Investigation of a MEMS piezoelectric energy harvester system with a frequency-widened-bandwidth mechanism introduced by mechanical stoppers

For a vibration-based PEH operating in the 3–1 mode, an applied mechanical stress σ₁ in the longitudinal direction, i.e., denoted as the 1-axis, induces an electrical displacement D₃ across the piezoelectric layer, i.e., an electrical field generated along the normal direction to the cantilever (3-axis). Meanwhile, the applied electrical field E₃ across the piezoelectric layer in turn affects the mechanical strain ξ₁. The relationship between the electrical displacement D₃ and the mechanical strain ξ₁ is given by the piezoelectric constitutive equations as

$$\begin{eqnarray} \displaystyle {D}_{3}={\varepsilon }_{33}{E}_{3}+{d}_{31}{\sigma }_{1}&&\displaystyle \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \displaystyle {\xi }_{1}={s}_{11}{\sigma }_{1}+{d}_{31}{E}_{3}&&\displaystyle \end{eqnarray} \tag{ 2 }$$

where s₁₁, ε₃₃ and d₃₁ are the axial elastic compliance under a constant electric field, the transverse dielectric coefficient measured at a constant stress and the transverse–axial piezoelectric constant, respectively.

The open circuit voltage can be derived [34] from equation (1) as

$$\begin{eqnarray} &&{V}_{\mathrm{oc}}=\frac{-{d}_{31}Y{t}_{\mathrm{e}}}{{\varepsilon }_{33}{l}_{\mathrm{b}}}\int \nolimits _{0}^{{l}_{\mathrm{b}}}{\xi }_{1}(x)\hspace{0.167em} \mathrm{d}x \end{eqnarray} \tag{ 3 }$$

where Y is the Young's modulus of the piezoelectric material, t_e is the thickness of the piezoelectric layer, l_b is the length of the supporting beam and ξ₁(x) is the strain distribution along the top surface of the supporting beam. Considering that PEH-B is subjected to a base acceleration, a concentrated force is assumed to be applied at the middle of the mass. The strain distribution ξ_1B(x) in terms of the mass tip displacement δ_B of PEH-B is given by

$$\begin{eqnarray} \displaystyle {\xi }_{1\mathrm{B}}(x)=\frac{3{t}_{\mathrm{b}}}{{l}_{\mathrm{b}}}\left(\frac{2{l}_{\mathrm{b}}+{l}_{\mathrm{m}}-2 x}{4{l}_{\mathrm{b}}^{2}+9{l}_{\mathrm{b}}{l}_{\mathrm{m}}+6{l}_{\mathrm{m}}^{2}}\right){\delta }_{\mathrm{B}}&&\displaystyle \end{eqnarray} \tag{ 4 }$$

where x refers to the variable distance from the beam anchor to the beam tip, l_m is the proof mass length, l_b and t_b are the supporting beam length and thickness, respectively. It is assumed that PEH-T is subjected to a concentrated force applied at the free end of the supporting beam. The strain distribution ξ_1T(x) in terms of the beam tip displacement δ_T of PEH-T is given by

$$\begin{eqnarray} &&{\xi }_{1\mathrm{T}}(x)=\frac{3{t}_{\mathrm{b}}({l}_{\mathrm{b}}-x)}{2{l}_{\mathrm{b}}^{3}}{\delta }_{\mathrm{T}}. \end{eqnarray} \tag{ 5 }$$

For detailed derivations of the strain distributions for PEH-B and PEH-T the reader is referred to [34–36]. The open circuit voltages in terms of the mass tip displacement of PEH-B and the beam tip displacement of PEH-T are expressed as

$$\begin{eqnarray} \displaystyle {V}_{\mathrm{ocB}}=\frac{-{d}_{31}Y{t}_{\mathrm{e}}}{{\varepsilon }_{33}{l}_{\mathrm{b}}}\frac{3({l}_{\mathrm{b}}+{l}_{\mathrm{m}}){t}_{\mathrm{b}}{\delta }_{\mathrm{B}}}{4{l}_{\mathrm{b}}^{2}+9{l}_{\mathrm{b}}{l}_{\mathrm{m}}+6{l}_{\mathrm{m}}^{2}}&&\displaystyle \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \displaystyle {V}_{\mathrm{ocT}}=\frac{-{d}_{31}Y{t}_{\mathrm{e}}}{{\varepsilon }_{33}}\frac{3{t}_{\mathrm{b}}{\delta }_{\mathrm{T}}}{4{l}_{\mathrm{b}}^{2}}.&&\displaystyle \end{eqnarray} \tag{ 7 }$$

