Multiple-Scale Analysis of a Tunable Bi-Stable Piezoelectric Energy Harvester

Qian, Feng; Abaid, Nicole; Zuo, Lei

doi:10.1115/1.4046961

Abstract

This paper presents the theoretical modeling and multiple-scale analysis of a novel piezoelectric energy harvester composed of a metal cantilever beam, piezoelectric films, and an axial preload spring at the moveable end. The harvester experiences mono- and bi-stable regimes as the stiffness of preload spring increases. The governing equations are derived with two high-order coupling terms induced by the axial motion. The literature shows that these high-order coupling terms lead to tedious calculations in the stability analysis of solutions. This work introduces an analytical strategy and the implementation of the multiple-scale method for the harvester in either the mono- or bi-stable status. Numerical simulations are performed to verify the analytical solutions. The influence of the electrical resistance, excitation level, and the spring pre-deformation on the voltage outputs and dynamics are investigated. The spring pre-deformation has a slight influence on the energy harvesting performance of the mono-stable system, but a large effect on that of the bi-stable system.

1 Introduction

Considerable interest in vibration energy harvesting has emerged in the last decade with the rapid development of wireless sensors and low-power electrical devices [1–3]. Piezoelectric materials are extensively studied for mechanical-to-electrical energy conversion due to the high power densities and easy fabrication [4–6]. One critical issue of linear piezoelectric vibration energy harvesters (PEHs) is the limited frequency bandwidth [7,8]. A linear PEH needs to be deliberately designed to match its natural frequency with the excitation frequency to achieve resonance. A slight shift in the excitation frequency or any defects in the operation could lead to the frequency mismatch and thus a significant power reduction [9].

Nonlinearities are exploited to improve the operation frequency bandwidth, because nonlinear PEHs are insensitive to the change in excitation frequency due to the tilted frequency curves [10]. Nonlinear PEHs include mono-stable [1,4,9,11,12], bi-stable [13,14], and tri-stable harvesters [15,16]. Among them, bi-stable harvesters are preferable for their large power output during the larger-amplitude snap-through vibrations. Bi-stable PEHs have been shown to achieve a 300% improvement in the open-circuit root-mean-square voltage [17] and 13.1 times more power output [18]. Bi-stability can be realized by various mechanisms, such as repulsive magnets [19] and composite structures [20]. The nonlinearities usually make the analytical analysis become a daunting task, especially for systems with high-order coupling terms [21–23]. The harmonic balance method [11,12] has shown good accuracy for bi-stable systems. However, research shows the stability analysis of the harmonic balance solutions can be very complicated, even analytically impossible for systems with higher-order coupling terms [24]. On the other hand, the stability of multiple-scale solutions can be easily determined from the Jacobian matrix. Although the method of multiple scales [4,25,26] has been widely used for nonlinear systems, very few articles systematically apply this technique to systems with tunable potential wells and higher-order coupling terms. In particular, specific strategies are needed to handle the negative stiffness of bi-stable systems.

This letter presents the multiple-scale solutions of a broadband PEH with a tunable potential function, which consists of a metal cantilever beam, piezoelectric films, and an axial preload spring. The spring plays the similar role of repulsive magnets in literature to achieve the buckling mechanism. Advantages of this strategy lie in avoiding the magnetic interference with wireless sensor nodes and reducing the total mass [27]. The main contributions of this work include the following: (1) systematically introducing the technical strategies of the method of multiple scales for mono- and bi-stable systems; (2) obtaining the analytical frequency responses, phase angle, and phase portraits of the mono-stable and bi-stable systems; and (3) investigating the effect of the external resistance, excitation level, and the spring pre-deformation on the voltage outputs and dynamics of the systems.

2 Theoretical Model

Figure 1(a) shows the proposed PEH, where one end of the harvester is clamped and the other end connected to the spring is moveable along the guide rails. When the spring is compressed, the beam is exposed to the axial spring force. When the spring has a small pre-deformation, not enough to buckle the beam, the PEH is mono-stable; otherwise, the PEH enters into the bi-stable regime with two stable equilibria and one unstable equilibrium, as shown in Fig. 1(a). The two piezoelectric films on the beam are connected in parallel and the generated electricity is delivered to a resistive load. The PEH can be deemed as a current source and a capacitor [28], resulting in the simplified electrical circuit in Fig. 1(b). Assume the PEH experiences a base excitation of $\ddot{a} = A \cos (Ω t)$ ⁠, where t is time, A and Ω are the excitation level and frequency.

Fig. 1

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(a) The nonlinear PEH and (b) equivalent electrical circuit

Let b and L be the width and length, h_s and h_p the thicknesses, ρ_s and ρ_p the densities of the beam and the piezoelectric films. The equivalent mass of per unit length is m = ρ_sbh_s + 2ρ_pbh_p. The effect of the electrode layers on the stiffness and mass is ignored because they are very thin and light. Let w(x, t), u(x, t), and q(t) be the transverse, axial displacements, and the generated charge over the surfaces of the electrode layers. For simplicity, the transient displacement is approximated by the first mode as w(x, t) = η(t)φ(x), where η(t) is the first modal coordinate in the time domain and φ(x) = [1 − cos(2πx/L)]/2 is the first mode shape function [18]. The inextensible assumption requires (1 + u′)² + w′² = 1, and thus

u (x, t) = - 1 / 2 \int_{0}^{x} {[w^{'} (x, t)]}^{2} d x

[4,11,23], where the prime indicates the first derivative with respect to the coordinate x. The kinetic energy, potential energy, and work done by generalized forces are

T = \frac{m}{2} \int_{0}^{L} [{\dot{η}}^{2} φ^{2} (x) + 2 \dot{η} \dot{a} φ (x) + {\dot{a}}^{2}] d x + \frac{m}{2} (η \dot{η})^{2} \int_{0}^{L} {(\int_{0}^{x} {[φ^{'} (x)]}^{2} d x)}^{2} d x

