A nonlinear galloping energy harvester with quasi-zero stiffness

Abstract

One of the design requirements to enhance the performance of a galloping piezoelectric energy harvester (GPEH) is to have a low natural frequency. A nonlinear system with a zero local linearised stiffness about the working point, known as a Quasi-Zero-Stiffness (QZS) can be employed to achieve ultra-low natural frequency. The energy is harvested by attaching a piezoelectric sheet to the transverse degree of freedom of the harvester. In this paper, an aero-electro-mechanical model of the QZS nonlinear GPEH is obtained, with the QZS mechanism formed by adding two transverse springs to the GPEH. Numerical integration demonstrates that a QZS nonlinear GPEH with a lower and flatter stiffness-displacement curve can harvest more power than one with a higher and steeper curve. An analytical solution using the harmonic balance method (HBM) was obtained and used to optimise the system. It is shown that three optimal values of the mechanical to electrical time-constant ratio exist for the QZS nonlinear GPEH. Also, a relative performance study was performed, and it is shown that the QZS GPEH has the potential to harvest more power than the conventional GPEH.

Introduction

Flow induced oscillation (FIO) energy harvesters have gained increasing popularity in recent years for applications in micro-electromechanical systems (MEMS; Abdelkefi, 2016; Daqaq et al., 2019; Wen et al., 2021). The FIO phenomena that find application in energy harvesting include vortex-induced vibrations (Bernitsas et al., 2008; Facchinetti et al., 2004), galloping (Abdelkefi et al., 2013c; Barrero-Gil et al., 2010), wake galloping (Abdelkefi et al., 2013b; Alhadidi and Daqaq, 2016), flutter (Erturk et al., 2010; Hiroaki and Watanabe, 2021), and buffeting (Armandei and Fernandes, 2016; Canfield et al., 2008). Energy harvesting from galloping has attracted considerable interest from researchers (Abdelkefi, 2016; Nabavi and Zhang, 2016; Wang et al., 2020a; Watson et al., 2019). The principle of operation of galloping-based energy harvesters is well established and occurs when a fluid flows past an elastic bluff body. Once the instability threshold is exceeded, an oscillatory motion develops transverse to the flow, with increasing amplitude, until the energy dissipated per cycle by mechanical damping balances the energy input from the flow (Barrero-Gil et al., 2010). The energy available to the mechanical oscillator can be transformed into electricity via an electromechanical transduction mechanism which is mainly piezoelectric or electromagnetic.

Most of the recent research in GPEHs is aimed towards improving the performance via minimising the cut-in flow speed and maximising the power output. The cut-in flow speed and the power output, which are the main performance indicators of a given GPEH with a given bluff body, depend on the damping forces, electromechanical coupling, and the strength of the restoring forces. Generally, for a GPEH, the magnitude of the electromechanical coupling of a piezoelectric transducer is constant while the inherent damping is beyond a designer’s control, the main design parameter which can be used to improve the performance of GPEHs is the spring restoring force.

A large body of research has considered the restoring force in GPEHs as linear (Abdelkefi, 2016; Abdelkefi et al., 2013a; Daqaq et al., 2019; Ewere and Wang, 2013; Tang et al., 2015; Wen et al., 2021; Zhao et al., 2012). To increase the size of the bluff body of a single degree-of-freedom (SDOF) GPEH with a linear restoring force, the natural frequency should be lowered to increase the harvested power (Barrero-Gil et al., 2010) which necessitates a reduction in the stiffness. However, a low linear stiffness results in excessive deflection at high flow velocities, which can make the harvester unstable, narrow its safe operating velocity range, and cause significant static deflection (Chen and Zhao, 2023).

Nonlinear restoring forces have been introduced to enhance the performance of SDOF GPEHs by Bibo et al. (2015). In Bibo et al. (2015), a galloping flow energy harvester with a quartic potential energy function was analysed, allowing the system to exhibit either a softening, hardening or bi-stable restoring force. The results showed that, for similar design parameters and equal magnitudes of effective linear stiffness and effective cubic stiffness, the bi-stable configuration yields superior performance compared to softening and hardening configurations as long as the inter-well oscillations are activated. However, when the motion of the bi-stable configuration remains confined to a single potential well, the one with a softening restoring force outperforms the other designs.