Hence, the average power delivered to the connected load is

$$\begin{eqnarray} &&{P}_{\mathrm{rms}}=\frac{1}{2}\frac{{V}_{\mathrm{oc}}^{2}}{\mathop{({Z}_{\mathrm{P}}+{Z}_{\mathrm{L}})}\nolimits ^{2}}{Z}_{\mathrm{L}} \end{eqnarray} \tag{ 8 }$$

where Z_P and Z_L are the complex impedances of the piezoelectric capacitor and connected load, respectively. The maximum power transfer occurs when the load impedance Z_L matches with the piezoelectric impedance, i.e., Z_L = Z_P. In a situation where the connected load is purely real, i.e., Z_L = R_L, the maximum average power transfer occurs when the load resistance matches the magnitude of the piezoelectric impedance, i.e., R_L = |Z_P|.

Figure 2 illustrates a piecewise linear model of the impact-based PEH system with stoppers on two sides. PEH-B, which is modeled as a primary suspension system, consists of a proof mass m suspended by a spring k₀ and a damper c₀. PEH-T and the metal package are considered as secondary suspension systems and are assumed to have spring stiffnesses of k₁ and k₂ and damping factors of c₁ and c₂, respectively. PEH-T is mounted above the proof mass with a top-stopper distance of d₁ while the metal package is mounted below the proof mass of PEH-B with a bottom-stopper distance of d₂. The top-stopper distance d₁ is assumed to be smaller than the bottom-stopper distance d₂, i.e., d₁ < d₂. These secondary suspension systems limit the relative movement of the mass and prevent the mass from excessive travel. In the model, the base excitation y(t) causes the proof mass to move relative to the housing as z(t). The relative motion of the proof mass can be divided into three stages. In the first stage (stage I), assuming that the relative motion of the mass is smaller than the stopper distances of d₁ and d₂, the system retains an overall stiffness and damping of k₀ and c₀, respectively. When the relative mass motion exceeds d₁ but is smaller than d₂, the top-stopper, i.e., PEH-T, will be engaged (stage II). The overall stiffness and damping of the system are then increased to k₀ + k₁ and c₀ + c₁, respectively. In the third stage (stage III), when the relative mass motion exceeds d₂, the top-stopper and the bottom-stopper (the metal package) will both be engaged. The overall stiffness and damping will then be increased to k₀ + k₂ and c₀ + c₂ as the downward motion exceeds d₂ and change to k₀ + k₁ and c₀ + c₁ as the upward motion exceeds d₁.

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3.3.1. Stoppers on two sides.

The differential equation of motion of the impact-based PEH system with stoppers engaged on two sides can be written as [37]

$$\begin{eqnarray} &&\displaystyle \begin{array}{@{}l@{}} \displaystyle \fl m\ddot{z}+({c}_{0}+{c}_{1})\dot {z}+({k}_{0}+{k}_{1})z-{k}_{1}{d}_{1}=-m\ddot{y}\\ \displaystyle (z\geq {d}_{1})\\ \displaystyle \fl m\ddot{z}+{c}_{0}\dot {z}+{k}_{0}z=-m\ddot{y}\tqs (-{d}_{2}\lt z\lt {d}_{1})\\ \displaystyle \fl m\ddot{z}+({c}_{0}+{c}_{2})\dot {z}+({k}_{0}+{k}_{2})z-{k}_{2}{d}_{2}=-m\ddot{y}\\ \displaystyle (z\leq -{d}_{2}). \end{array}&&\displaystyle \end{eqnarray} \tag{ 9 }$$

