An experimentally validated model for geometrically nonlinear plucking-based frequency up-conversion in energy harvesting

The overall goal in this section is to obtain a relation between the overlap length d and the release height h for a given plectrum length L. To this end, it is required to implement a geometrically nonlinear model for the quasistatic flexible plectrum deformation. The homogeneous differential equations governing the static deflection of a thin cantilever (flexible plectrum in figure 1) with von Kármán-type geometric nonlinearity [41] can be given by

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}N}{{\rm{d}}x}=0\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}}^{2}M}{{\rm{d}}{x}^{2}}+\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}\left(N\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right)=0\end{eqnarray} \tag{ 2 }$$

where the resultant axial force (N) and the bending moment (M) terms are

$$\begin{eqnarray}&&N(x)=EA\left[\displaystyle \frac{{\rm{d}}u}{{\rm{d}}x}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right)}^{2}\right]\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&M(x)=-EI\displaystyle \frac{{{\rm{d}}}^{2}w}{{\rm{d}}{x}^{2}}.\end{eqnarray} \tag{ 4 }$$

Here, u and w are the longitudinal and transverse displacement variables (as shown in figure 1(b)), respectively, EA is the axial rigidity and EI is the flexural rigidity of the plectrum.

The resultant transverse shear force term can be extracted from equation (2) as

$$\begin{eqnarray}&&\begin{array}{l}Q(x)=\displaystyle \frac{{\rm{d}}M}{{\rm{d}}x}+N\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}=-\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}\left(EI\displaystyle \frac{{{\rm{d}}}^{2}w}{{\rm{d}}{x}^{2}}\right)\\ \,\,\,+EA\left[\displaystyle \frac{{\rm{d}}u}{{\rm{d}}x}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right)}^{2}\right]\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}.\end{array}\end{eqnarray} \tag{ 5 }$$

Next, we introduce the quasistatic tip force that is associated with the plucking excitation at the instant of release depicted in figure 1(b). For the clamped-free plectrum beam with a transversely applied force (P) at the free end, the boundary conditions at the clamped $(x=0)$ and the free $(x=L)$ ends are

$$\begin{eqnarray}&&u(0)=0\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&w(0)=0\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right|}_{x=0}=0\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&N(L)=EA{\left[\displaystyle \frac{{\rm{d}}u}{{\rm{d}}x}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right)}^{2}\right]}_{x=L}=0\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&M(L)={\left.-EI\displaystyle \frac{{{\rm{d}}}^{2}w}{{\rm{d}}{x}^{2}}\right|}_{x=L}=0\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\begin{array}{l}Q(L)={\left[\displaystyle \frac{{\rm{d}}M}{{\rm{d}}x}+N\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right]}_{x=L}=\left\{-\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}\left(EI\displaystyle \frac{{{\rm{d}}}^{2}w}{{\rm{d}}{x}^{2}}\right)\right.\\ \,\,\,+{\left.EA\left[\displaystyle \frac{{\rm{d}}u}{{\rm{d}}x}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right)}^{2}\right]\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right\}}_{x=L}=P.\end{array}\end{eqnarray} \tag{ 11 }$$

It should be noted that this quasistatic transverse tip force at the instant of the release is the same tip force acting on the harvester (in the opposite direction) that yields the initial displacement for each plucking motion.

From equation (1), the resultant axial force is constant $(N(x)=C)$ throughout the plectrum, but from the boundary condition given by equation (9), $N(L)=0.$ Thus, $N(x)=0$ throughout the length and it follows from equation (3) that

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}u}{{\rm{d}}x}=-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}\right)}^{2}.\end{eqnarray} \tag{ 12 }$$

Using equation (12) along with the boundary condition given by equation (6),

$$\begin{eqnarray}&&u(x)=-\displaystyle \frac{1}{2}\displaystyle {\int }_{0}^{x}{\left(\displaystyle \frac{{\rm{d}}w}{{\rm{d}}\eta }\right)}^{2}{\rm{d}}\eta .\end{eqnarray} \tag{ 13 }$$