The QZS nonlinearity is another nonlinear restoring force that finds application mainly in low-frequency vibration isolation (Abolfathi, 2024; Carrella et al., 2007, 2012; Kovacic et al., 2008). The QZS mechanism is designed to achieve a localised zero stiffness at equilibrium by combining a matched nonlinear negative-stiffness structure in parallel with a positive-stiffness structure (Hu and Zhou, 2022). An important characteristic of the QZS nonlinearity which can be explored to enhance the performance of GPEHs, is its ability to maintain a low dynamic natural frequency. Chen and Zhao (2023) investigated a QZS two degree-of-freedom (2DOF) nonlinear galloping oscillator designed for aeroelastic energy harvesting at ultra-low wind speeds. A distributed-parameter aero-electro-mechanical model was developed, which was validated through wind tunnel tests on a fabricated prototype. The results demonstrated that the QZS 2DOF GPEH achieved approximately $200\%$ increase in voltage at a wind velocity of $2.5$ m/s compared to a linear SDOF GPEH, and around $100\%$ improvement over a linear 2DOF GPEH. The QZS nonlinearity in the system was implemented using magnets. In a sister paper to this study, the authors investigated a 2DOF GPEH in which both bluff bodies are excited by the incoming flow, while the springs are considered linear (Major et al., 2024). To the best knowledge of the authors, the dynamics and performance of a SDOF QZS nonlinear GPEH have not been studied.

This paper investigates a SDOF QZS nonlinear GPEH in which the QZS nonlinearity is achieved using transverse springs. In section 2, a physics based aero-electro-mechanically coupled theoretical model of the harvester, assuming a piezoelectric transduction mechanism and a quasi-steady aerodynamic flow field, is developed. The effect of different combinations of stiffness ratio and geometrical parameters that give rise to stable QZS, on the transverse displacement and output power is presented in Section 3. An approximation to the QZS restoring force is introduced in the mathematical model of the harvester, and an approximate analytical solution of the resulting simplified model using the HBM, is presented in Section 4. In Section 5, the performance of the QZS GPEH is compared to that of the linear, hardening, softening, and bi-stable GPEHs. In Section 6, the results are discussed, while the main conclusions are presented in Section 7.

Mathematical modelling

The coupled analytical aero-electro-mechanical model of the QZS SDOF GPEH should consider the aerodynamic force acting on the bluff body, the nonlinear restoring force due to the nonlinear mechanism which can exhibit QZS, and the electromechanical coupling effect. A 3D representation of the QZS SDOF GEPH is shown in Figure 1(a) and its mathematical model is shown in Figure 1(b). The QZS mechanism is constructed by two transverse linear springs (each of stiffness $k_o$ ) in parallel with the beam which is modelled as a linear spring with effective stiffness $K$ as shown in Figure 2. In the case of the QZS GPEH, the elastically mounted bluff body is placed in a uniform air flow with a mean flow velocity, $U$ . When the flow velocity exceeds the critical velocity of galloping, the harvester undergoes steady-state limit cycle oscillations in the direction $x$ , perpendicular to the flow direction.

Figure 1.

(a) A 3-D model of a QZS nonlinear GPEH and (b) lumped-parameter model of the QZS GPEH being investigated in this study (top view, $\mathrm{xz}$ plane).

Figure 2.

Schematic representation of the three-spring mechanism at the static equilibrium position which can exhibit quasi-zero stiffness.

The ordinary differential equations governing the dynamics of the lumped-parameter model shown in Figure 1(b) can be written as

$$M\stackrel{··}{x}+C\stackrel{·}{x}+f_n-\mathrm{\Theta }V=F_a,$$

(1a)

$$C_p\stackrel{·}{V}+\frac{V}{R}+\mathrm{\Theta }\stackrel{·}{x}=0,$$

(1b)

where the dot denotes differentiation with respect to time, $t.M$ is the effective mass of the harvester; $x$ is the displacement of the harvester in the direction normal to the wind flow; $C=2{\zeta }_{m}M{\omega }_n$ , is the linear damping coefficient of the harvester; ${\zeta }_m$ is the mechanical damping ratio; ${\omega }_n=\sqrt{K/M}$ , is the natural frequency of the system at short-circuit conditions without the transverse springs; $\mathrm{\Theta }$ is the electromechanical coupling coefficient; $V$ is the generated voltage across the electrical load $R$ ; $C_p$ is the capacitance of the piezoelectric transducer; $f_n$ is the nonlinear restoring force of the three-spring mechanism when the harvester is displaced from the static equilibrium position; and $F_a$ is the vertical component of the aerodynamic force acting on the bluff body.