Equation (9) can be rearranged as follows:

$$\begin{eqnarray} &&\displaystyle \begin{array}{@{}l@{}} \displaystyle \fl \ddot{z}+(2{\xi }_{0}{\omega }_{0}+2{\xi }_{1}{\omega }_{1})\dot {z}+({\omega }_{0}^{2}+{\omega }_{1}^{2})z-{\omega }_{1}^{2}{d}_{1}=-\ddot{y}\\ \displaystyle (z\geq {d}_{1})\\ \displaystyle \fl \ddot{z}+2{\xi }_{0}{\omega }_{0}\dot {z}+{\omega }_{0}^{2}z=-\ddot{y}\tqs (-{d}_{2}\lt z\lt {d}_{1})\\ \displaystyle \fl \ddot{z}+(2{\xi }_{0}{\omega }_{0}+2{\xi }_{2}{\omega }_{2})\dot {z}+({\omega }_{0}^{2}+{\omega }_{2}^{2})z-{\omega }_{2}^{2}{d}_{2}=-\ddot{y}\\ \displaystyle (z\leq -{d}_{2}) \end{array}&&\displaystyle \end{eqnarray} \tag{ 10 }$$

where y(t) = Ysin(ωt), Y is the amplitude of the base excitation, ω is the excitation frequency, ξ₀ and ω₀ are the primary suspension damping and frequency characteristics, and ξ₁, ξ₂ and ω₁, ω₂ are the secondary suspension damping and frequency characteristics, which can be further defined as $2{\xi }_{0}{\omega }_{0}=\frac{{c}_{0}}{m}$, $2{\xi }_{1}{\omega }_{1}=\frac{{c}_{1}}{m}$, $2{\xi }_{2}{\omega }_{2}=\frac{{c}_{2}}{m}$, ${\omega }_{0}^{2}=\frac{{k}_{0}}{m}$, ${\omega }_{1}^{2}=\frac{{k}_{1}}{m}$, ${\omega }_{2}^{2}=\frac{{k}_{2}}{m}$. In order to study the frequency response of the impact-based PEH system, we use dimensionless variables

$$\begin{eqnarray*} \displaystyle \tau ={\omega }_{0}t,\tqs \rho =\frac{\omega }{{\omega }_{0}},\tqs {\rho }_{1}=\frac{{\omega }_{1}}{{\omega }_{0}},&&\displaystyle \end{eqnarray*}$$

$$\begin{eqnarray*} \displaystyle {\rho }_{2}=\frac{{\omega }_{2}}{{\omega }_{0}},\tqs u=\frac{z}{Y},\tqs v=\frac{y}{Y}=\sin (r \tau ),&&\displaystyle \end{eqnarray*}$$

$$\begin{eqnarray*} \displaystyle {\delta }_{1}=\frac{{d}_{1}}{Y},\tqs {\delta }_{2}=\frac{{d}_{2}}{Y}&&\displaystyle \end{eqnarray*}$$

to obtain the following dimensionless equation of the mass motion:

$$\begin{eqnarray} &&\ddot{u}+2{\xi }_{0}\dot {u}+u={\rho }^{2}\sin (\rho \tau )+f(u,\dot {u}) \end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray} \displaystyle \fl f(u,\dot {u})=\left\{\begin{array}{@{}ll@{}} \displaystyle -2{\rho }_{1}{\xi }_{1}\dot {u}-{\rho }_{1}^{2}u+{\rho }_{1}^{2}{\delta }_{1}&\displaystyle \tqs (u\geq {\delta }_{1})\\ \displaystyle 0&\displaystyle \tqs (-{\delta }_{2}\lt u\lt {\delta }_{1})\\ \displaystyle -2{\rho }_{2}{\xi }_{2}\dot {u}-{\rho }_{2}^{2}u+{\rho }_{2}^{2}{\delta }_{2}&\displaystyle \tqs (u\leq -{\delta }_{2}). \end{array}\right.&&\displaystyle \\ \displaystyle &&\displaystyle \end{eqnarray} \tag{ 12 }$$

The frequency response function, which describes the dimensionless amplitude a with respect to frequency ρ, is obtained as

$$\begin{eqnarray} &&{\pi }^{2}{\rho }^{4}={X}_{1}^{2}+{X}_{2}^{2} \end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray} \displaystyle \fl {X}_{1}=-2{\xi }_{0}a \rho \pi -{\rho }_{1}{\xi }_{1}a \rho (\pi -2{\varphi }_{1}-\sin 2{\varphi }_{1})&&\displaystyle \\ \displaystyle -~{\rho }_{2}{\xi }_{2}a \rho (\pi -2{\varphi }_{2}-\sin 2{\varphi }_{2})&&\displaystyle \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \displaystyle \fl {X}_{2}=\pi a(1-{\rho }^{2})-[\frac{1}{2}{\rho }_{1}^{2}a(2{\varphi }_{1}-\sin 2{\varphi }_{1}-\pi )&&\displaystyle \\ \displaystyle +~\frac{1}{2}{\rho }_{2}^{2}a(2{\varphi }_{2}-\sin 2{\varphi }_{2}-\pi )+2{\rho }_{1}^{2}{\delta }_{1}\cos {\varphi }_{1}&&\displaystyle \\ \displaystyle +~2{\rho }_{2}^{2}{\delta }_{2}\cos {\varphi }_{2}]&&\displaystyle \end{eqnarray} \tag{ 15 }$$

