Analysis of a micro piezoelectric vibration energy harvester by nonlocal elasticity theory

Abstract

A theoretical model of a micro piezoelectric energy harvester is proposed based on the nonlocal elasticity theory, which is operated in the flexural mode for scavenging ambient vibration energy. A nonlocal scale is defined as the product of internal characteristic length and a constant related to the material. The dependences of performance of the harvester upon the nonlocal scale and the scale ratio of the nonlocal scale to the external characteristic parameter are investigated in detail. Numerical results show that output power of the harvester decreases, and resonance frequency reduces gradually at first then increases rapidly when nonlocal scale increases. The results of nonlocal elasticity theory are compared with that of classic beam theory. All the results are helpful for material and structure design of the micro piezoelectric energy harvester.

Introduction

Harvesting energy from mechanical vibrations by piezoelectric structure has received a considerable amount of interest as a means of wireless power supply for the mechanical systems to be self-sensing and self-support. Moreover, small vibration on the order of nanometers poses challenges for the improving precision of ultra-precision machine tools and metrology equipment.^1–3 Therefore, harvesting redundant energy from ambient environment not only can scavenge mechanical energy into electricity for wireless devices but also is conducive to the stability of the structure and reduces the small vibration from the machine. The flexibility associated with piezoelectric materials makes it very attractive for power harvesting, such as high energy densities, no electromagnetic inference, and integration potential. ⁴ Hence, the piezoelectric device is a premium choice for the design of self-powered wireless sensors embedded in mechanical systems, which is called piezoelectric vibration energy harvester. ⁵

With the further development of microelectronic devices and modern micro-sensing technology toward micro and nano-scale dimensions, the need for efficient piezoelectric energy harvesting micro devices is becoming more and more critical. ⁶ Size effect of the small length scales such as lattice spacing between individual atoms and grain size is needed to be considered when the structural scale reaches to micron or nano level. In addition, the mechanical properties of materials will result in an exception because of the heterogeneity of the internal microstructure of materials and show the obvious size effect. ⁷ Thus, size effects in the electromechanical conversion of piezoelectric power harvesters should be taken into account. ⁸

Since controlled experiments in micro and nano-scale are both difficult and expensive, the development of appropriate mathematical models for nanostructures is an important issue concerning approximate analysis of nanostructures. ⁹ Although solution through molecular dynamics (MD) simulation is a possibility for such problems, its large computational cost prohibits its use for a general analysis. Moreover, the conventional continuum models cannot handle this issue since without taking scale effects into account. Hence, the best alternatives are those methods which can both provide the simplicity of continuum models and incorporate the scale effect into such chosen continuum models. Strain gradient theory can calculate the scale effect by considering the high-order effects. ⁸ But the high-order material constants are difficult to measure and cannot be expressed directly in relation to the scale. Eringen’s nonlocal elasticity theory is one of the mathematical models to account for scale effect in the nanostructures by introducing an intrinsic length scale.^10–12 The internal length scale is introduced into the constitutive equations as a material parameter by the nonlocal elasticity theory. In recent years, the studies of nanostructures using the nonlocal elasticity theory have been an area of active research, which also includes the piezoelectric structures.

In this article, the theoretical model of the micro piezoelectric energy harvester is first proposed based on the nonlocal elasticity theory. The piezoelectric vibration energy harvesters have the most efficiency of the energy collection when resonance occurs under the ambient vibration. ¹³ The size effects on performance of the micro piezoelectric energy harvester and the resonance frequency are focused on investigation.

Introduction to theory of nonlocal elasticity

In Eringen’s nonlocal elasticity theory,^10–12 the stress at a point X in a body depends not only on the strain at that point but also on those at all points of the body. Recently, Zhou et al.^14–16 extended Eringen’s nonlocal elasticity theory to the piezoelectric materials. Based on the nonlocal elasticity theory, the constitutive equations for a homogeneous and elastic piezoelectric solid without body force can be written as

$$T_{\mathrm{ij}}(\mathit{X})=\underset{V}{\int }\alpha (|\mathit{X}\prime -\mathit{X}|,\tau )[C_{\mathrm{ijkl}}{S}_{\mathrm{kl}}(\mathit{X}\prime )-e_{\mathrm{kij}}{E}_k(\mathit{X}\prime )]d\mathit{X}\prime$$

