study the resulting Duffing’s equation. R. Ramlan et al. [
7] studied nonlinear energy harvesters with two types of non-linear stiffnesses: the first system considered a non-linear bistable snap-through mechanism, while in the second system, nonlinearity is invoked with the hardening of the spring. Daqaq [
8] theoretically investigated the monostable Duffing’s oscillator for colored excitations and Gaussian white noise. The authors also studied other types of nonlinearities such as damping and inertial nonlinearities, capable of enhancing the performance of the harvester’s operation. Stanton C. et al. [
9] investigated the hardening and softening response of piezoelectric energy harvesters, which were invoked by the tuning of magnetic interaction. The study concludes that a nonlinear energy harvester is suitable for ambient excitation with slow varying frequencies. Kang-Qi Fan [
10] proposed a design and experimental verification of a compact bi-directional nonlinear energy harvester. The proposed design has two magnetically coupled piezoelectric cantilever beams with orthogonal directions of deflection. Experimental verifications spotlighted the advantages of a nonlinear piezoelectric energy harvester over the linear ones.
On the other side, scientists and researchers explored advanced manufacturing technologies to improve the mechanical performance or control numerous properties in a desirable direction. These materials are called functionally graded materials (FGMs), in which properties are controlled in the volume using grading index (
n) [
11,
12,
13,
14]. The merit of using these materials is that the mechanical properties can be controlled at the layer interface and make smooth stress variation at the interface of the ceramic layer (piezoelectric) and metal layer to avoid debonding at the interface [
15,
16].
Theoretical and, up to some extent, experimental studies of the functionally graded materials have been reported in the literature for energy harvesting and active control system applications [
17,
18,
19,
20]. However, FGMs’ performance was not explored for cantilever-based nonlinear harvesters. In this paper we report the performance of the functionally graded piezoelectric nonlinear energy harvester and compare it with the linear functionally graded piezoelectric energy harvester. It is to note that FGMs are difficult to prepare. Most of the studies are reported with FGMs, which are almost difficult to fabricate. On the other hand, authors have considered FGM, which has already been experimentally fabricated by some other group. Hence, the present study is closer to actual performance. Dynamic studies have been carried out in time and frequency domains. However, a frequency domain has limited use for nonlinear systems (only for excitations with one frequency and one harmonic function), as the superposition principle is not valid anymore. For other types of excitation, e.g., random excitation, some linearization techniques may be required.
4. Numerical Studies
To study the functionally graded piezoelectric material energy harvester, a cantilever structure has been adopted. To introduce the nonlinearity in the structure, one magnet is used at the free end of the cantilever and another at a distance
d0, as shown in
Figure 1. Magnets are placed such that there are repulsive forces between the two magnets. Due to the repulsive force, the position of the magnet is stable at two extreme points along the direction of vibration. The force between the magnets can be altered by changing the distance between them. To support the piezoelectric materials, aluminum is used for the host layer. PZT-Pt (Pb(Zr,Ti)O
3-Pt) was selected as FGM for numerical studies. Takagi et al. [
16] attempted a fabrication of the above mentioned FGM and have reported physical properties up to PZT-30%Pt. In this study, the top surface of the upper piezoelectric layer is considered as PZT-0%Pt, while PZT-20%Pt is considered for the bottom surface. Similarly, the top surface of the lower piezoelectric layer is made of PZT-20%Pt and PZT-0%Pt for the bottom surface. The grading index (
n) controls the distribution of piezoelectric properties in the thickness direction of the piezoelectric layers according to Equation (1). Geometrical and material properties are listed in
Table 3 and
Table 4, respectively. For these material properties, variations of Young’s modulus for different grading indexes has been shown in
Figure 7. For grading index
n = 0, there is a ductile–brittle interface between the host layer (ductile) and piezoelectric material (brittle), which is prone to debonding failure. However, as grading index (
n) increases, the ductility increases at the interface due to addition of the PZT-20%Pt material. At grading index
n = 1000, the piezoelectric material has more ductility, which enhances the interface bonding but reduces its piezoelectric properties.
To verify the advantage of the functionally graded piezoelectric material, a static study has been carried out for a linear harvester. A point force is applied at the tip of the cantilever to induce bending of the cantilever, which generates stresses in the cantilever along the length.