φ₁ = sin⁻¹(δ₁/a) and φ₂ = sin⁻¹(δ₂/a) are the phase angles when the proof mass engages the top- and bottom-stoppers, respectively. Detailed derivations are shown in the appendix.

3.3.2. Stopper on one side.

In the situation where only a stopper on one side is involved in the impact-based PEH system, the dimensionless differential equation of motion can be rewritten as

$$\begin{eqnarray} &&\ddot{u}+2{\xi }_{0}\dot {u}+u={\rho }^{2}\sin (\rho \tau )+{f}_{i}(u,\dot {u}) \end{eqnarray} \tag{ 16 }$$

where i = 1 or 2 represents the situation where the mass motion engages either the top-stopper or the bottom-stopper, respectively.

$$\begin{eqnarray} \displaystyle \fl {f}_{1}(u,\dot {u})=\left\{\begin{array}{@{}ll@{}} \displaystyle -2{\rho }_{1}{\xi }_{1}\dot {u}-{\rho }_{1}^{2}u+{\rho }_{1}^{2}{\delta }_{1}&\displaystyle \tqs (u\geq {\delta }_{1})\\ \displaystyle 0&\displaystyle \tqs (u\lt {\delta }_{1}) \end{array}\right.&&\displaystyle \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} \displaystyle \fl {f}_{2}(u,\dot {u})=\left\{\begin{array}{@{}ll@{}} \displaystyle 0&\displaystyle \tqs (u\gt -{\delta }_{2})\\ \displaystyle -2{\rho }_{2}{\xi }_{2}\dot {u}-{\rho }_{2}^{2}u+{\rho }_{2}^{2}{\delta }_{2}&\displaystyle \tqs (u\leq -{\delta }_{2}). \end{array}\right.&&\displaystyle \end{eqnarray} \tag{ 18 }$$

An implicit equation for the amplitude a as a function of the excitation frequency ρ is given by

$$\begin{eqnarray} &&{\pi }^{2}{\rho }^{4}={Z}_{1}^{2}+{Z}_{2}^{2} \end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray} \displaystyle \fl {Z}_{1}=-2{\xi }_{0}a \rho \pi -{\rho }_{i}{\xi }_{i}a \rho (\pi -2{\varphi }_{i}-\sin 2{\varphi }_{i})&&\displaystyle \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \displaystyle \fl {Z}_{2}=\pi a(1-{\rho }^{2})-[\frac{1}{2}{\rho }_{i}^{2}a(2{\varphi }_{i}-\sin 2{\varphi }_{i}-\pi )&&\displaystyle \\ \displaystyle +~2{\rho }_{i}^{2}{\delta }_{i}\cos {\varphi }_{i}]&&\displaystyle \end{eqnarray} \tag{ 21 }$$

φ_i = sin⁻¹(δ_i/a) is the phase angle when the proof mass engages the top- or bottom-stopper.

3.4.1. Stopper on one side.

In the case where the impact-based PEH system employs a stopper on one side, the frequency response can be solved analytically using equation (19). The metal package is assumed to be the only stopper with a bottom-stopper distance of 1 mm in the simulation. The base acceleration is set as 0.4g; the damping ratios are assumed to be ξ₀ = 0.025 and ξ₂ = 0.1; the frequency characteristics are supposed to be f₀ = 35.8 and f₂ = 100. According to the simulated results, the frequency response of the mass motion is divided into two stages as shown in figure 3. Initially, the mass motion follows the frequency response of a linear spring–mass–damper model and increases monotonically from A to B as the excitation frequency increases (stage I). At point B, the relative motion reaches a displacement of 1 mm and the proof mass starts to engage with the bottom-stopper, hence the mass motion behavior transforms to a piecewise linear model with a stopper on one side where the frequency response follows the trace from B to C (stage II). The overall stiffness and damping in stage II are much higher than those in stage I, thus the operating bandwidth is significantly extended beyond the original frequency bandwidth. When the excitation frequency sweeps to point C, the mass motion amplitude drops immediately to point D, and reverts to the original trace of the linear model (without stopper) in stage I. Subsequently, the mass motion amplitude decreases monotonically from D to E along with up-sweeping frequencies.