For the three-spring mechanism, the restoring force is given by (Wang et al., 2020b)

$$f_n=\mathrm{Kx}+2k_{o}x(1-\frac{L_o}{\sqrt{x^2+a^2}}),$$

(2)

where $a$ is the length of each transverse spring at the static equilibrium position and $L_o$ is their length when unloaded. The stiffness of the three-spring mechanism is obtained by differentiating the force with respect to the displacement to give

$$K_n=K+2k_o[1-\frac{L_o{a}^2}{{(x^2+a^2)}^{\frac{3}{2}}}].$$

(3)

The vertical component of the aerodynamic force acting on the bluff body is given by (Barrero-Gil et al., 2009)

$$F_a=\frac{1}{2}\rho U^2\mathrm{LD}{C}_y,$$

(4)

where $\rho$ is the air density, $L$ and $D$ are, respectively, the cross-flow length and width of the bluff body. $C_y$ is the instantaneous fluid force coefficient in the normal direction to the incident flow. In the static case, the instantaneous force coefficient (perpendicular to the incident flow) can be expanded in powers of the angle of attack, $\psi$ (see Figure 1(b)),

$$C_y\left(\psi \right)=\sum _{j=0}^n{s}_j{\psi }^j.$$

(5)

Assuming small values of the velocity ratio $\stackrel{·}{x}/U$ and expanding $\psi$ in Taylor series, $\psi ={\tan }^{-1}(\stackrel{·}{x}/U)\approx \stackrel{·}{x}/U$ , then equation (5) becomes

$$C_y\left(\psi \right)=\sum _{j=0}^n{s}_j{(\stackrel{·}{x}/U)}^j.$$

(6)

The coefficients $s_j$ accounts for the different geometries and aspect ratios of the bluff body. The instantaneous fluid force coefficient $C_y$ is directly related to the lift and drag coefficients $C_l$ and $C_d$ , respectively, by $C_y=-[C_l\cos (\psi )+C_d\sin \psi ]$ . Barrero-Gil et al. (2010) showed that the instantaneous fluid force coefficient can be approximated by a cubic polynomial expansion of $\stackrel{·}{x}/U$ . Hence, the transverse aerodynamic force can be written as

$$F_a=\frac{1}{2}\rho U^2\mathrm{LD}[s_1\frac{\stackrel{·}{x}}{U}+s_3{\left(\frac{\stackrel{·}{x}}{U}\right)}^3].$$

(7)

According to Den Hartog stability criterion (Den Hartog, 1985), a section of a structure on a flexible support is susceptible to galloping when the linear coefficient $s_1$ is positive. The nonlinear coefficient $s_3$ is always negative, because as the angle of attack increases, $C_y$ increases, attains a maximum value, and then decreases (Abdelkefi et al., 2012).

The dimensionless form of equations (1a) and (1b) can be obtained by introducing the following nondimensional variables and parameters:

$$\begin{array}{c}\multicolumn{1}{c}{\overline{x}=\frac{x}{D},\overline{V}=\frac{C_{p}V}{\mathrm{\Theta }D},{\overline{L}}_o=\frac{L_o}{D},\beta =\frac{k_o}{K},}\\ \multicolumn{1}{c}{\overline{U}=\frac{U}{{\omega }_{n}D},\overline{m}=\frac{M}{\rho L{D}^2},\kappa =\frac{{\mathrm{\Theta }}^2}{M{\omega }_n^2{C}_p},\alpha ={\omega }_{n}R{C}_p,}\end{array}$$

where $\overline{x}$ and $\overline{V}$ represent the dimensionless transverse displacement and voltage, respectively; ${\overline{L}}_o$ is the dimensionless initial length of the transverse spring; $\beta$ is the ratio of the spring stiffnesses; $\overline{U}$ is the reduced flow velocity; $\overline{m}$ is the flow to harvester mass ratio; $\kappa$ is the dimensionless electromechanical coupling; and $\alpha$ is the mechanical to electrical time-constant ratio. Equations (1a) and (1b) can be written in terms of the nondimensional variables and parameters as

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{x}}^{″}+2{\zeta }_m{\overline{x}}^{\prime }-\frac{1}{2}\frac{\overline{U}{s}_1}{\overline{m}}{\overline{x}}^{\prime }+\frac{s_3}{2\overline{m}\overline{U}}{\left({\overline{x}}^{\prime }\right)}^3+\overline{x}}\\ \multicolumn{1}{c}{+2\beta \overline{x}[1-\frac{1}{\sqrt{{\left(\frac{\overline{x}}{{\overline{L}}_o}\right)}^2+{\gamma }^2}}]-\kappa \overline{V}=0,}\end{array}$$

(8a)

$${\overline{V}}^{\prime }+\frac{\overline{V}}{\alpha }+{\overline{x}}^{\prime }=0,$$

(8b)

where the prime denotes a derivative with respect to nondimensional time $\tau ={\omega }_{n}t$ , and

$$\gamma =\frac{a}{L_o}=\cos {\theta }_0,$$

(9)

is a geometrical parameter, and ${\theta }_0$ is defined as the initial angle of inclination of the transverse springs (Wang et al., 2020b).