(1)

W = - c η \dot{η} \int_{0}^{L} φ^{2} (x) d x + q (t) v (t)

(2)

\begin{aligned} U & = η^{2} [\frac{E I}{2} \int_{0}^{L} {[φ^{''} (x)]}^{2} d x - \frac{k_{d} Δ}{2} \int_{0}^{L} {[φ^{'} (x)]}^{2} d x] \\ + η^{4} [\frac{E A}{8} \int_{0}^{L} {[φ^{'} (x)]}^{4} d x - \frac{k_{d}}{8} {(\int_{0}^{L} {[φ^{'} (x)]}^{2} d x)}^{2}] \\ - η b e_{31} v (t) (h_{s} + h_{p}) \int_{0}^{L_{e}} φ^{''} (x) d x - \frac{1}{2} C_{p} v^{2} (t) \end{aligned}

(3)

where E₃ = −v(t)/h_p has been used and c denotes the damping coefficient. E_s and E_p are Young’s modulus of the beam and the piezoelectric films, respectively. k_d and Δ are the stiffness and the pre-deformation of the spring. EA = E_sbh_s + 2E_pbh_p and

E I = E_{s} b h_{s}^{3} / 12 + E_{p} b h_{p} (4 h_{p}^{2} + 6 h_{p} h_{s} + 3 h_{s}^{2}) / 6

are the equivalent axial and bending stiffness.

C_{p} = 2 ϵ_{33}^{s} b L / h_{p}

is the capacitance of the piezoelectric films. e₃₁ and

ϵ_{33}^{s}

are the piezoelectric constant and relative permittivity measured at a constant strain. The governing equations are then extracted from the Lagrangian method as follows:

\ddot{η} + 2 ξ ω_{n} \dot{η} + β (η {\dot{η}}^{2} + η^{2} \ddot{η}) + (k_{1} - k_{s}) η + (k_{2} - k_{3}) η^{3} - θ v = - μ \ddot{a}

\dot{v} + λ v + \bar{θ} \dot{η} = 0

where

ω_{n} = \sqrt{k_{1}}

⁠, and ξ is the mechanical damping ratio. The Gauss current law

\dot{q} (t) = I (t) = - v (t) / R

has been used in the derivation of the electrical equation . All the parameters in the governing equations are evaluated by

\begin{aligned} ξ & = \frac{c}{2 m ω_{n}}, β = \frac{2 \int_{0}^{L} {(\int_{0}^{x} {[φ^{'} (x)]}^{2} d x)}^{2} d x}{m \int_{0}^{L} φ^{2} (x) d x}, k_{1} = \frac{E I \int_{0}^{L} {[φ^{''} (x)]}^{2} d x}{m \int_{0}^{L} φ^{2} (x) d x}, k_{s} = \frac{k_{d} Δ \int_{0}^{L} {[φ^{'} (x)]}^{2} d x}{m \int_{0}^{L} φ^{2} (x) d x}, \\ k_{2} = \frac{3 E A {(\int_{0}^{L} {[φ^{'} (x)]}^{2} d x)}^{2}}{8 L m \int_{0}^{L} φ^{2} (x) d x}, k_{3} = \frac{k_{d} {(\int_{0}^{L} {[φ^{'} (x)]}^{2} d x)}^{2}}{8 m \int_{0}^{L} φ^{2} (x) d x}, θ = \frac{e_{31} b (h_{s} + h_{p}) \int_{0}^{l_{e}} φ^{''} (x) d x}{4 m \int_{0}^{L} φ^{2} (x) d x}, \\ μ = \frac{\int_{0}^{L} φ (x) d x}{\int_{0}^{L} φ^{2} (x) d x}, λ = \frac{1}{C_{p} R}, \bar{θ} = \frac{e_{31} b h_{p} (h_{s} + h_{p})}{4 C_{p}} \int_{0}^{l_{e}} φ^{''} (x) d x \end{aligned}

Defining the dimensionless frequency and time ω = Ω/ω_n and τ = ω_nt, Eqs. and can be rearranged as

η^{''} + 2 ξ η^{'} + β [η {(η^{'})}^{2} + η^{2} η^{''}] + χ η + ϑ η^{3} - α v = - Λ \cos (ω τ)

v^{'} + \bar{λ} v + \bar{θ} η^{'} = 0

where the prime denotes the derivative with respect to τ,

χ = (1 - k_{s} / ω_{n}^{2})

⁠,

ϑ = (k_{2} - k_{3}) / ω_{n}^{2}

⁠,

α = θ / ω_{n}^{2}

⁠,

\bar{λ} = λ / ω_{n}

⁠, and

Λ = μ A / ω_{n}^{2}

⁠. The governing equations have two higher-order coupling terms, the product of the displacement and squared velocity, and the product of the squared displacement and acceleration. Those terms significantly complicate the derivation of analytical solutions. Setting η″ = η′ = 0 and ignoring the effect of electrical term, the equilibria can be obtained as

η_{1}^{*} = 0, η_{2, 3}^{*} = \pm \sqrt{- χ / ϑ} = \pm \sqrt{- (k_{1} - k_{s}) / (k_{2} - k_{3})}

(6)

The parameter $ϑ$ is always positive because the scaled axial stiffness k₂ is much larger than the equivalent spring stiffness k₃. Thus, the existence of the equilibria $η_{2, 3}^{*}$ depends on the effective spring stiffness k_s. The system has one equilibrium η₁ = 0 when Δ is small (k_s < k₁), and the PEH is mono-stable. As Δ increases to a certain level (k_s = k₁), the axial compressive force equals the critical buckling load of the beam. Then, the PEH begins to enter into the buckling regime.