Using equation (12) and in view of the fixed flexural rigidity (EI) of the uniform plectrum, equation (11) reduces to

$$\begin{eqnarray}&&{\left.-EI\displaystyle \frac{{{\rm{d}}}^{3}w}{{\rm{d}}{x}^{3}}\right|}_{x=L}=P.\end{eqnarray} \tag{ 14 }$$

From the boundary conditions defined by equations (7), (8), (10) and (14), the transverse deflection in terms of the applied force can be obtained as

$$\begin{eqnarray}&&w(x)=\displaystyle \frac{P}{6EI}(3L{x}^{2}-{x}^{3})\end{eqnarray} \tag{ 15 }$$

yielding

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}w}{{\rm{d}}x}=\displaystyle \frac{P}{2EI}(2Lx-{x}^{2}).\end{eqnarray} \tag{ 16 }$$

Substituting equation (16) into equation (13) gives

$$\begin{eqnarray}&&u(x)=-\displaystyle \frac{1}{8}{\left(\displaystyle \frac{P}{EI}\right)}^{2}\displaystyle {\int }_{0}^{x}{(2L\eta -{\eta }^{2})}^{2}{\rm{d}}\eta \\ &&\,\,=-\displaystyle \frac{1}{8}{\left(\displaystyle \frac{P}{EI}\right)}^{2}\displaystyle {\int }_{0}^{x}(4{L}^{2}{\eta }^{2}-4L{\eta }^{3}+{\eta }^{4}){\rm{d}}\eta \\ &&\,\,=-\displaystyle \frac{1}{8}{\left(\displaystyle \frac{P}{EI}\right)}^{2}\left(\displaystyle \frac{4{L}^{2}{x}^{3}}{3}-L{x}^{4}+\displaystyle \frac{{x}^{5}}{5}\right)\\ &&\,\,=-{\left(\displaystyle \frac{P}{EI}\right)}^{2}\left(\displaystyle \frac{20{L}^{2}{x}^{3}-15L{x}^{4}+3{x}^{5}}{120}\right)\,.\end{eqnarray} \tag{ 17 }$$

From equation (15),

$$\begin{eqnarray}&&\displaystyle \frac{P}{EI}=\displaystyle \frac{6w(x)}{3L{x}^{2}-{x}^{3}}\end{eqnarray} \tag{ 18 }$$

which can be substituted into equation (17) to give

$$\begin{eqnarray}&&\begin{array}{l}u(x)=-{\left(\displaystyle \frac{6w(x)}{3L{x}^{2}-{x}^{3}}\right)}^{2}\left(\displaystyle \frac{20{L}^{2}{x}^{3}-15L{x}^{4}+3{x}^{5}}{120}\right)\\ \,\,=-\displaystyle \frac{3}{10}\left(\displaystyle \frac{20{L}^{2}{x}^{3}-15L{x}^{4}+3{x}^{5}}{9{L}^{2}{x}^{4}-6L{x}^{5}+{x}^{6}}\right){w}^{2}(x).\end{array}\end{eqnarray} \tag{ 19 }$$

Equation (19) relates the longitudinal deflection $u(x)$ of the plectrum neutral axis to its geometrically large transverse deflection $w(x).$ Therefore, at the free end of the plectrum, the axial deflection due to transverse deflection can be given by

$$\begin{eqnarray}&&u(L)=-\displaystyle \frac{3}{5}\displaystyle \frac{{w}^{2}(L)}{L}\end{eqnarray} \tag{ 20 }$$

where, expectedly, $u(L)\lt 0$ regardless of the sign of $w(L).$

$$\begin{eqnarray}&&w(L)=\sqrt{-\displaystyle \frac{5L}{3}u(L)}.\end{eqnarray} \tag{ 21 }$$

Hence, if the overlap length of the plectrum on the stiff harvester beam is given by d, then its tip deflection at the instant of release (i.e. the release height in figure 1) is

$$\begin{eqnarray}&&h=\sqrt{\displaystyle \frac{5}{3}dL}.\end{eqnarray} \tag{ 22 }$$

Note that the transverse tip force P at the instant of release can be obtained via equation (22) by setting $x=L$ and $w(L)=h.$

Since the bimorph harvester is much stiffer than the plectrum exciting it (a reasonable underlying assumption), the harvester behavior is approximated by the linear cantilevered bimorph piezoelectric energy harvester model of Erturk and Inman [6, 14]. Note that under large stress fields (i.e. very hard excitation of the harvester by the plectrum due to larger overlap length, plectrum thickness, or elastic modulus, etc), material nonlinearities could be pronounced even for a stiff and geometrically linear piezoelectric bimorph, and those nonlinearities can be accommodated as needed using a materially nonlinear framework [42] (which is beyond the scope of the current work).