Figure 8a shows the variation of the stress (
σx) in the thickness direction of the harvester for different grading indexes (
n). For this study,
n is varying from
n = 0 to
n ∞, where
n = 1000 has been considered as
n ∞. From Equation (1), it is easy to calculate the variation of PZT-0%Pt to PZT-20%Pt in the piezoelectric layer in thickness direction. In the cantilever, the piezoelectric layers are completely made from PZT-0%Pt for
n = 0 and PZT-20%Pt for
n = 1000. Hence, the proportion of PZT-20%PT increases with the value of the grading index. From
Figure 8, it can be concluded that the grading index has an impact on the stress distribution in the thickness direction. Stress variations at the layer interface become smoother with an increase in grading index. If the piezoelectric layer has grading index zero (
n = 0), then stress variation increases at the interface of the layers between the host layer and piezoelectric layer due to 100% piezoelectric (PZT-0%Pt) and 100% metal (Aluminum) bonding. It is to be noted that in
Figure 8, a sharp bend in the stress variation curve for
n = 0 is observed inside the host structure. This is due to the fact that in finite element formulation, we calculate stress values at Gauss integration points, which lie inside the thickness of a particular layer. A similar conclusion can be drawn from
Figure 8b, which shows the strain energy variation in thickness direction. There is a smooth strain energy variation in the cantilever along the thickness.
After performing the static analysis, simulations for a dynamic study have been performed for the energy harvester. The base of the harvester is accelerated sinusoidally with an amplitude of 1 g and varying frequency in the range of 20 Hz to 100 Hz. In this study, mechanical damping (0.2%) considered as mechanical loss during harmonic force. The linear harvester is only effective when the excitation frequency is close to its natural frequency (52 Hz) with any magnetic force. The power drastically decreases as the excitation frequency is away from the resonance frequency. Here, a wider frequency range is chosen due the unpredictable nature of the ambient vibrations. Simulations are performed for different energy harvesters which are having different grading indices (
n) in the piezoelectric layer as well as different distances between the magnets varying from 0.4 mm to 1 mm. Due to the magnetic force, nonlinearity is introduced in the energy harvester. The magnitude of the magnetic force can be altered with the help of the distance between the magnets. The energy harvester has more than one equilibrium if the force is applied beyond a certain load. Like in
Figure 1, the cantilever can be stable only in an upper bending or lower bending position due to repulsive force between magnets. The nominal equilibrium will lose stability, and new stable equilibrium positions will appear at a particular distance between two magnets.
Figure 9 shows the voltage vs. frequency plots for different grading indexes and distances between the magnets. From
Figure 9, we can conclude that energy harvesters can act as linear harvesters or nonlinear harvesters, depending on the distance between the magnets. For higher values of
d0, the energy harvester is a linear harvester, because the force due to the magnets is small. However, as two magnets come closer, the generated repulsive magnetic force increases, which changes the linear harvester into a nonlinear harvester. In this study, for a distance
d0 = 0.4 mm, the magnetic force is higher, due to which the system is bistable as shown in
Figure 9a. A second observation from
Figure 9a is that as the grading index (
n) increases, the nonlinearity of the harvester increases and reduces the voltage output. For different grading index values (
n), the harvester responses are quite different for a fixed magnetic distance due to the change in stiffness of the harvesters. The voltage peak is a maximum for
n = 0 while it is a minimum for
n = 1000. A change in voltage is observed due the change in the material characteristics of the harvester. We can conclude that the electric performance decreases while the mechanical performance increases at layer interfaces by increasing the grading index value (
n). As the distance between magnets increases, the nonlinear effect decreases and moves towards the linear behavior as shown in
Figure 9b–f. For distance
d0 = 1 mm and
d0 = 10 mm, the voltage plots are almost similar, which indicates that there is a negligible effect of magnetic force, and linear behavior of the harvester is observed.
However, for an energy harvesting application, the voltage is not a clear indicator of the performance of the harvester. The power across the load resistance (R) is an important output parameter of the energy harvester, which can be calculated using Equation (6). To harvest the maximum energy, it is required to use optimum resistance across the piezoelectric material. However, in the case of piezoelectric energy harvesting, optimum resistance can be written as 1/
ωcp, which is equal to the impedance of the piezoelectric layer [
26].