The potential function is $\bar{V} (η) = χ η^{2} / 2 + ϑ η^{4} / 4$ ⁠, implying the parameter χ determines if the PEH is unbuckled or buckled. When χ > 0 (k_s/k₁ < 1), the harvester is unbuckled, otherwise, it is buckled bi-stable. Figure 2 plots the potential for the varying Δ with a constant spring stiffness of k_d = 1000 N/m. The geometric and material properties of the PEH are listed in Table 1. The potential function has a flattened parabolic shape initially and turns to a double-well shape as Δ increases. This indicates the PEH turns from the mono-stable to the bi-stable state.

Fig. 2

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Potential of the system with different spring pre-deformations

Table 1

The geometric and material properties of the PEH

Beam (steel) and spring		Piezoelectric layer
Parameters	Value	Parameters	Value
Length, L	300 mm	Length, L_p	300 mm
Width, b	10 mm	Width, b	10 mm
Thickness, h_s	0.1 mm	Thickness, h_p	0.08 mm
Density, ρ_s	7850 kg m⁻³	Density, ρ_p	4000 kg m⁻³
Modulus, E_s	203 GPa	Modulus, E_p	40 GPa
Damping, ξ	0.042	d₃₁	−10 pC N⁻¹
Stiffness, k_d	2500 N m⁻¹	ɛ₃₃	8.854 × 10⁻¹⁰ F m⁻¹

Beam (steel) and spring		Piezoelectric layer
Parameters	Value	Parameters	Value
Length, L	300 mm	Length, L_p	300 mm
Width, b	10 mm	Width, b	10 mm
Thickness, h_s	0.1 mm	Thickness, h_p	0.08 mm
Density, ρ_s	7850 kg m⁻³	Density, ρ_p	4000 kg m⁻³
Modulus, E_s	203 GPa	Modulus, E_p	40 GPa
Damping, ξ	0.042	d₃₁	−10 pC N⁻¹
Stiffness, k_d	2500 N m⁻¹	ɛ₃₃	8.854 × 10⁻¹⁰ F m⁻¹

3 Mono-Stable Piezoelectric Energy Harvester

To approximate the analytical solutions using the method of multiple scales, the damping term, nonlinear higher-order terms, cubic term, and the excitation are assumed to be small. Therefore, the mechanical equation can be written as

η^{''} + 2 ε ξ η^{'} + ε β [η {(η^{'})}^{2} + η^{2} η^{''}] + χ η + ε ϑ η^{3} - ε α v = - ε Λ \cos (ω τ)

(7)

where ɛ is an small number. Introducing the fast and slow time variables T₀ = τ and T₁ = ɛτ, the solutions are expanded and retained to the first order of ɛ by the asymptotic series

{\begin{array}{l} η (τ) = η_{0} (T_{0}, T_{1}) + ε η_{1} (T_{0}, T_{1}) \\ v (τ) = v_{0} (T_{0}, T_{1}) + ε v_{1} (T_{0}, T_{1}) \end{array}

(8)

Substituting the multiple-scale solutions into Eqs. and , and equating the terms with the same order of ɛ yields

{\begin{array}{l} ε^{0} : D_{0}^{2} η_{0} + χ η_{0} = 0 \\ ε^{1} : D_{0}^{2} η_{1} + χ η_{1} = - Λ \cos (ω t) - 2 D_{0} D_{1} η_{0} - 2 ξ D_{0} η_{0} - \\ β [η_{0} {(D_{0} η_{0})}^{2} + η_{0}^{2} D_{0}^{2} η_{0}] - ϑ η_{0}^{3} + α v_{0} \end{array}

(9)

{\begin{array}{l} ε^{0} : D_{0} v_{0} + \bar{λ} v_{0} = - \bar{θ} D_{0} η_{0} \\ ε^{1} : D_{0} v_{1} + \bar{λ} v_{1} + \bar{θ} D_{0} η_{1} = - D_{1} v_{0} - \bar{θ} D_{1} η_{0} \end{array}

(10)

The solutions to the zeroth-order equations are assumed to be

{\begin{array}{l} η_{0} = A (T_{1}) e^{i \hat{ω} T_{0}} + \bar{A} (T_{1}) e^{- i \hat{ω} T_{0}} \\ v_{0} = B (T_{1}) e^{i \hat{ω} T_{0}} + \bar{B} (T_{1}) e^{- i \hat{ω} T_{0}} \end{array}

(11)

where

\hat{ω} = \sqrt{χ}

⁠,

i = \sqrt{- 1}

⁠, A and B are the complex amplitudes and are functions of T₁.

\bar{A}

and

\bar{B}

are the complex conjugates of A and B. Substituting the solutions into the first equation in Eq. , the voltage amplitude can be expressed as

B = \bar{θ} \hat{ω} A / (- \hat{ω} + i \bar{λ}) and \bar{B} = - \bar{θ} \hat{ω} \bar{A} / (\hat{ω} + i \bar{λ})

(12)

Assume

ω = \hat{ω} + ε σ

⁠, where σ is a detuning parameter. Substituting the resultant solutions into the second equation in Eq. and setting the coefficients of the secular terms associated with

e^{i \hat{ω} T_{0}}

and

e^{- i \hat{ω} T_{0}}

to zeros, one has

{\begin{array}{l} - \frac{Λ}{2} e^{i σ T_{1}} + 2 i \hat{ω} A^{'} + 2 ξ i \hat{ω} A - (2 β {\hat{ω}}^{2} - 3 ϑ) A^{2} \bar{A} - α \frac{\bar{θ} \hat{ω} A}{- \hat{ω} + i \bar{λ}} = 0 \\ - \frac{Λ}{2} e^{- i σ T_{1}} + 2 i \hat{ω} {\bar{A}}^{'} + 2 ξ i \hat{ω} \bar{A} + (2 β {\hat{ω}}^{2} - 3 ϑ) A {\bar{A}}^{2} - α \frac{\bar{θ} \hat{ω} \bar{A}}{\hat{ω} + i \bar{λ}} = 0 \end{array}

(13)

where

\cos [(\hat{ω} + ε σ) τ] = 1 / 2 [e^{i (\hat{ω} T_{0} + σ T_{1})} + e^{- i (\hat{ω} T_{0} + σ T_{1})}]

has been used. The prime denotes the derivative with respect to T₁.

