Abstract
An important issue in the field of energy harvesting through piezoelectric materials is the design of simple and efficient structures which are multi-frequency in the ambient vibration range. This paper deals with the experimental assessment of four fractal-inspired multi-frequency structures for piezoelectric energy harvesting. These structures, thin plates of square shape, were proposed in a previous work by the author and their modal response numerically analysed. The present work has two aims. First, to assess the modal response of these structures through an experimental investigation. Second, to evaluate, through computational simulation, the performance of a piezoelectric converter relying on one of these fractal-inspired structures. The four fractal-inspired structures are examined in the range between 0 and 100 Hz, with regard to both eigenfrequencies and eigenmodes. In the same frequency range, the modal response and power output of the piezoelectric converter are investigated.
1. Introduction
This paper investigates both computationally and experimentally the modal response of fractal-inspired multi-frequency structures for piezoelectric harvesting of ambient kinetic energy. In addition, a piezoelectric converter conforming to one of these structures is presented and preliminarily investigated through a computational model with regard to modal response and electric charge generation. The study of energy harvesting devices, which convert freely available ambient energy into electrical energy, has gained considerable attention in research and industrial communities in the last few years. The interest in these devices comes from the development of low-consumption wireless sensor nodes that can be self-powered through ambient energy.
For the aim of energy harvesting, one of the most suitable forms of ambient energy is kinetic energy, due to its peculiarity of being almost ubiquitous and easily accessible. Kinetic energy can be conveniently exploited for energy harvesting, since it is almost ubiquitous and easily accessible in the ambient. In particular, kinetic energy is typical of machines from household goods to industrial plant, where it can be found in the form of vibrations or impulsive forces. A critical challenge is the design of converter structures that exhibit a multi-frequency response in the range of ambient mechanical vibrations (mainly between 0 and 100 Hz [1]), so as to achieve maximum power output of the harvester.
Among the available technologies for energy harvesting, piezoelectric materials [2] have been intensively studied due to their high conversion efficiency and the simple configuration of the converters that they allow. In the literature, the most common piezoelectric converter structure is a cantilever beam configuration [3–7]. There are two main advantages of this structure. First, the large strains that are generated; and second, easy tuning at the desired resonant frequency [5, 6]. By assembling a batch of cantilevers, a multi-frequency energy converter can be obtained, where each cantilever is tuned at a specific resonant frequency [3, 4, 7]. The main drawback of this configuration is that each resonant frequency pertains only to a specific cantilever of the batch. Therefore, at each resonant frequency the electric charge generation is limited to the cantilever of the batch that is deformed by the specific eigenmode, thus remarkably reducing the global efficiency of the harvester.
In a previous work [8], the author presented four fractal-inspired conceptual solutions for energy converters, which achieve a broadband multi-frequency response below 100 Hz. These structures are simple thin sheets where the fractal geometry is obtained through cuts in the plate that create either a straight or a convoluted path between the external constraints. The computational analyses performed in [8] show that these fractal-inspired geometries are able to generate up to four times the electric charge generated by a batch of cantilevers having the same eigenfrequencies and surface area. This higher efficiency is imputable to the fact that the proposed fractal-inspired structures include and combine both the eigenfrequencies of the structure as a whole and the ones of the single branches.
This work has two aims. First, to assess experimentally the modal response of the two most promising solutions presented in the previous work [8]. Second, to investigate computationally a piezoelectric converter inspired by one of these structures. Both structures have a square shape with a side length of 100 mm and are investigated at two fractal iteration levels. The work is organized in two steps. The first step examines experimentally and numerically the eigenfrequencies and eigenmodes of the fractal-inspired structures, in the frequency range between 0 and 100 Hz. The second step investigates, through a computational model, the modal response and the power output of a piezoelectric converter inspired by one of the structures examined in the first step. The converter consists of a supporting steel plate (0.2 mm thick) and an upper thin piezoelectric sheet (0.267 mm thick) of lead zirconate titanate.
The results provided by the experimental tests appear to be in good agreement with those from the computational models, in terms of both eigenfrequencies and eigenmodes. Moreover, the proposed piezoelectric converter conforming to one of these fractal-inspired structures provides four eigenfrequencies below 100 Hz, and seems to be able to generate a significant electric charge.