(1)

$$D_i(\mathit{X})=\underset{V}{\int }\alpha (|\mathit{X}\prime -\mathit{X}|,\tau )[e_{\mathrm{ikl}}{S}_{\mathrm{kl}}(\mathit{X}\prime )+{\epsilon }_{\mathrm{ik}}{E}_k(\mathit{X}\prime )]d\mathit{X}\prime$$

(2)

where physical quantity with X as a variable parameter is the physical macroscopic (classical) physical tensor at X which is related to the linear physical tensor at any point X′ in the body. T_ij, S_kl, D_i, and E_k are the components of the stress, strain, electric displacement, and electric field, respectively; C_ijkl, e_kij, and ε_ik are the elastic constants, piezoelectric constants, and dielectric constants, respectively; $\alpha (|\mathit{X}\prime -\mathit{X}|,\tau )$ is the attenuation function which depends on the distance |X′- X | and a scale ratio τ which denotes the ratio of the nonlocal scale e₀a to the external characteristic length l (e.g. crack length and wavelength), that is, τ = e₀a/l. a represents internal characteristic length (e.g. lattice parameter and granular distance), and e₀ is a constant appropriate to each material. ¹⁷ It is also possible to represent the integral constitutive relations in an equivalent differential form as ¹⁸

$$T_{\mathrm{ij}}(\mathit{X})-{\left(e_{0}a\right)}^2{\nabla }^2{T}_{\mathrm{ij}}(\mathit{X})=C_{\mathrm{ijkl}}{S}_{\mathrm{kl}}(\mathit{X})-e_{\mathrm{kij}}{E}_k(\mathit{X})$$

(3)

$$D_i(\mathit{X})-{\left(e_{0}a\right)}^2{\nabla }^2{D}_i(\mathit{X})=e_{\mathrm{ikl}}{S}_{\mathrm{kl}}(\mathit{X})+{\epsilon }_{\mathrm{ik}}{E}_k(\mathit{X})$$

(4)

where ${\nabla }^2$ is the Laplacian operator and the nonlocal scale length e₀a considers the influences of small scale on the response of nanostructures.

A model of the micro piezoelectric bimorph harvester

A model of the piezoelectric bimorph (PB) harvester is shown in Figure 1: one end of the micro beam is cantilevered by a vibrator, which vibrates along the vertical direction harmonically with a known amplitude A at a given frequency ω, and the free end is attached by a mass m₀. Two piezoelectric layers with same thickness h are poled along the thickness direction. L and b denote length and width of the PB, respectively. The bimorph is connected with a load circuit, whose impedance can be simplified by Z_L.

Figure 1.

A micro cantilevered piezoelectric bimorph power harvester.

Based on the linear piezoelectric theory, ¹⁹ the electric field in the ceramic layers corresponding to the electrode configurations, shown in Figure 1, is of the following components

$$E_1=0,E_2=0,E_3=-{\phi }_{,3}$$

(5)

where φ is the electric potential of each piezoelectric layer, and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with index is used. We now consider the flexural motion of the bimorph in the x₃ direction, assuming that the bimorph is slim, that is, the length L is much larger than the thickness 2h and the width b. According to the Bernoulli–Euler beam assumption, ²⁰ the axial strain S₁ of the flexural motion of such a thin bimorph can be expressed as

$$S_1=-x_3{u}_{3,11}$$

(6)

where u₃ is the displacement of the PB in the x₃ direction.

Despite the existence of shearing force, the shear strain is ignored in Euler beam. The constitutive relation for the piezoelectric layers can be written as ¹⁹

$$\begin{array}{c}{S}_1=s_{11}{T}_1+d_{31}{E}_3\hfill \\ D_3=d_{31}{T}_1+{\epsilon }_{33}{E}_3\hfill \end{array}$$

(7)

where s₁₁, ε₃₃, and d₃₁ denote the axial elastic compliance measured with fixed electric field, the transverse dielectric constant measured with fixed stress, and the transverse-axial piezoelectric coefficient. From equation (7), we solve for the axial stress T₁ and the transverse electric displacement D₃ and obtain

$$\begin{array}{c}{T}_1=s_{11}^{-1}{S}_1-s_{11}^{-1}{d}_{31}{E}_3\hfill \\ D_3=s_{11}^{-1}{d}_{31}{S}_1+{\overline{\epsilon }}_{33}{E}_3\hfill \end{array}$$

(8)

where ${\overline{\epsilon }}_{33}={\epsilon }_{33}(1-k_{31}^2)$ , and $k_{31}^2=d_{31}^2/({\epsilon }_{33}{s}_{11})$ stands for the electromechanical coupling coefficient.