Solving any of the two equations in Eq. gives the solutions of the amplitude and phase angle. We assume the complex amplitudes have the following polar forms

A = Γ (T_{1}) / 2 e^{i φ (T_{1})} and \bar{A} = Γ (T_{1}) / 2 e^{- i φ (T_{1})}

(14)

where Γ and φ are the amplitude and phase and are real functions of time T₁. Substituting Eq. into the first equation in Eq. and separating the real and imaginary parts yields

{\begin{array}{l} Γ^{'} = - \frac{Λ}{2 \hat{ω}} \sin (σ T_{1} - φ) - ξ Γ - a_{0} \bar{λ} Γ \\ Φ_{0}^{'} = - \frac{Λ}{2 \hat{ω} Γ} \cos (σ T_{1} - φ) + σ + a_{1} Γ^{2} - a_{0} \hat{ω} \end{array}

(15)

where Φ₀ = σT₁ − φ,

a_{0} = α \bar{θ} / [2 ({\hat{ω}}^{2} + {\bar{λ}}^{2})]

⁠,

a_{1} = (2 β {\hat{ω}}^{2} - 3 ϑ) / (8 \hat{ω})

⁠. By setting Γ′ = Φ′ = 0, the frequency response function can be derived by

(ξ + a_{0} \bar{λ})^{2} + (σ + a_{1} Γ^{2} - a_{0} \hat{ω})^{2} = [Λ / (2 \hat{ω} Γ)]^{2}

(16)

The corresponding phase angle is

Φ_{0} = ta n^{- 1} [- (ξ + a_{0} \bar{λ}) / (σ + a_{1} Γ^{2} - a_{0} \hat{ω})]

(17)

The amplitude Γ can be solved for different values of the detuning parameter σ from Eq. . The voltage amplitude can be attained from the second equation in Eq. , which is

V = \bar{θ} \hat{ω} Γ / \sqrt{{\hat{ω}}^{2} + {\bar{λ}}^{2}}

(18)

Multiple solutions of the amplitude Γ can be obtained from Eq. for a given detuning parameter σ. These solutions include stable solutions in both high and low branches and unstable trivial solutions. The stability analysis of the solutions can be achieved by analyzing the real part of the eigenvalues of the Jacobian matrix. If the real parts of all the eigenvalues are negative, the solution is stable. Otherwise, it is unstable. The Jacobian matrix can be derived from Eq. as

J = [\begin{matrix} - (ξ + a_{0} \bar{λ}) & - Γ (σ + a_{1} Γ^{2} - a_{0} \hat{ω}) \\ \frac{1}{Γ} (σ + 3 a_{1} Γ^{2} - a_{0} \hat{ω}) & - (ξ + a_{0} \bar{λ}) \end{matrix}]

(19)

Eliminating the secular terms, the first-order solution can be obtained as

η_{1} = (2 β {\hat{ω}}^{2} - ϑ) / (8 {\hat{ω}}^{2}) (A^{3} e^{3 i \hat{ω} T_{0}} + {\bar{A}}^{3} e^{- 3 i \hat{ω} T_{0}})

(20)

Plugging Eq. into Eqs. and , then substituting the results into Eq. , the approximate analytical solution of multiple scales in the time domain can be written as

η (τ) = Γ \cos (ω τ - Φ) + ε \frac{(2 β {\hat{ω}}^{2} - ϑ) Γ^{3}}{32 {\hat{ω}}^{2}} \cos 3 (ω τ - Φ)

(21)

where the relationship Φ = σT₁ − φ has been used.

4 Case Study of the Mono-Stable Piezoelectric Energy Harvester

The voltage frequency responses are derived under three excitation levels of Y = 0.005 g, 0.01 g, and 0.02 g. To verify the analytical solutions, numerical simulations are conducted at discrete frequency points. The initial displacement, velocity, and voltage are assumed to be zeros. The preload spring is soft, limited to k_s < k₁ so that the PEH is mono-stable. Figure 3(a) presents the voltage frequency responses under the open-circuit condition, which has the typical characteristics of a mono-stable nonlinear system, consisting of high and low branches and tilting to the right hand. When the frequency is low (ω < 0.8), the system has a single stable solution at the high branch, while two stable solutions concurrently appear at the low and high branches at the high frequency. The numerical simulations only get the low branch solutions in the frequency range of multiple stable solutions due to the zero initial conditions. The phase portraits at ω = 0.57 are plotted in Fig. 3(b), which demonstrates the approximate solutions match well with the numerical simulations. Figure 4(a) shows the voltage frequency responses under different resistive loads at the excitation level of 0.005 g. Both the high and low branch VFRs decrease as the external resistance becomes smaller.

Fig. 3

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(a) Voltage frequency responses of the mono-stable PEH and (b) phase portraits at ω = 0.57

Fig. 4

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Effects of (a) the electrical resistive load, (b) excitation level, and (c) spring pre-deformation on the voltage output

The effects of the excitation level and the spring pre-deformation on the voltage output are studied at two frequencies of ω = 0.4 and 1.5. The spring pre-deformation is Δ = 0.05%L for the case of varying excitation level. Figure 4(b) shows only the high branch oscillation happens at the small excitation frequency of ω = 0.4 and the voltage increases along with the excitation level. Both the stable high and low branch solutions exist at lower excitation level (Y = 0∼0.3 g) for ω = 1.5. Figure 4(c) shows the voltage has a very slight increment as the spring pre-deformation increases for a fixed excitation level. This suggests that the spring pre-deformation has insignificant influence on the voltage output. This is because the linear and cubic stiffness of the mono-stable PEH is dominated by the bending and axial stiffness of the beam, when the spring pre-deformation is small.