In view of the tip deflection-based deformation shape (predominantly in the first bending vibration mode), assuming a single-mode solution for the vibration response yields the following approximation:

$$\begin{eqnarray}&&{w}_{h}({L}_{h},t)=\displaystyle \sum _{r=1}^{\infty }{\phi }_{r}({L}_{h}){\eta }_{r}(t)\cong {\phi }_{1}({L}_{h}){\eta }_{1}(t)\end{eqnarray} \tag{ 23 }$$

where ${w}_{h}({L}_{h},t)$ is the transverse displacement of the bimorph harvester (of overhang length ${L}_{h}$), ${\phi }_{1}({L}_{h})$ is the mass normalized eigenfunction (based on the Euler–Bernoulli theory) evaluated at the cantilever tip, and ${\eta }_{1}(t)$ is the modal coordinate. Then, the governing electromechanical equations for free vibrations are:

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}}^{2}{w}_{h}({L}_{h},t)}{{\rm{d}}{t}^{2}}+2{\zeta }_{1}{\omega }_{1}\displaystyle \frac{{\rm{d}}{w}_{h}({L}_{h},t)}{{\rm{d}}t}+{\omega }_{1}^{2}{w}_{h}({L}_{h},t)\\ &&\,\,-{\theta }_{1}{\phi }_{1}({L}_{h})v(t)=0\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{C}_{p}\displaystyle \frac{{\rm{d}}v(t)}{{\rm{d}}t}+\displaystyle \frac{v(t)}{{R}_{{\rm{l}}}}+\displaystyle \frac{{\theta }_{1}}{{\phi }_{1}({L}_{h})}\displaystyle \frac{{\rm{d}}{w}_{h}({L}_{h},t)}{{\rm{d}}t}=0\end{eqnarray} \tag{ 25 }$$

where ${C}_{p}$ is the equivalent capacitance of the bimorph, ${R}_{{\rm{l}}}$ is the load resistance, $v(t)$ is the voltage across the resistance, ${\zeta }_{1}$ is the viscous damping ratio, ${\omega }_{1}$ is the natural frequency of the linear bimorph harvester for the first bending mode in short circuit, and ${\theta }_{1}$ is the modal electromechanical coupling that depends on the cross-section and material parameters of the bimorph [6, 14]. The initial conditions due to quasistatic plucking excitation are

$$\begin{eqnarray}&&{w}_{h}({L}_{h},0)=\displaystyle \frac{P{L}_{h}^{3}}{3E{I}_{{\rm{h}}}},\,\displaystyle \frac{{\rm{d}}{w}_{h}({L}_{h},0)}{{\rm{d}}t}=0,\,v(0)=0.\end{eqnarray} \tag{ 26 }$$

Here, $E{I}_{{\rm{h}}}$ is the flexural rigidity of the bimorph harvester and $P$ is the quasistatic transverse tip force on the harvester due to the plucking deflection h in figure 1, which is the same as the tip force of the plectrum under quasistatic conditions:

$$\begin{eqnarray}&&P=\displaystyle \frac{3EI}{{L}^{3}}\sqrt{\displaystyle \frac{5}{3}dL}\end{eqnarray} \tag{ 27 }$$

where EI is the flexural rigidity of the plectrum as in the derivation given in the previous section.

Equations (24) and (25) can then be cast into first-order form and simulated in time domain using the initial conditions given by equation (26). Note that an implicit assumption is that the quasistatic external stimulus on the plectrum (due to walking, etc) is large enough to enable plucking excitation, which can easily be checked by comparing the external load on the plectrum to the load at the release given by equation (27).