Following the above assumptions, the piezoelectric constitutive of nonlocal elasticity theory can be expressed as

$$\begin{array}{c}{T}_1-{\left(e_{0}a\right)}^2{T}_{1,11}=s_{11}^{-1}(-x_3{u}_{3,11})-s_{11}^{-1}{d}_{31}{E}_3\hfill \\ D_3-{\left(e_{0}a\right)}^2{D}_{3,11}=s_{11}^{-1}{d}_{31}(-x_3{u}_{3,11})+{\overline{\epsilon }}_{33}{E}_3\hfill \end{array}$$

(9)

Performance of the micro PB harvester

In this section, we apply precise electric field method ¹⁹ to get the electric field first, then obtain general solution of the displacement by solving the equation of motion, and finally combine the boundary conditions with circuit equation to solve the output power density of the harvester.

Electric field

From the charge equation of electrostatics D_3,3 = 0, differential equation of the electric displacement in equation (9) can be written as

$$-s_{11}^{-1}{d}_{31}{u}_{3,11}+{\overline{\epsilon }}_{33}{E}_{3,3}=0$$

(10)

The electric field can be obtained by integrating on equation (10) with respect to x₃. Then from equation (5), the electrical potential can be written as

$$\phi =-\int E_{3}d{x}_3=-\frac{1}{2}\frac{d_{31}}{s_{11}{\overline{\epsilon }}_{33}}{u}_{3,11}{x}_3^2-e_1{x}_3+e_2$$

(11)

where e₁ and e₂ are integration constant independent of x₃. Let $\phi =0$ for the grounded electrodes at $x_3=0$ ; $\phi =V$ for the electrodes at $x_3=\pm h$ , and note that $V$ is the output voltage. Thus, the electric field can be solved as

$$E_3=\{\begin{array}{c}\frac{d_{31}}{s_{11}{\overline{\epsilon }}_{33}}{u}_{3,11}(x_3-\frac{h}{2})-\frac{V}{h},0\le x_3\le h\hfill \\ \multicolumn{1}{c}{\frac{d_{31}}{s_{11}{\overline{\epsilon }}_{33}}{u}_{3,11}(x_3+\frac{h}{2})+\frac{V}{h},-h\le x_3\le 0}\end{array}$$

(12)

General solution of the displacement

The equation of motion for the slim beam can be expressed in terms of the bending moment as

$$M_{,11}=m{\stackrel{··}{u}}_3$$

(13)

with M as the bending moment of the cross section, and m = 2hbρ represents the mass per unit length of the PB, where ρ is the density of the piezoelectric layer. By analyzing the cross section of the bimorph, the bending moment and shear force can be calculated as

$$\begin{array}{c}M=\underset{A}{\int }{x}_3{T}_1\mathrm{dA}\hfill \\ N=M_{,1}\hfill \end{array}$$

(14)

From equations (9), (12), (13), and (14), we have

$$M=-D{u}_{3,11}+s_{11}^{-1}{d}_{31}\mathrm{bhV}+{\left(e_{0}a\right)}^{2}m{\stackrel{··}{u}}_3$$

(15)

where generalized bending stiffness

$$D=(\frac{2}{3}+\frac{1}{6}{\overline{k}}_{31}^2)s_{11}^{-1}b{h}^3,{\overline{k}}_{31}^2=\frac{d_{31}^2}{{\overline{\epsilon }}_{33}{s}_{11}}$$

Substituting equation (15) into equation (13), the equation of motion can be expressed by u₃ as follows

$$m{\stackrel{··}{u}}_3=-D{u}_{3,1111}+{\left(e_{0}a\right)}^{2}m{\stackrel{··}{u}}_{3,11}$$

(16)

For harmonic motions, we use the complex notation

$$\{u_3\left(x\right),V,Q_e,I\}=\mathrm{Re}\left\{[U\left(x\right),\overline{V},{\overline{Q}}_e,\overline{I}]\exp \left(i\omega t\right)\right\}$$

(17)

where i is an imaginary unit. The equation of motion (16) can be simplified as

$$U_{,1111}+{\left(e_{0}a\right)}^2\alpha U_{,11}-\alpha U=0$$

(18)

where $\alpha =m{\omega }^2/D$ . The general solution to equation (18) can be written as

$$\begin{array}{cc}\multicolumn{1}{c}{U=}& B_1\sinh \left({\beta }_1{x}_1\right)+B_2\cosh \left({\beta }_1{x}_1\right)\hfill \\ \multicolumn{1}{c}{}& +B_3\sin \left({\beta }_2{x}_1\right)+B_4\cos \left({\beta }_2{x}_1\right)\hfill \end{array}$$