5 Bi-Stable Piezoelectric Energy Harvester

As the effective spring stiffness k_s exceeds the scaled linear stiffness k₁, the overall linear stiffness χ becomes negative and the system enters into the bi-stable regime. The bi-stable PEH exhibits both small-amplitude intra-well oscillation and large-amplitude inter-well vibration. For the intra-well oscillation, a coordinate transform is performed to estimate the solutions around the local equilibria. For the inter-well oscillation, a strategy has to be taken to handle the negative stiffness. This section presents the strategies and solutions of the multiple scales for the approximate analytical analysis.

5.1 Intra-Well Oscillation.

To analyze the intra-well dynamics, the original origin at the unstable equilibrium needs to be shifted to one of the stable equilibria

η_{2, 3}^{*}

[29]. Defining a new coordinate

y = η_{2, 3}^{*} - \sqrt{- χ / ϑ}

and substituting it into the governing Eq. , one has

\begin{aligned} y^{''} & + 2 ε^{2} \bar{ξ} y^{'} - 2 \bar{χ} y + ε^{2} \bar{β} [(y + κ) {(y^{'})}^{2} + (y^{2} + 2 y κ) y^{''}] \\ + 3 ε \bar{ϑ} κ y^{2} + ε^{2} \bar{ϑ} y^{3} - ε^{2} \bar{α} v = - ε^{2} \bar{Λ} \cos (ω τ) \end{aligned}

(22)

where

\bar{ξ} = ξ / [ε^{2} (1 + β κ^{2})]

⁠;

\bar{β} = β / [ε^{2} (1 + β κ^{2})]

⁠;

\bar{χ} = χ / (1 + β κ^{2})

⁠;

\bar{ϑ} = ϑ / [ε^{2} (1 + β κ^{2})]

⁠;

\bar{α} = α / [ε^{2} (1 + β κ^{2})]

⁠;

\bar{Λ} = Λ / [ε^{2} (1 + β κ^{2})]

⁠;

κ = \sqrt{- χ / ϑ}

⁠. Introducing the multiple time variables T₀ = τ, T₁ = ɛτ, and T₂ = ɛ²τ, the solutions can be approximated by the following asymptotic series:

\begin{aligned} {\begin{array}{l} y (τ) = y_{0} (T_{0}, T_{1}, T_{2}) + ε y_{1} (T_{0}, T_{1}, T_{2}) + ε^{2} y_{2} (T_{0}, T_{1}, T_{2}) \\ v (τ) = v_{0} (T_{0}, T_{1}, T_{2}) + ε v_{1} (T_{0}, T_{1}, T_{2}) + ε^{2} v_{2} (T_{0}, T_{1}, T_{2}) \end{array} \end{aligned}

(23)

Plugging Eqs. into , keeping the order to ɛ² and equating the like-order terms of ɛ, one has

{\begin{array}{l} ε^{0} : D_{0}^{2} y_{0} - 2 \bar{χ} y_{0} = 0 \\ ε^{1} : D_{0}^{2} y_{1} - 2 \bar{χ} y_{1} = - 2 D_{0} D_{1} y_{0} - 3 \bar{ϑ} κ y_{0}^{2} \\ ε^{2} : 2 D_{0} D_{1} y_{1} + 2 D_{0} D_{2} y_{0} + D_{0}^{2} y_{2} + D_{1}^{2} y_{0} - 2 \bar{χ} y_{2} + \\ 2 \bar{ξ} D_{0} y_{0} + \bar{β} [y_{0} {(D_{0} y_{0})}^{2} + κ {(D_{0} y_{0})}^{2} + y_{0}^{2} D_{0}^{2} y_{0} + \\ 2 κ y_{0} D_{0}^{2} y_{0}] + \bar{ϑ} y_{0}^{3} + 6 \bar{ϑ} κ y_{0} y_{1} - \bar{α} v_{0} = - \bar{Λ} \cos (ω τ) \end{array}

(24)

Similarly, substituting the approximate solutions into Eq. and collecting the like-order terms of ɛ yield

{\begin{array}{l} ε^{0} : D_{0} v_{0} + \bar{λ} v_{0} = - \bar{θ} D_{0} y_{0} \\ ε^{1} : D_{0} v_{1} + \bar{λ} v_{1} = - D_{1} v_{0} - \bar{θ} D_{1} y_{0} - \bar{θ} D_{0} y_{1} \\ ε^{2} : D_{0} v_{2} + \bar{λ} v_{2} = - D_{0} y_{2} - D_{1} y_{1} - D_{2} y_{0} - D_{1} v_{1} - D_{2} v_{0} \end{array}

(25)

Assuming the solutions have the following forms:

{\begin{array}{l} y_{0} (τ) = A (T_{1}, T_{2}) e^{i \hat{ω} T_{0}} + \bar{A} (T_{1}, T_{2}) e^{- i \hat{ω} T_{0}} \\ v_{0} (τ) = B (T_{1}, T_{2}) e^{i \hat{ω} T_{0}} + \bar{B} (T_{1}, T_{2}) e^{- i \hat{ω} T_{0}} \end{array}

(26)

where

\hat{ω} = \sqrt{- 2 χ}

⁠; the complex amplitudes A and B, and their complex conjugates

\bar{A}

and

\bar{B}

depend on T₁ and T₂. Substituting the solutions into the first-order equations, one has

\begin{aligned} D_{0}^{2} y_{1} - 2 \bar{χ} y_{1} & = 2 i \hat{ω} (- \frac{\partial A}{\partial T_{1}} e^{i \hat{ω} T_{0}} + \frac{\partial \bar{A}}{\partial T_{1}} e^{- i \hat{ω} T_{0}}) \\ - 3 \bar{ϑ} κ (A^{2} e^{i 2 \hat{ω} T_{0}} + 2 A \bar{A} + {\bar{A}}^{2} e^{- i 2 \hat{ω} T_{0}}) \end{aligned}