Figure 10 shows the power output vs. frequency for different grading indexes and magnetic distance. Patterns of the graphs are similar to the voltage. As the distance (
d0) increases, the nonlinear effect in the structure reduces. By decreasing the nonlinear effect, the peak of the power output increases while the effective harvesting frequency bandwidth decreases for each grading index. For example, at
d0 = 0.4 mm and
n = 1, the effecting harvesting frequency range is 20 Hz to 85 Hz, while at
d0 = 10 mm the effective harvesting frequency range is 35 Hz to 65 Hz. There is an approximately 216% increase of effective harvesting frequency bandwidth at a distance of 0.4 mm as compared to the bandwidth at a distance of
d0 = 10 mm. The peak voltage and power out of a nonlinear energy harvester at different magnetic distances and grading indexes have been summarize in
Table 5.
In this study we further extend the simulations in a time domain. For time domain simulations, we fix the excitation frequency to 5 Hz, which is far away from its natural frequency (about 51 Hz), for each grading index and compare the voltage output for the bistable and linear harvester. Obtained results are plotted in
Figure 11.
Figure 11a,c,e,g,i are plotted for velocity vs. amplitude, and
Figure 11b,d,f,h,j are plotted for voltage vs. time at different distances between the magnets. From the figure it can be observed that for the nonlinear harvester the velocity and voltage output are much higher as compared to the linear harvester. Velocity vs. displacement plots verify the bistability of the harvester for a nonlinear behavior of the harvester, while the second plot shows the voltage comparison for linear and bistable harvesters. As the grading index (
n) increases the output, voltage decreases, which confirms that at a higher grading index the electric performance of the harvester decreases. From the
Figure 11, we can conclude that the nonlinear harvester is much more effective than the linear harvester, for excitation frequencies away from its resonance frequency.
5. Conclusions
In the present approach, we have converted a multi-degree system into single-degree system to solve with a harmonic balance method. Mechanical system equations are coupled with an electric equation due to piezoelectric constitutive equations. The distance between magnets also has a great impact on the solution of coupled equations. Energy harvesting using functionally graded piezoelectric material has been explored using the finite element method for a linear and a nonlinear energy harvester. Non-linearity has been introduced using two magnets that represent a Duffing’s oscillator. The simple harmonic balance method has been employed to obtain the harvester response in a frequency domain. In addition, voltage and power responses have also been computed in a time domain. It is established that as the distance between magnets decreases, the degree of non-linearity between the system increases, and the effective working frequency bandwidth also increases significantly. The effective harvesting frequency range for d0 = 0.4 mm and n = 1 is observed to be 20 Hz to 85 Hz, while it was only between 35 Hz to 65 Hz for d0 = 10 mm, therefore yielding a 216% increase in the frequency bandwidth. By introducing functional grading in piezoelectric layers, a smooth stress variation is ensured at the layer interfaces, minimizing the possibility of debonding. However, a decrease in power output is recorded with grading in material properties. The decrease in electrical performance varies with the magnetic distance and the value of grading index; e.g., a decrease of 10% to 50% is seen for n = 0.5 to 1000, for a value of d0 = 0.4 mm, while for d0 = 1 mm this decrease ranges from 12% to 28% for n = 0.5 to 1000. Under different case studies, the peak output power varied from 2 mW (d0 = 0.4 mm and n = ∞) to 6 mW (d0 = 10 mm and n = 0). It is true the power decreases with increasing the ratio of PZT-20%PT to PZT-0%PT, but it also decreases the delamination at the layers interface. This is also an important factor to consider, the life of the energy harvester system, as these systems are potential substitutes for batteries. Based on the application and nature of vibration, the right selection can be done based on the grading index and distance between magnets (degree of nonlinear system in the harvester). In the present study, we focused on energy harvesting using functionally graded piezoelectric material. This study highlights the selection of a harvester parameter based on working conditions and applications. For example, a linear harvester is good where the exciting frequency is not much varying, and there is less possibility of the delamination of the piezoelectric layer. However, if there is a possibility of the delamination of layers due to environmental conditions, then the selection of the linear harvester is not good. In the energy harvester, delamination can occur between the piezoelectric and host layers due to the mechanical properties difference at the contact surface. Due to their difference in properties, both layers behave differently at the contact source. For example, if there is a change in operating temperature frequently, delamination can be possible due to a thermal expansion difference. Based on the working condition, parameters of the harvester can be optimized using a grading index and degree of nonlinearity in the system.