(19)

where B_j (j = 1, 2, 3, 4) are undetermined constants, and

$$\begin{array}{c}{\beta }_1=\sqrt{\frac{\sqrt{{\left(e_{0}a\right)}^4{\alpha }^2+4\alpha }-{\left(e_{0}a\right)}^2\alpha }{2}},\hfill \\ {\beta }_2=\sqrt{\frac{\sqrt{{\left(e_{0}a\right)}^4{\alpha }^2+4\alpha }+{\left(e_{0}a\right)}^2\alpha }{2}}\hfill \end{array}$$

(20)

At the cantilevered end of the beam, the boundary conditions are

$$u_3(0,t)=A\exp \left(i\omega t\right)$$

(21)

$$u_{3,1}(0,t)=0$$

(22)

At the right end, we have no bending moment but a shear force balancing the inertia force of the attached mass, that is

$$M(L,t)=0$$

(23)

$$-N(L,t)=m_0{\stackrel{··}{u}}_3(L,t)$$

(24)

The undetermined constants B_j (j = 1, 2, 3, 4) can be determined from the boundary conditions of the cantilevered beam, under the premise of finding the expression of voltage V which can be got from electrical condition.

Output power density

At first, we solve the expression of electrical displacement from equation (9) to obtain the expression of voltage V. For the upper piezoelectric layer, the electrical displacement, which is independent of x₃, can be obtained from equations (9) and (12) as

$$\begin{array}{c}{D}_3-{\left(e_{0}a\right)}^2{D}_{3,11}\hfill \\ =-s_{11}^{-1}{d}_{31}\frac{h}{2}[B_1{\beta }_1^2\sinh \left({\beta }_1{x}_1\right)+B_2{\beta }_1^2\cosh \left({\beta }_1{x}_1\right)]\hfill \\ +s_{11}^{-1}{d}_{31}\frac{h}{2}[B_3{\beta }_2^2\sin \left({\beta }_2{x}_1\right)+B_4{\beta }_2^2\cos \left({\beta }_2{x}_1\right)]-{\overline{\epsilon }}_{33}\frac{\overline{V}}{h}\hfill \end{array}$$

(25)

where the expression of the general solution of displacement (19) has been used. The general solution of equation (25) consists of the general solution of homogeneous linear equation and special solution. Hence, the general solution of electric displacement can be written as

$$\begin{array}{c}{D}_3=f_1\sinh (\lambda x_1)+f_2\cosh (\lambda x_1)+m_1{B}_1\sinh \left({\beta }_1{x}_1\right)\hfill \\ +m_1{B}_2\cosh \left({\beta }_1{x}_1\right)+m_2{B}_3\sin \left({\beta }_2{x}_1\right)\hfill \\ +m_2{B}_4\cos \left({\beta }_2{x}_1\right)-{\overline{\epsilon }}_{33}\frac{\overline{V}}{h}\hfill \end{array}$$

(26)

where $f_1$ and $f_2$ are undetermined coefficients, and

$$\lambda =\frac{1}{e_{0}a},m_1=\frac{d_{31}{\beta }_1^{2}h{\lambda }^2}{2s_{11}({\beta }_1^2-{\lambda }^2)},m_2=\frac{d_{31}{\beta }_2^{2}h{\lambda }^2}{2s_{11}({\beta }_2^2+{\lambda }^2)}$$

From electric boundary conditions that D₃ = 0 for the electric displacement at x₁ = 0 and x₁ = L, we have

$$D_3\left(0\right)=f_2+m_1{B}_2+m_2{B}_4-{\overline{\epsilon }}_{33}\frac{\overline{V}}{h}=0$$

(27)

$$\begin{array}{cc}\multicolumn{1}{c}{D_3\left(L\right)=}& f_1\sinh (\lambda L)+f_2\cosh (\lambda L)\hfill \\ \multicolumn{1}{c}{}& +m_1{B}_1\sinh \left({\beta }_{1}L\right)-{\overline{\epsilon }}_{33}\frac{\overline{V}}{h}\hfill \\ \multicolumn{1}{c}{}& +m_1{B}_2\cosh \left({\beta }_{1}L\right)+m_2{B}_3\sin \left({\beta }_{2}L\right)\hfill \\ \multicolumn{1}{c}{}& +m_2{B}_4\cos \left({\beta }_{2}L\right)=0\hfill \end{array}$$