(27)

Eliminating the secular terms yields

\partial A / \partial T_{1} = \partial \bar{A} / \partial T_{1} = 0

⁠, implying A and

\bar{A}

only depend on T₂, and solving Eq. gives

y_{1} = \frac{\bar{ϑ} κ}{{\hat{ω}}^{2}} (A^{2} e^{i 2 \hat{ω} T_{0}} + {\bar{A}}^{2} e^{- i 2 \hat{ω} T_{0}} - 6 A \bar{A})

(28)

Substituting the solutions into the second-order equation in Eq. , defining

ω = \hat{ω} + ε^{2} σ

⁠, and setting the coefficients of the secular terms to zero, one has

{\begin{array}{l} 2 i \hat{ω} A^{'} + (2 i \bar{ξ} \hat{ω} - \frac{\bar{α} \bar{θ} \hat{ω}}{- \hat{ω} + i \bar{λ}}) A - Ψ \bar{A} A^{2} + \frac{\bar{Λ}}{2} e^{i σ T_{2}} = 0 \\ 2 i \hat{ω} {\bar{A}}^{'} + (2 i \bar{ξ} \hat{ω} - \frac{\bar{α} \bar{θ} \hat{ω}}{\hat{ω} + i \bar{λ}}) \bar{A} + Ψ A {\bar{A}}^{2} - \frac{\bar{Λ}}{2} e^{- i σ T_{2}} = 0 \end{array}

(29)

where

Ψ = (2 \bar{β} {\hat{ω}}^{2} + 30 \frac{{(\bar{ϑ} κ)}^{2}}{{\hat{ω}}^{2}} - 3 \bar{ϑ})

⁠. Both the above equations will give the same solution of A. Assuming

A = Γ (T_{2}) e^{i φ (T_{2})} / 2

and

\bar{A} = Γ (T_{2}) e^{- i φ (T_{2})} / 2

⁠, substituting into the first equation in Eq. , and setting the real and imaginary parts to zeros yield

{\begin{array}{l} Γ^{'} = - \bar{ξ} Γ - a_{2} \bar{λ} Γ - \bar{Λ} \sin Φ_{1} / (2 \hat{ω}) \\ {Φ^{'}}_{1} = σ - a_{2} {\hat{ω}}^{2} + a_{3} Γ^{2} - \bar{Λ} \cos Φ_{1} / (2 \hat{ω} Γ) \end{array}

(30)

where Φ₁ = σT₂ − φ,

a_{2} = \bar{α} \bar{θ} / [2 ({\hat{ω}}^{2} + {\bar{λ}}^{2})]

⁠, and a₃ =

[2 \bar{β} {\hat{ω}}^{2} + 30 {(\bar{ϑ} κ)}^{2} / {\hat{ω}}^{2} - 3 \bar{ϑ}] / (8 \hat{ω})

⁠. Let

Γ^{'} = Φ_{1}^{'} = 0

⁠, one can obtain the steady-state amplitude and phase angle as

[2 σ - a_{2} {\hat{ω}}^{2} + 2 a_{3} Γ^{2}]^{2} + (2 \bar{ξ} + a_{2} \bar{λ})^{2} = {(\frac{\bar{Λ}}{\hat{ω} Γ})}^{2}

(31)

Φ_{1} = ta n^{- 1} [(- 2 \bar{ξ} - a_{2} \bar{λ}) / 2 σ - a_{2} {\hat{ω}}^{2} + 2 a_{3} Γ^{2}]

(32)

The voltage amplitude equation is the same as Eq. . The Jacobian matrix can be obtained from Eq. as

J = [\begin{matrix} - \bar{ξ} - a_{2} \bar{λ} & Γ (- σ + a_{2} {\hat{ω}}^{2} - a_{3} Γ^{2}) \\ (σ - a_{2} {\hat{ω}}^{2} + 3 a_{3} Γ^{2}) / Γ & - \bar{ξ} - a_{2} \bar{λ} \end{matrix}]

(33)

After eliminating the secular terms, one has

\begin{aligned} y_{2} & = \frac{- 1}{(9 {\hat{ω}}^{2} + 2 \bar{χ})} (2 \bar{β} {\hat{ω}}^{2} - \bar{ϑ} - \frac{6 {(\bar{ϑ} κ)}^{2}}{{\hat{ω}}^{2}}) (A^{3} e^{i 3 \hat{ω} T_{0}} + {\bar{A}}^{3} e^{- i 3 \hat{ω} T_{0}}) \\ - \frac{3 \bar{β} κ {\hat{ω}}^{2}}{2 (2 {\hat{ω}}^{2} + \bar{χ})} (A^{2} e^{i 2 \hat{ω} T_{0}} + {\bar{A}}^{2} e^{- i 2 \hat{ω} T_{0}}) + \frac{2 \bar{β} κ {\hat{ω}}^{2}}{\bar{χ}} A \bar{A} \end{aligned}

(34)

Combing Eqs. , , , and , the approximate analytical solution can be obtained in the time domain as

\begin{aligned} y (τ) & = Γ \cos (ω τ - Φ_{1}) + ε \frac{\bar{ϑ} κ Γ^{2}}{2 {\hat{ω}}^{2}} [\cos 2 (ω τ - Φ_{1}) - 3] \\ + ε^{2} [\frac{- Γ^{3}}{4 (9 {\hat{ω}}^{2} + 2 \bar{χ})} (2 \bar{β} {\hat{ω}}^{2} - \bar{ϑ} - \frac{6 {(\bar{ϑ} κ)}^{2}}{{\hat{ω}}^{2}}) \cos 3 (ω τ - Φ_{1}) - \frac{3 \bar{β} κ {\hat{ω}}^{2} Γ^{2}}{4 (2 {\hat{ω}}^{2} + \bar{χ})} \cos 2 (ω τ - Φ_{1}) + \frac{\bar{β} κ {\hat{ω}}^{2} Γ^{2}}{2 \bar{χ}}] \end{aligned}