2. Method
Figure 1 shows sketches of the four fractal-inspired structures for the energy converters examined in the present work. Each row in figure 1 presents two configurations of the same conceptual solution, which differ only in a different number of fractal iterations (F1 and F2). Chosen among those proposed and studied numerically in [8], these structures are thin plates constrained similarly to a cantilever. The side length L of the structures is equal to 100 mm while the length of the inner cuts is obtained according to the following relationship:
where i = 1 and 2 in the structures in figures 1(a) and (c) (F1) while i = 1, 2 and 3 in the structures in figures 1(b) and (d) (F2). An equal distance between the inner cuts was assumed for all the structures.
Figure 1. Sketches of the fractal-inspired structures examined in the present work.
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Standard imageThe work is organized in two steps. First, the modal response of the proposed structures for energy converters (figure 1) is assessed both experimentally, on an electro-dynamic shaker, and through a computational model. Second, a piezoelectric converter, conforming to structure 1 at fractal iteration F1 (figure 1(a)), is designed and examined through a computational model, with regard to modal response and power output.
2.1. Assessment of the fractal-inspired structures
2.1.1. Experimental tests.
Figure 2 shows the experimental setup of the four thin-sheet structures corresponding, from (a) to (d), to the ones presented in figure 1. All the structures were made, through laser-jet cutting, from a sheet of S235JR with a thickness of 0.8 mm. The modal response of each structure was investigated in the range between 0 and 100 Hz through an electro-dynamic shaker (Data Physics BV400). Through the shaker control unit, a sinusoidal acceleration signal was applied, sweeping the above frequency range with a constant amplitude of 9.81 m s−2. A closed loop control was implemented on the system through three miniature accelerometers (MMF KS94B100 [9]), fixed by their magnetic bases. One of them (not visible in figure 2) was fixed to the vibrating table of the shaker. The remaining two accelerometers were fixed in different positions on the structure under testing (figure 2) so as to register the dynamic response of the structure. The main drawback of accelerometers is that they represent an added mass for the structure, thus causing frequency shift of the cantilever branches to which they are attached. A laser Doppler vibrometer would be a better solution to investigate the modal response of these structures, but it was not available in the laboratory where the tests were performed.
Figure 2. Experimental setup of the structures under investigation.
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Standard imageThe shaker was managed by an eight channel Abacus controller, equipped with proprietary software (Signal Star) installed on a PC, which performed both control of the system and data acquisition.
2.1.2. Computational analysis.
The modal responses of all the four thin-sheet structures shown in figure 1 were investigated, in addition, through a computational model developed using the commercial finite element software ABAQUS [10]. The model described in detail the experimental configuration, including also the miniature accelerometers applied to the structures (figure 2). The computational model described the structures through a four noded, doubly-curved, thin shell element (S4R5), with reduced integration and hourglass control [10]. According to the convergence procedure described in [8], the average side length of the elements was set equal to 2 mm. The material was described as linear elastic with a Young's modulus of 206 GPa and a Poisson's ratio of 0.3. The miniature accelerometers were described in terms of point mass and inertia. Through an internal kinematic constraint they were linked to the surface on the same cantilevers as in figure 2, keeping the same positions as in the experimental tests. The mass and inertia properties of the accelerometers were calculated according to the data provided in [9] (figure 3). Each structure was constrained through a built-in boundary condition applied along the edges, according to the constraints in figure 1. The modal analysis was performed adopting the Lanczos algorithm, in the range between 0 and 100 Hz.
Figure 3. Sketch and properties of the miniature accelerometers.
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Standard image2.2. Assessment of the piezoelectric converter
Figure 4 shows a sketch of the piezoelectric converter, conforming to structure 1 at iteration level F1 (figure 1(a)). The piezoelectric converter consists of a support plate and an upper layer of piezoelectric material (grey area in figure 4), which is placed closed to the constrained side of the structure. The support plate is made of a thin sheet of steel (S235JR), with a thickness of 0.2 mm. This value of the thickness of the support plate was chosen as the best trade-off between the need to provide an adequate mechanical strength and the need to obtain a large number of eigenfrequencies in the range below 100 Hz.
Figure 4. Sketch of the piezoelectric converter.