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In this model, certain parameters such as the base acceleration a, secondary suspension damping ξ₂, secondary suspension frequency characteristics ω₂ and bottom-stopper distance d₂ show strong influences on the frequency response. As shown in figure 4, each of these four parameters has been studied separately by keeping the other three parameters fixed. The other parameters in the simulation, such as ξ₀, f₀, remain the same as in the case of figure 3. Figure 4(a) shows that, for fixed values of ξ₂, ω₂, and d₂, the base acceleration has a strong influence on the frequency operating bandwidth. For instance, when the base acceleration increases from 0.4g to 0.6g, the operating bandwidth is widened from 7 to 14 Hz. Likewise, figure 4(b) shows the frequency response with different frequency characteristics of the bottom-stopper, i.e., 70, 100 and 130 Hz. Since the frequency characteristic is related to the spring stiffness according to ${\omega }_{2}^{2}=\frac{{k}_{2}}{m}$, it is seen that a higher stiffness of the bottom-stopper results in a wider frequency bandwidth. However, as the stiffness increases, the rate of amplitude increas decreases. In figure 4(c), as the damping ratio of the bottom-stopper increases from 0.05 to 0.1 to 0.2, the frequency bandwidth decreases from 11 to 7 to 4 Hz. Hence, a lower damping ratio is necessary to realize a wider frequency bandwidth. The frequency response with various bottom-stopper distances is shown in figure 4(d). As can be seen, a lower stopper distance results in a wider frequency bandwidth at the expense of a reduction in the relative mass motion. From the above observation, it is seen that the FWB behavior is strengthened by a decrease in the damping and an increase in the stiffness of the stopper. In addition, a high base acceleration is also preferred to realize a better performance (wider operating bandwidth and higher power output). There is a trade-off for the stopper distance, since it affects the frequency bandwidth and mass motion amplitude with opposite trend.

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3.4.2. Stoppers on two sides.

For the impact-based PEH system with stoppers on two sides, the frequency response can be solved analytically by using equations (13) and (19). Similarly to the simulation for a stopper on one side, initially the base acceleration is set as 0.4g; the damping ratios are assumed to be ξ₀ = 0.025, ξ₁ = 0.03, ξ₂ = 0.1; the frequency characteristics are supposed to be f₀ = 35.8, f₁ = 35.8, f₂ = 100; the top- and bottom-stopper distances are set to be 0.5 mm and 1 mm, respectively. As shown in figure 5, the relative mass motion of PEH-B is divided into three stages. In stage I, the mass motion follows the frequency response of a linear model and increases monotonically from A to B with up-sweeping frequency. At point B (d₁ = 0.5 mm), the mass starts to engage with the top-stopper. Hence, the mass motion transforms into a piecewise linear model with a stopper on one side in stage II. The mass motion amplitude increases gradually from point B until it impacts the bottom-stopper at point C with d₂ = 1 mm. At this stage, the mass motion transforms into a piecewise linear model with stoppers on two sides in stage III. Since the overall stiffness and damping factor in stage III are higher than those in stage II, the mass motion amplitude increases slightly from C to D. At point D, the mass motion amplitude drops immediately to point E and reverts to the original trace of the linear model in stage I and subsequently the mass motion amplitude decreases monotonically to point F along with up-sweeping frequencies.

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Figure 6 shows the frequency response of the system with stoppers on two sides for various parameters. Similarly to the simulation for a stopper on one side in Figure 4, one of the four parameters (a, ξ₁, f₁, d₁) is varied while maintaining the other three constant. The stiffness and damping are varied only for the top-stopper. The other parameters in the simulation, such as ξ₀, ξ₂, f₀, f₂, d₂, remain the same as in figure 5. Figure 6(a) shows the frequency responses under accelerations of 0.4g and 0.6g. A higher base acceleration results in a wider operating bandwidth which is manifested mainly in stage III. In figure 6(b), as the stiffness of the top-stopper (in terms of the frequency characteristic) increases from 35.8 to 50 to 70 Hz, the operating bandwidth in stage II is broadened from 4 Hz (34–38 Hz) to 7 Hz (34–41 Hz) to 13 Hz (34–47 Hz), respectively, while the starting frequencies of the operating bandwidth in stage III are shifted accordingly from the ending frequencies in stage II, i.e., 38, 41 and 47 Hz. Figure 6(c) shows that the lower the damping of the top-stopper is, the wider the operating bandwidth as reflected in stage III is. Figure 6(d) shows the frequency responses of different top-stopper distances. As can be seen, a smaller stopper distance results in a wider operating bandwidth in stage II and a shift of the stage III operating bandwidth to a higher frequency range. A higher stiffness and a larger stopper distance will cause a larger bandwidth shift in stage III. In addition, a higher base acceleration and a lower damping will result in a larger operating bandwidth and mass motion amplitude.