(35)

5.2 Case Study of the Intra-Well Oscillation.

The spring pre-deformation needs to satisfy k_s > k₁ to achieve the bi-stable configuration. Therefore, Δ = 0.5%L is chosen in the following study unless otherwise stated. The analytical and numerical VFRs under the open-circuit condition are plotted in Fig. 5(a). The numerical results match with the analytical solutions. Different from that of the mono-stable PEH in Fig. 3(a), the intra-well VFRs of the bi-stable PEH bend to the left-hand side for the soft stiffness. The bi-stable PEH shows no advantage in voltage outputs compared with the mono-stable PEH at Y = 0.01 g.

Fig. 5

View large Download slide

Intra-well (a) voltage frequency responses, (b) phase portraits at ω = 3.46, and (c) phase angle

The phase portraits at ω = 3.46 and different excitation levels are presented in Fig. 5(b). The analytical solutions agree well with the numerical results. Figure 5(b) shows that the harvester is confined in one of the local potential wells. Figure 5(c) plots the phase angle along with the excitation frequency at Y = 0.05 g. The bifurcation can be observed from the phase angle as the excitation frequency varies.

The voltage frequency responses at different resistive loads are presented in Fig. 6(a) for Y = 0.03 g, which shows the resistive load has a significant influence on the voltage output. The voltage decreases as the resistive load dwindles. The voltage output is shown in Fig. 6(b) for varying excitation levels at ω = 2.9 and 3.3. Unlike the case of the mono-stable PEH, only a single stable solution appears and augments as the excitation level becomes large at ω = 3.3. For ω = 2.9, the system has a single stable low branch solution under small excitation levels, but has two stable solutions of both high and low branches as the excitation level increases. There is only one stable high branch solution at high excitation levels.

Fig. 6

View large Download slide

Effect of the (a) electrical resistive load, (b) excitation level, and (c) spring pre-deformation on the voltage output

The open-circuit voltage outputs at varying spring pre-deformations are plotted in Fig. 6(c) under the three excitation levels and σ = 0. The reason that the detuning parameter is set to zero is because the natural frequency of the system varies as the spring pre-deformation changes. Only one stable solution exists over the considered range of the spring pre-deformation. The voltage output decreases as the spring-deformation increases. We conclude that the spring-deformation has evident influence on the voltage output of the bi-stable PEH. This is attributed to the fact that the spring force has a remarkable contribution to the system stiffness.

5.3 Inter-Well Oscillation.

Strategies are needed to handle the negative stiffness issue to apply the method of multiple scales. A fictitious positive stiffness term is introduced into Eq. to shift the linear negative stiffness to a positive one. By adding and subtracting the fictitious term

{\hat{ω}}^{2} η

to the left-hand side of Eq. , one has

\begin{aligned} η^{''} + 2 ε ξ η^{'} + {\hat{ω}}^{2} η + ε β [η {(η^{'})}^{2} + η^{2} η^{''}] + ε [\bar{χ} η + ϑ η^{3}] - ε α v \\ = - ε Λ \cos (ω τ) \end{aligned}

(36)

where

\hat{ω} = ω - ε σ

⁠, and

\bar{χ} = χ - {\hat{ω}}^{2}

⁠. The damping, high-order terms, electrical coupling terms, excitation, and the stiffness terms

\bar{χ} η + ϑ η^{3}

are assumed to be small. The solutions are assumed to be the same with those in Eq. . Substituting the solutions into Eqs. and , one has

{\begin{array}{l} ε^{0} : D_{0}^{2} η_{0} + {\hat{ω}}^{2} η_{0} = 0 \\ ε^{1} : D_{0}^{2} η_{1} + {\hat{ω}}^{2} η_{1} = - Λ \cos (ω τ) - 2 D_{0} D_{1} η_{0} - 2 ξ D_{0} η_{0} \\ - β [η_{0} {(D_{0} η_{0})}^{2} + η_{0}^{2} D_{0}^{2} η_{0}] - \bar{χ} η_{0} - ϑ η_{0}^{3} + α v_{0} \end{array}

(37)

{\begin{array}{l} ε^{0} : D_{0} v_{0} + \bar{λ} v_{0} = - \bar{θ} D_{0} η_{0} \\ ε^{1} : D_{0} v_{1} + \bar{λ} v_{1} = - \bar{θ} D_{0} η_{1} - D_{1} η_{0} - D_{1} v_{0} \end{array}

(38)

Substituting the zeroth-order solutions in Eq. into the first equation in Eq. , one can find the relationship between the displacement and the voltage. Plugging the solutions into the second equation in Eq. , and setting the coefficients of the secular terms to zeros, and substituting the complex conjugate amplitudes A and

\bar{A}

into one of the resultant equation, one has

{\begin{array}{l} Γ^{'} = - Λ \sin Φ_{2} / (2 \hat{ω}) - (ξ + a_{0} \bar{λ}) Γ \\ {Φ^{'}}_{0} = σ - Λ \cos Φ_{2} / (2 \hat{ω} Γ) - [\bar{χ} / (2 \hat{ω}) + a_{0} \hat{ω}] + a_{1} Γ^{2} \end{array}

(39)

where Φ₀, a₀, and a₁ are defined in Sec. 3 and Γ is the real displacement amplitude. Let

Γ^{'} = Φ_{0}^{'} = 0

⁠, one can obtain the steady-state amplitude and phase angle as

[2 \hat{ω} + σ - (\bar{χ} + 2 a_{0} {\hat{ω}}^{2}) + 2 a_{1} \hat{ω} Γ^{2}]^{2} + 4 (ξ + a_{0} \bar{λ})^{2} {\hat{ω}}^{2} = [\bar{Λ} / (\hat{ω} Γ)]^{2}