The bimorph harvester used in the experiments (Piezo Systems, Inc.—T215-H4-503X) is shown in figure 2(a) along with its experimental setup. This base excitation setup is employed to ensure that the parameters in the bimorph energy harvester model are accurate prior to the plucking experiments, and to identify or update parameters such as the mechanical damping ratio (in short circuit) and static capacitance. The resulting fundamental short-circuit resonance frequency of the harvester is around 74.6 Hz, which becomes 78.8 Hz in open circuit as observed in figures 2(b) and (c). The voltage and vibration frequency response function (FRF) simulations are shown for a range of resistive electrical loads, revealing very good agreement with the experimental data. Therefore, bimorph harvester parameters based on the model in [6, 14] can be combined with the nonlinear plectrum model to simulate the plucking experiments performed next.

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The experimental setup for plucking-based energy harvesting is shown in figure 3(a) and a close-up of the plucking excitation is given in figure 3(b). The plectrum is made of stainless steel and it has a thickness of around 0.05 mm (much thinner than the bimorph harvester), overhang length of 45 mm, and its width of 31.8 mm is identical to that of the harvester. As can be seen from figure 3, the plectrum deformation can be quite nonlinear (which is necessary for the release and plucking to take place). Simulations of the voltage output and the tip velocity of the harvester obtained using the combined nonlinear plectrum–linear harvester model are shown in figure 4 for two select overlap lengths (2 and 10 mm—which are the two extremes considered here). The load resistance in figure 4 is the optimal value (20 kΩ) among the set of resistors used in the experiments. Overall, the voltage and tip velocity time series of the harvester shown in figure 4 reveal very good agreement between the model simulations and experimental measurements.

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Next, the average electrical power delivered to the resistive load is analyzed by using the time series of the harvester response under plucking excitation for a range of load resistance values. Note that the root-mean-square voltage and the average power are easily calculated using the voltage time series (such as those in figure 4) for a time window that covers until the oscillations die out for a given load resistance. For a given plectrum and harvester combination, the main input to the nonlinear plectrum and linear harvester combination is the overlap length d. We recall that a critical aspect of the problem is the relationship between the geometric parameters d (overlap length) and h (released height), as shown in figure 1. As can be anticipated, the overlap length (d) between the plectrum and the harvester alters the amount of force on the harvester (discussed further in the next section), and therefore affects the harvested power output, as shown in figure 5. The overall results are shown in this figure for various load resistance and overlap length values, where the model simulations (solid lines) agree well with the experimental data points. Note that a significant part of the mismatch is due to the difficulty in achieving zero initial velocity condition in the experiments.

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Having validated the electromechanical model, the effects of critical system parameters, such as the plectrum thickness and overlap length, on the tip force and deflection as well as the harvested power are simulated for the same system shown in figure 3. First, the quasistatic transverse tip force transmitted to the harvester at the instant of the release and the resulting harvester tip displacement are simulated as a function of plectrum thickness and overlap length. As expected, it is observed from figure 6 that the tip displacement and force follow similar trends when the plectrum thickness and overlap length are changed. The maximum harvester tip deflection of 0.44 mm and tip force of 0.21 N are seen when the plectrum thickness and overlap are at their maximum simulated values among the parameter ranges in figure 6. Either the plectrum thickness (in design) or the overlap length can be increased in order to increase the force transmitted to the harvester. It is important to note that, in these simulations, the tip displacement of the harvester does not exceed a value that would result in the failure of the harvester (which can easily be shown by analyzing the maximum stress at the root), which would be a typical design constraint.

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Increasing the overlap length and plectrum thickness increases the average power output from the harvester. Figures 7(a) and (b) show the average power variation with plectrum thickness at the two extreme cases of overlap length (2 and 10 mm). This effect can be seen for all overlap lengths in figure 8(a). The average power at optimal load increases by almost four orders of magnitude (from 0.15 to 108 μW) as the overlap length is increased from 2 to 10 mm, which can be seen in figure 8(b). From the design and application standpoint, for a given plectrum thickness, one can increase the overlap length to increase the power output. Alternatively, the plectrum thickness can be tailored to increase the power output under the constraint of a fixed overlap length. The effects of several other parameters, such as the plectrum material (i.e. elastic modulus), can be simulated using the proposed model in order to optimize the harvested power, while avoiding a mechanical failure in the harvester.

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