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To study the FWB and FUC characteristics of the PEH system experimentally, a fine-adjustment (FA) mechanism was fabricated to arrange PEH-B and PEH-T such that the top-stopper distance d₁ could be precisely adjusted. As shown in figure 7(a), the FA mechanism consists of top and bottom L-shaped aluminum plates mounted on a microstage, such that the relative position of these two plates can be finely adjusted in the x- and z-directions. PEH-B and PEH-T are fixed in the bottom and top breadboards respectively and further attached to the bottom and top L-shaped plates. Under such an assembly, the lateral overlap distance and vertical stopper distance of PEH-B and PEH-T can be varied accurately. The entire FA mechanism is mounted on a vibration shaker as shown in figure 7(b). The vibration frequency and amplitude of the shaker are controlled by a dynamic signal analyzer through an amplifier. The output voltages of PEH-B and PEH-T are recorded separately by the dynamic signal analyzer and an oscilloscope. The capacitances of the PZT layers of PEH-B and PEH-T are 4.3 nF and 0.72 nF, respectively. The internal impedances of the dynamic signal analyzer and oscilloscope are both 1 MΩ. Therefore, the measured output voltages of PEH-B and PEH-T are considered as the load voltages instead of the open circuit voltages. In the following sections, the PEH systems with stoppers on one side and two sides are studied in detail.

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In configuration I, only the bottom-stopper, i.e., metal package of PEH-B, is employed, with a bottom-stopper distance of 1.1 mm, as shown in figure 8(a). The output rms voltages against excitation frequencies from 20 to 60 Hz with different accelerations are shown in figure 8(b). At a low acceleration of 0.1g, PEH-B oscillates freely and does not engage with the bottom-stopper, and a maximum output rms voltage of 65 mV is generated at a low resonant frequency of 36 Hz. As the base acceleration increases, the vibration amplitude of PEH-B increases accordingly. When the base acceleration increases to 0.2g, the proof mass of PEH-B impacts the bottom-stopper. The frequency response exhibits a broad operating bandwidth in the neighborhood of its original resonant frequency. The operating bandwidth continues to widen with increasing acceleration and widens to a frequency range of 10 Hz (32–42 Hz) at an acceleration of 0.6g. The output rms voltage steadily increases from 83 to 106 mV within this frequency range.

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In configuration II as shown in figure 9, PEH-T is employed as a top-stopper and mounted above the proof mass of PEH-B with different top-stopper distances of 0.75 and 0.5 mm. The overlap distance between the proof mass tip of PEH-B and the beam tip of PEH-T is 0.1 mm. Figure 10 shows the output rms voltages of PEH-B against frequencies under accelerations of 0.2g, 0.4g and 0.6g. In figure 10(a), the top- and bottom-stopper distances to the proof mass are 0.75 mm and 1.1 mm, respectively. At a base acceleration of 0.2g, as the excitation frequency sweeps up, the output rms voltage increases monotonically until the proof mass impacts the top-stopper where the vibration amplitude of the proof mass reaches 0.75 mm. Thereafter, the frequency response transforms into the stage II behavior for the model with a stopper on one side described previously, and the output rms voltage increases steadily from 60 mV at 34 Hz to 83 mV at 40 Hz. At the base acceleration of 0.4g, when the vibration amplitude of PEH-B reaches 0.75 mm, the proof mass starts to engage the top-stopper (stage II). Thereafter, the vibration amplitude increases continuously until it reaches 1.1 mm at 40 Hz, when the proof mass engages the bottom-stopper as well (stage III). In stage II, the output rms voltage increases from 60 to 83 mV as the frequency sweeps from 32 to 40 Hz. In stage III, the output rms voltage increases slightly from 83 to 92 mV as the frequency sweeps from 40 to 46 Hz. Since the stiffness of the bottom-stopper is much higher than that of the top-stopper, the voltage increment in stage III is significantly lower than that in stage II. For an acceleration of 0.6g, the operating bandwidths in stages II and III are broadened to 31–40 Hz and 40–49 Hz, respectively, while the corresponding output rms voltages are increased to 60–83 mV and 83–97 mV. When the top-stopper distance is reduced to 0.5 mm as shown in figure 10(b), the maximum rms voltages in stage II are not significantly reduced except at the initial operating phase of stage II. The voltage increment in stage II becomes steeper and the frequency range is widened significantly from both sides. At around the excitation frequency of 44 Hz, PEH-B begins to engage with the bottom-stopper (stage III) and the output rms voltage curve becomes relatively flat. As a result, decreasing the top-stopper distance would increase the operating frequency range in stage II and shift the onset of stage III to a higher frequency range.