(40)

Φ_{0} = ta n^{- 1} {- 2 (ξ + a_{0} \bar{λ}) \hat{ω} / [2 \hat{ω} σ - (\bar{χ} + 2 a_{0} {\hat{ω}}^{2}) + 2 a_{1} \hat{ω} Γ^{2}]}

(41)

The Jacobian matrix can be derived from Eq. as follows:

J = [\begin{matrix} - (ξ + a_{0} \bar{λ}) & - Γ σ + Γ (\bar{χ} / 2 \hat{ω} + a_{0} \hat{ω}) - a_{1} Γ^{3} \\ \frac{σ}{Γ} - \frac{1}{Γ} (\bar{χ} / 2 \hat{ω} + a_{0} \hat{ω}) + 3 a_{1} Γ & - (ξ + a_{0} \bar{λ}) \end{matrix}]

(42)

The first-order solution η₁ is obtained after eliminating the secular terms. Together with the zeroth-order solution η₀, the analytical solution of the inter-well oscillation is

η = Γ \cos (ω τ - Φ_{0}) - ε \frac{(2 β ω^{2} - ϑ)}{8 ω^{2}} Γ^{3} \cos 3 (ω τ - Φ_{0})

(43)

5.4 Case Study of the Inter-Well Oscillation.

The analytical voltage frequency responses at Y = 0.03 g, 0.05 g, and 0.15 g are presented in Fig. 7(a). The excitation levels are higher than those for the intra-well oscillations to activate the inter-well oscillation. Comparing the voltage frequency responses under Y = 0.03 g and 0.05 g in Fig. 7(a) with those in Fig. 5(a), it shows the inter-well oscillation outperforms the intra-well oscillation in voltage output. The numerical simulation results are not given in Fig. 7(a) as we did above because the inter-well oscillation cannot be always activated under the zero initial conditions.

Fig. 7

View large Download slide

Inter-well (a) voltage frequency responses, (b) numerical results and analytical solutions, and (c) phase angle

To validate the analytical solution, upward and downward frequency sweeps are numerically performed to the governing equations within the frequency range of ω ∈ [0,7] and the excitation amplitdue of Y = 0.15 g. The numerical results are presented in Fig. 7(b) together with the analytical solutions. The amplitude of the numerical voltage response matches with the analytical solutions. Nevertheless, the analytical method overestimates the solutions at the higher frequency range. The numerical simulations are confined to the two local equilibria as the excitation frequency increases. The phase angle at the open circuit and Y = 0.05 g in Fig. 8(b) shows the bifurcation at ω = 0.32.

Fig. 8

View large Download slide

Effect of the (a) external resistive load, (b) excitation level, and (c) spring pre-deformation on the voltage output

The voltage frequency responses of the inter-well oscillation under Y = 0.05 g and different resistive loads are plotted in Fig. 8(a). Similarly, the voltage output becomes smaller as the external resistive load decreases. The voltage outputs over varying excitation levels are plotted in Fig. 8(b) for two frequencies of ω = 1.0 and 0.3. The high voltage output from the large amplitude inter-well oscillation happens when the excitation level is greater than 0.27 g for ω = 1.0. However, it is not available at the low excitation levels (0g ∼ 0.27 g) because of failing to activate the inter-well oscillation. For ω = 0.3, the excitation amplitude has a similar influence on the voltage output with that of the intra-well oscillation. Figure 8(c) presents the voltage output over varying spring pre-deformations at the three excitation levels. The trend of the voltage outputs along with the varying spring pre-deformation is the same with these in Fig. 7(a). This is because the system stiffness is dominated by the effective spring stiffness k_s induced by the spring pre-deformation. The increase in the spring pre-deformation increases the linear stiffness.

6 Conclusion

A novel piezoelectric beam energy harvester with tunable potential function is developed and analytical modeled. An axial preload spring is connected to one moveable end of the beam to achieve the mono-stable or bi-stable systems. The method of multiple scales is systematically implemented to solve for the analytical frequency responses and phase angles. To handle the negative stiffness, technical strategies are introduced to analyze the intra-well and inter-well dynamics. Numerical simulations are performed to confirm the analytical solutions. The effects of the electrical resistive load, excitation level, and spring pre-deformation on the dynamics and voltage output are studied. The voltage output of both the mono-stable and bi-stable systems increase along with the excitation level. The spring pre-deformation has slight influence on the energy harvesting performance of mono-stable system, but considerable effect on that of the bi-stable system.

Acknowledgment

This research was partially supported by National Science Foundation under Grants Nos. 1508862 and 1935951.

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Multiple-Scale Analysis of a Tunable Bi-Stable Piezoelectric Energy Harvester

Abstract

1 Introduction

2 Theoretical Model

3 Mono-Stable Piezoelectric Energy Harvester

4 Case Study of the Mono-Stable Piezoelectric Energy Harvester

5 Bi-Stable Piezoelectric Energy Harvester

5.1 Intra-Well Oscillation.

5.2 Case Study of the Intra-Well Oscillation.

5.3 Inter-Well Oscillation.

5.4 Case Study of the Inter-Well Oscillation.

6 Conclusion

Acknowledgment

References

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Multiple-Scale Analysis of a Tunable Bi-Stable Piezoelectric Energy Harvester

Abstract

1 Introduction

2 Theoretical Model

3 Mono-Stable Piezoelectric Energy Harvester

4 Case Study of the Mono-Stable Piezoelectric Energy Harvester

5 Bi-Stable Piezoelectric Energy Harvester

5.1 Intra-Well Oscillation.

5.2 Case Study of the Intra-Well Oscillation.

5.3 Inter-Well Oscillation.

5.4 Case Study of the Inter-Well Oscillation.

6 Conclusion

Acknowledgment

References

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