(28)

From equation (26), the electric charge on the top electrode at x₃ = h is given by

$$\begin{array}{cc}\multicolumn{1}{c}{Q_e}& =-b\underset{0}{\overset{L}{\int }}{D}_3(x_3=h)d{x}_1\hfill \\ \multicolumn{1}{c}{}& =-b\{\frac{f_1}{\lambda }[\cosh (\lambda L)-1]+\frac{f_2}{\lambda }\sinh (\lambda L)+\frac{m_1{B}_1}{{\beta }_1}[\cosh ({\beta }_{1}L)-1]\hfill \\ \multicolumn{1}{c}{}& +\frac{m_1{B}_2}{{\beta }_1}\sinh ({\beta }_{1}L)+\frac{m_2{B}_3}{{\beta }_2}[1-\cos ({\beta }_{2}L)]+\frac{m_2{B}_4}{{\beta }_2}\sin ({\beta }_{2}L)-{\overline{\epsilon }}_{33}\frac{\overline{V}}{h}L\}\hfill \end{array}$$

(29)

The current flowing out of the electrode is

$$I=-{\stackrel{·}{Q}}_e$$

(30)

As indicated in Figure 1, double current flows to the circuit load because two piezoelectric layers are in parallel. Thus, the current and voltage satisfy Ohm’s law

$$2I=\frac{V}{Z_L}$$

(31)

From equations (29) to (31), we have

$$\begin{array}{c}2\overline{I}=2i\omega b\{\frac{f_1}{\lambda }[\cosh (\lambda L)-1]+\frac{f_2}{\lambda }\sinh (\lambda L)+\frac{m_1{B}_1}{{\beta }_1}[\cosh ({\beta }_{1}L)-1]+\frac{m_1{B}_2}{{\beta }_1}\sinh ({\beta }_{1}L)\hfill \\ +\frac{m_2{B}_3}{{\beta }_2}[1-\cos ({\beta }_{2}L)]+\frac{m_2{B}_4}{{\beta }_2}\sin ({\beta }_{2}L)\}-2i\omega C_0\overline{V}=\frac{\overline{V}}{Z_L}\hfill \end{array}$$

(32)

where $C_0={\overline{\epsilon }}_{33}\mathrm{bL}/h$ represents the equivalent static capacitance of one piezoelectric layer. There are seven equations, equations (21)–(24), (27), (28), and (32), for seven undetermined coefficients B_i (i = 1, 2, 3, 4), f₁, f₂, and $\overline{V}$ . Similar to the study by Hu et al. ⁸ and Jiang et al., ²¹ the output electrical power density p with the complex notation is defined to measure the performance of the harvester as

$$p=\frac{1}{4\Omega }(\overline{I}{\overline{V}}^{*}+{\overline{I}}^{*}\overline{V})$$

(33)

where an asterisk represents complex conjugate and volume of the harvester $\Omega =2\mathrm{bhL}$ .

Numerical results

To illustrate the size effect on the performance of scavenge energy from ambient vibration of the harvester, the piezoelectric material PMN-33% PT is chosen for numerical calculation. Its density $\rho =8060\mathrm{kg}/{\mathrm{m}}^3$ and the following elastic, piezoelectric, and dielectric constants ²²

$$\begin{array}{c}{s}_{11}=69\times {10}^{-12}{\mathrm{m}}^2/N,\hfill \\ d_{31}=-1330\times {10}^{-12}\mathrm{C}/\mathrm{N},{\epsilon }_{33}=8200{\epsilon }_0\hfill \end{array}$$

(34)

For the piezoelectric materials, damping is always introduced by replacing the elastic compliance s₁₁ with a complex form $s_{11}(1-i{Q}^{-1})$ . The quality factor Q is chosen to be 100 in the calculation. In order to analyze conveniently, the parameters of the PB b = 1 µm, h = 0.1 µm, $m_0=m\times 5\mu \mathrm{m}$ , and the acceleration amplitude ${\omega }^{2}A=1\mathrm{m}/{\mathrm{s}}^2$ are fixed in all calculations. The normalized impedance of load circuit $Z_L=i{Z}_0$ , where $Z_0=1/i\omega C_0$ is defined as the equivalent impedance of one piezoelectric layer.