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Standard imageThe piezoelectric layer consists of four rectangular patches made of PSI-5H4E [11], a commercial sheet of lead zirconate titanate with a thickness of 0.267 mm. The length of the piezoelectric patches was chosen as the best trade-off between energy conversion efficiency and the need to adopt commercially available piezoelectric patches. In particular, this second issue is aimed at reducing the cost of the converter. Table 1 reports the mechanical and electrical properties of the piezoelectric patches which include nickel electrodes and connecting wires on both sides.
Table 1. Electrical and mechanical properties of PSI-5H4E.
Piezoelectric strain coefficient, d31 (m V −1) | − 320 × 10−12 |
Relative dielectric constant, k3 | 3800 |
Mass density, ρ (kg m−3) | 7800 |
Young's modulus, E (GPa) | 62 |
Poisson's ratio, ν | 0.3 |
Structural damping | 0.02 |
Each piezoelectric patch is bonded to one of the inner cantilevers through Hysol 9466 [12], a two-key epoxy which acts as an excellent electrical insulator. The thickness of the adhesive layer equals 0.1 mm, in order to guarantee a good electrical insulation.
The piezoelectric converter was examined computationally as described in section 2.2.1
2.2.1. Computational analysis.
The piezoelectric converter (figure 4) was examined through a computational model, using the commercial finite element software ABAQUS [10]. The computational model, like the one in section 2.1.2, described the support plate through semi-structural thin shell elements placed on its mid-plane. By contrast, the piezoelectric patches and the adhesive layer were described as three-dimensional, through solid elements. In particular, the same thin shell elements (S4R5) as in the previous modal analysis of section 2.1.2 were used to describe the support plate (figure 5). The piezoelectric layer was assumed as a homogeneous material, disregarding the top and bottom layers of nickel which act as electrodes, since they are extremely thin. Therefore, each piezoelectric patch (on the top in figure 5) was described through three layers of piezoelectric hexahedral elements (C3D8E). The peculiarity of these elements is an electrical conduction degree of freedom, in addition to the translational ones [10]. Also the adhesive layer was described in the computational model, since its thickness is of the same order of magnitude of that of the support plate and piezoelectric layer. The adhesive (in the middle in figure 5) was described through two layers of linear hexahedral elements (C3D8). The average side length of both the hexahedral elements and the shells was set equal to 0.5 mm, in order to obtain a correct aspect ratio.
Figure 5. Image of the computational model of the piezoelectric converter with a close-up view of the different layers.
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Standard imageBoth the steel of the support plate and the adhesive were described as linear elastic with Young's modulus and Poisson's ratio of 206 GPa and 0.3 for the steel and 1718 MPa and 0.3 for the adhesive respectively. Similarly, the piezoelectric material was described as linear elastic with structural damping, in order to take into account the damping effect due to the generated electric charge. Its piezoelectric properties were defined according to the data reported in table 1. The structure was constrained through a built-in boundary condition along the left edge of the support plate, according to the constraints shown in figure 4. On the bottom face of the piezoelectric layer the electric potential was set to zero through an internal constraint. This was aimed at simulating an electrical connection of the piezoelectric patches with an external oscilloscope.
The computational analysis was organized in two steps. First, a modal analysis was performed adopting the subspace algorithm [10], in the range between 0 and 100 Hz, with the aim of investigating the eigenfrequencies and eigenmodes of the converter. Second, a direct-solution steady state dynamic analysis [10] was performed to examine the response of the system under periodic excitation in the range between 0 and 100 Hz. This analysis, which applied the loading at a number of different frequencies in the above range, was aimed at investigating the power output of the converter, under a periodic sinusoidal excitation with constant amplitude of 9.81 m s−2. All the analyses were performed with a PC equipped with an Intel i7 processor and 12 GB of RAM.
3. Results
3.1. Assessment of the fractal-inspired structures
3.1.1. Experimental tests.
Figure 6 shows diagrams of the frequency response function for structure 1 (figures 6(a) and (b)) and structure 2 (figures 6(c) and (d)) provided by the experimental tests (solid dotted black lines). The two solid dotted lines in the diagrams correspond to the results provided by different accelerometers, which were placed as shown in figure 2. Figure 7 shows the first and second experimental eigenmodes of structure 1 at iteration level F1 (figures 7(a) and (c)), and at iteration level F2 (figures 7(e) and (g)), respectively. Figure 8 shows the same results for structure 2 at iteration level F1 (figures 8(a) and (c)) and iteration level F2 (figures 8(e) and (g)). All these pictures were taken with a 6 Mpx digital photo camera.