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Figure 11(a) shows the real-time output voltages of PEH-B and PEH-T at an excitation frequency of 38 Hz and with a base acceleration of 0.6g and d₁ = 0.75 mm and d₂ = 1.1 mm. As can be seen, PEH-B oscillates according to the base excitation at 38 Hz. During each vibration cycle, PEH-B impacts the supporting beam of PEH-T, resulting in a self-oscillation of PEH-T at its high resonant frequency of up to 618 Hz. The average peak-to-peak voltages of PEH-B and PEH-T are 226 mV and 76 mV, respectively. From figure 11(a), the beam tip displacement of PEH-T and the mass tip displacement of PEH-B are calculated by using equations (6) and (7) and are shown in figure 11(b). Critical positions in each oscillation cycle of PEH-B as shown in figure 9 are also indicated in figure 11(b). The dashed lines indicate the corresponding positions of PEH-T in figure 9. We consider a particular instant of an oscillation cycle when the proof mass is at its lowest point at position ①. It then starts to move to position ② where the proof mass starts impacting the supporting beam of PEH-T. Then the proof mass forces the supporting beam to bend upward until position ③, where the proof mass reaches its maximum amplitude and the supporting beam reaches its maximum upward deflection. Subsequently, the proof mass together with the supporting beam move downward to position ④, where the supporting beam is released at its maximum downward deflection. Thereafter, the proof mass continues its downward movement to position ① during which the supporting beam of PEH-T self-oscillates at its high resonant frequency. The cycle is repeated as the proof mass moves toward position ② again. Such impact-based FUC behavior is realized in each oscillation cycle as long as the vibration amplitude of the proof mass is larger than the top-stopper distance (0.75 mm). If the vibration amplitude is increased further to 1.1 mm, the proof mass will also impact its metal package base. In figure 11(b), the average peak amplitudes of PEH-B and PEH-T are derived as 0.78 mm and 0.03 mm, respectively, which are in good agreement with the desired values.

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By using equation (8), the optimal power outputs against frequency for configurations I and II are calculated with a base acceleration of 0.6g as shown in figure 12(a). For configuration II (d₁ = 0.75 mm and d₂ = 1.1 mm), the output power varies from 34 to 100 nW within a wideband range of 30–48 Hz. For configuration I, the output power is relatively higher from 72 to 114 nW within a narrower bandwidth ranging from 32 to 42 Hz. Figure 12(b) shows the optimal power at an excitation frequency of 38 Hz derived from the real-time voltages of configuration II (figure 11(a)) at 0.6g. The average peak powers of PEH-B and PEH-T are around 140 and 20 nW respectively. The results indicate that PEH-T has a much lower output than PEH-B. This is mainly due to the much smaller displacement of PEH-T and the energy loss during the impact process. On the other hand, we define the power efficiency as the mean value of output power divided by the tip displacement of the cantilever. Therefore, from the average peak amplitudes of PEH-B and PEH-T, it is observed that the power efficiency of PEH-T at 667 nW mm⁻¹ is higher than that of PEH-B at 186 nW mm⁻¹. Hence, PEH-T would generate a considerably higher power output than PEH-B under the same amplitude, i.e., tip displacement of the cantilever. A power electronic converter [38] can be used to condition the outputs of PEH-B and PEH-T and to provide the required DC output to electronic loads such as wireless sensor nodes or microsystems.

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