We calculate in Figure 2 the relationship between the output power density and the driving frequency for different nonlocal scale e₀a, where L = 10 µm. It can be seen that output power density reaches maximum when resonance occurs. For further illustration, Figure 3 illustrates the maximum of output power density and resonance frequency versus the nonlocal scale e₀a. As can be seen, the maximum output power density of the piezoelectric power harvester decreases dramatically when the nonlocal scale e₀a increases, but its corresponding resonance frequency reduces gradually at first and then increases rapidly. The computational results of output power density and resonance frequency of nonlocal elasticity theory both have the same order as those of strain gradient theory. ⁸

Figure 2.

Power density versus the driving frequency for different nonlocal scale e₀a where L = 10 µm.

Figure 3.

The maximum power density and its corresponding resonance frequency f_r versus nonlocal scale e₀a where L = 10 µm.

We plot the output power density versus length of the PB for different nonlocal scale ratio e₀a/L = 0.05, 0.25, and 0.45 in Figure 4(a), respectively, and for different nonlocal scale e₀a = 1.5, 3.0, and 4.5 µm in Figure 5(a), respectively. It can be seen that output power densities calculated by classical beam theory and nonlocal electricity theory both increases with increasing length. Moreover, the curves of nonlocal electricity theory diverge farther away from that of classical beam theory with increasing e₀a or e₀a/L. It means that the smaller the structure size or the larger the nonlocal scale e₀a, the more obvious the size effect. Furthermore, comparing Figures 4(b) and 5(b), we can note that the curves of output power density ratio p_N/p_C of nonlocal elasticity theory to classic beam theory versus the length are almost horizontal lines when e₀a/L are fixed; but when e₀a are fixed, the curves become steeper. It is because that e₀a/L decreases with increasing length when e₀a is fixed, the output power density of nonlocal elasticity theory is close to that of classic beam theory more rapidly. It can be inferred from Figure 5(b) that these curves of p_N/p_C will be close to one when e₀a/L is close to zero.

Figure 4.

(a) Power density and (b) the ratio p_N/p_C of nonlocal power density to classical power density versus length for different scale ratio e₀a/L.

Figure 5.

(a) Power density and (b) the ratio $p_{\mathrm{N}}/p_{\mathrm{C}}$ of nonlocal power density to classical power density versus length for different nonlocal scale e₀a.

Resonance frequency is another important factor of the vibration energy harvester since at the frequency the harvester can scavenge the most energy from environment. Figures 6(a) and 7(a) demonstrate that resonance frequency decreases with increasing length when e₀a/L or e₀a is fixed. Figures 6(b) and 7(b) illustrate that the resonance frequency ratio f_rN/f_rC of nonlocal elasticity theory to classic beam theory versus the length of the PB when e₀a/L or e₀a is fixed, respectively. We can note from Figure 6(b) that classic beam theory overestimates the resonance frequency of the PB when e₀a/L is equal to 0.05 or 0.25. However, if e₀a/L is fixed 0.45, classic beam theory overestimates the resonance frequency when the length L is less than 20 µm and then underestimates the resonance frequency when the length is larger than 20 µm. Furthermore, Figure 7(b) shows that the ratio f_rN/f_rC is close to 1 with increasing length when e₀a is fixed. However, the classic beam theory may underestimate the resonance frequency when nonlocal scale e₀a and the length are with the same order, such as e₀a = 4.5 µm and L < 6 µm. These conclusions are consistent with that of Figure 3.

Figure 6.

(a) Resonance frequency and (b) the ratio f_rN/f_rC of nonlocal resonance frequency to classical resonance frequency versus length for different scale ratio e₀a/L.

Figure 7.

(a) Resonance frequency and (b) the ratio f_rN/f_rC of nonlocal resonance frequency to classical resonance frequency versus length for different nonlocal scale e₀a.

Conclusion

The performance of a vibration energy harvester with PB operated in the flexural mode is analyzed based on the nonlocal elasticity theory. The dependence of the performance of the harvester upon the nonlocal scale and scale ratio is investigated. When nonlocal scale increases, output power of the harvester decreases and resonance frequency reduces gradually at first and then increases rapidly. Classic beam theory overestimates the output power without considering size effect. When the scale ratio is fixed, the output power ratio of nonlocal elasticity theory to classic beam theory is independent of the length. Furthermore, the output power ratio of nonlocal elasticity theory to classic beam theory increases to 1 when the scale ratio decreases to 0. Classic beam theory may overestimate or underestimate the resonance frequency when the scale ratio is close to 1.