Figure 6. Diagrams of the experimental frequency response function ( accelerometer 1 and accelerometer 2) and numerical eigenfrequencies (– – –) of structure 1 ((a)–(b)) and structure 2 ((c)–(d)).
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Standard imageFigure 7. Experimental versus computational eigenmodes of structure 1.
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Standard imageFigure 8. Experimental versus computational eigenmodes of structure 2.
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Standard image3.1.2. Computational analysis.
Figure 6 shows the eigenfrequencies provided by the computational analyses for each of the four structures examined (red dashed lines in the diagrams). Figure 7 shows the first and second computational eigenmodes of structure 1 at iteration level F1 (figures 7(b) and (d)), and at iteration level F2 (figures 7(f) and (h)), respectively. Figure 8 shows the same results for structure 2 at iteration level F1 (figures 8(b) and (d)) and iteration level F2 (figures 8(f) and (h)). In both cases all the pictures were in the same scale of deformation.
3.2. Assessment of the piezoelectric converter
3.2.1. Computational analysis.
Table 2 reports the eigenfrequencies of the piezoelectric converter, in the range between 0 and 100 Hz. Figure 9 reports the eigenmodes of the converter from the one corresponding to the lower eigenfrequency (figure 9(a)) up to the higher eigenfrequency (figure 9(d)). The bar chart in figure 10 shows the total value of electric charge generated by the piezoelectric converter at each of the four eigenfrequencies in the range between 0 and 100 Hz. This electric charge was calculated from the results of the computational model according to the following relationship [13]:
where b is the width of a single branch of the converter, Ep is the Young's modulus of the piezoelectric layer, εx(x) is the longitudinal strain on the mid-surface of the piezoelectric patches, and d31 is the piezoelectric strain coefficient of the material (table 1). The longitudinal strain εx(x) was read on the mid-plane of the piezoelectric patch, and is a function of the longitudinal coordinate x in order to take into account the stress gradient from the built-in region to the free end. Hence, the total electric charge was calculated on each piezoelectric patch and then their absolute values were added.
Figure 9. Eigenmodes of the piezoelectric converter in the range between 0 and 100 Hz.
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Standard imageFigure 10. Bar chart of the total electric charge generated by the piezoelectric converter at the four eigenfrequencies below 100 Hz.
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Standard imageTable 2. Eigenfrequencies of the piezoelectric converter.
f1 (Hz) | 38.6 |
f2 (Hz) | 39.4 |
f3 (Hz) | 46.8 |
f4 (Hz) | 47.6 |
4. Discussion
4.1. Assessment of the fractal-inspired structures
By examining the diagrams in figure 6 it appears that the proposed fractal-inspired structures exhibit many eigenfrequencies in the range below 100 Hz. The experimental tests showed that structure 1 has four eigenfrequencies at both iteration levels (figures 6(a) and (b)). Similarly, structure 2 provided the same number of eigenfrequencies at iteration level F1 while eight eigenfrequencies were registered experimentally at iteration level F2 (figures 6(c) and (d)). For both structures, the signals of the two accelerometers are slightly different, since they were applied to different places. However, both accelerometers identify the same eigenfrequencies.
The computational modal analysis provides results in close agreement with the experimental ones at the lower eigenfrequencies. By contrast, some disagreements can be observed at the higher eigenfrequencies, where the computational model is stiffer, as occurs, in particular, in structure 1 (figures 6(a) and (b)) and at iteration level F1 of structure 2 (figure 6(c)). This can be imputed to the fact that the computational model did not describe the inertial effects of the wires connecting the accelerometers to the controller. In addition, the computational model provides some eigenfrequencies which were not clearly retrieved in the experimental campaign (figures 6(b)–(d)). Probably, additional experimental tests with the accelerometers in different positions on the plate would be needed to observe these eigenfrequencies.
Figure 6 highlights that, at each iteration level, structure 2 has a larger bandwidth than structure 1. In particular, at both iteration levels, the fundamental frequency of structure 2 is significantly lower than for structure 1. With regard to the amplitude of the frequency response function, however, structure 1 exhibits values that are noticeably higher than structure 2. This can be imputed to a greater strain energy involved in the corresponding eigenmodes.
Figures 7 and 8 show a close agreement between the experimental and computational eigenmodes of both structures. This good agreement was observed also for other eigenmodes of the structure, not reported here for the sake of brevity. It is observed that, despite not being visible, the accelerometers were described in detail in the computational model, in the same positions as in the real prototype in figure 2. The eigenmodes in figures 7 and 8 provide two additional observations. First, in the case of structure 2 the eigenmodes have a more complex shape. Second, the eigenmodes of structure 2 exhibit larger displacements. This can be attributed to the intrinsic higher compliance of this structure.
4.2. Assessment of the piezoelectric converter
As reported in table 2, the piezoelectric converter exhibits four eigenfrequencies in the range of interest. In particular, these eigenfrequencies are lower than the corresponding values of structure 1 at iteration level F1 (figure 6(a)). This good result comes from the lowest total thickness of the converter with respect to the fractal-inspired structure in figure 1(a). In particular, the support plate is four times thinner than in the prototype in figure 1(a). Moreover, the piezoelectric layer has a Young's modulus similar to that of aluminum, a mass density equal to steel, and a thickness similar to that of the support plate.
Figure 9 highlights that the eigenmodes of the piezoelectric converter resembles the ones of structure 1 at iteration level F1 (figures 7(a)–(d)). The main differences are imputable to the lack of the accelerometers, which were not introduced in the computational analysis of the piezoelectric converter. In particular, the first eigenmode (figure 9(a)) involves the whole structure as a unique cantilever of equal length. The following eigenmodes (figures 9(b)–(d)) are a combination of the eigenmodes of the whole structure with those of the single branches. Therefore, the pervasive deformation characterizing this fractal-inspired converter for all the eigenmodes, which is a key factor for high efficiency in energy conversion, appears in figure 9.
The bar chart in figure 10 shows that when the converter is excited with a periodic load whose frequency corresponds to the first eigenfrequency, a higher electric charge is generated. A relatively high value of electric charge is obtained also when the frequency of the experimental excitation corresponds to the second eigenfrequency. By contrast, the generated electric charge is very low at the third and fourth eigenmodes. This result comes from the different amplitudes and damping that usually characterize each eigenmode. According to equation (2), the highest charge by the first eigenmode comes from the highest deformation and consequently highest strain.
As a general remark it is observed that the fractal-inspired geometry of the structures proposed in this work makes it possible to obtain many eigenfrequencies in a narrow range (i.e. below 100 Hz). This modal response is a peculiarity of these structures which exhibit both the eigenfrequencies of the geometry as a whole (figures 7(a) and (b), 8(a) and (b), and 9(a)) and the ones of the sub-elements that are part of the structure (figures 7(c)–(h), 8(c)–(h), and 9(b)–(d)). In particular, in figure 9(a) the typical first eigenmode of a traditional cantilever plate is clearly visible. However, with respect to a cantilever plate whose second eigenfrequency would be well above 100 Hz, the subsequent eigenfrequencies of these fractal-inspired solutions are quite close to the fundamental eigenfrequency.
With regard to the piezoelectric converter studied in section 2.2, the outcome of this preliminary computational investigation is encouraging and proves that this configuration deserves to be assessed also experimentally. A physical prototype of the piezoelectric converter will be built, in order to assess experimentally the multi-frequency behaviour and the power generation (i.e. electrical energy generated by the piezoelectric material) of an energy harvester conforming to the proposed solution.
5. Conclusions
This work is focused on two issues. First, experimental assessment of the modal responses of four fractal-inspired multi-frequency structures previously proposed by the author. Second, the preliminary design and computational investigation of a piezoelectric converter inspired by one of the proposed structures. In the first step, the proposed structures, square plates with a side length of 100 mm, are examined in the range between 0 and 100 Hz on an electro-dynamic shaker and through a detailed computational analysis. The results show a significant multi-frequency response of the proposed structures and a good agreement between the results of the experimental tests and computational analyses. In the second step, the analysis of the piezoelectric converter shows good multi-frequency behaviour of the converter and significant power output at the first and second eigenmodes.