The nondimensional form of the stiffness of the three-spring mechanism can be expressed as

$${\overline{K}}_n=1+2\beta \{1-\frac{{\gamma }^2}{{[{\left(\frac{\overline{x}}{{\overline{L}}_o}\right)}^2+{\gamma }^2]}^{\frac{3}{2}}}\}$$

(10)

At the static equilibrium position of the harvester, the three-spring mechanism is expected to have zero stiffness. This occurs at $\overline{x}=0$ , where the two transverse springs become horizontal as shown in Figure 2, at which the positive stiffness of the third linear spring is exactly balanced by the (maximum) negative stiffness from the transverse springs. If equation (10) is evaluated at the static equilibrium position and set to zero, then the value of $\beta$ that ensures QZS behaviour for a given $\gamma$ is

$${\beta }_{\mathrm{QZS}}=\frac{\gamma }{2(1-\gamma )}.$$

(11a)

Similarly, the value of $\gamma$ that realises quasi-zero-stiffness for a given value of $\beta$ is

$${\gamma }_{\mathrm{QZS}}=\frac{2\beta }{2\beta +1}.$$

(11b)

Figure 3 shows the nondimensional force-displacement and stiffness-displacement characteristics of the three-spring system for different combinations of the geometrical parameter and stiffness ratio that ensures QZS behaviour.

Figure 3.

(a) Force-displacement characteristics of the QZS mechanism and (b) stiffness of the QZS mechanism as a function of the nondimensional displacement for different values of the geometrical parameter and stiffness ratio. ${\overline{L}}_o=1$ .

Effects of the geometrical parameter and stiffness ratio on the output power and transverse displacement

The effects of the geometrical parameter and the stiffness ratio on the transverse displacement and output power are investigated in this section numerically. The numerical simulations are performed using ode45 function of MATLAB. The same parameters as in Bibo et al. (2015), in which a square-sectioned bluff body is considered, are considered here and are listed in Table 1. The output power $P$ , of the QZS GPEH is

$$P=\frac{V^2}{R}=\frac{{\left[\overline{V}\left(\frac{\mathrm{\Theta }D}{C_p}\right)\right]}^2}{\left(\frac{\alpha }{{\omega }_n{C}_p}\right)}=\frac{{\overline{V}}^2}{\alpha }\times \frac{{\left(\mathrm{\Theta }D\right)}^2{\omega }_n}{C_p},$$

(12)

which can be nondimensionalised to give $\overline{P}=P/P_0$ , where $P_0={\left(\mathrm{\Theta }D\right)}^2{\omega }_n/C_p$ is the reference power of the piezoelectric transducer at the natural frequency of the cantilever beam, and

$$\overline{P}=\frac{{\overline{V}}^2}{\alpha }$$

(13)

is the nondimensional output power of the QZS GPEH.

Table 1.

Nondimensional properties of a SDOF GPEH (Bibo et al., 2015).

Parameter (symbol)	Value
Mechanical damping ratio $({\zeta }_m)$	0.003
Harvester to flow mass ratio $(\overline{m})$	366
Electrical to mechanical time-constant ratio $(\alpha )$	0.333
Nondimensional electromechanical coupling $(\kappa )$	0.01
Linear aerodynamic coefficient $(s_1)$	2.5
Cubic aerodynamic coefficient $(s_3)$	130

The variations of the transverse displacement and the output power with respect to the reduced flow velocity of the harvester is shown in Figure 4(a) and (b), respectively. The plots show that increasing the value of the geometrical parameter $\gamma$ , with the corresponding value of the stiffness ratio $\beta$ , both of which dependently ensure QZS behaviour of the harvester results in decreasing response of the harvester. From Figure 4(a), it is observed that for $\overline{U}<3.3$ , the amplitude of the displacement is approximately the same for the different combinations of $\gamma$ and $\beta$ . However, for $\overline{U}>3.3$ , the combination of $\gamma$ and $\beta$ with the smallest values lead to the highest magnitude of the transverse displacement. From Figure 4(b), it is shown that lower values of the reduced flow velocity, $\overline{U}<7$ , the output power is approximately the same for the different combinations of $\gamma$ and $\beta$ . For $\overline{U}>7$ , there is a slight difference in the level of harvested power between the different combinations of $\gamma$ and $\beta$ . The harvested power decreases with increasing values of the combination of $\gamma$ and $\beta$ . This can be seen more clearly in Figure 4(c). The parameters of the harvester used for the analysis presented so far are arbitrary parameters chosen based on previous studies. It is computationally costly to investigate the entire parameter space for the system’s response using numerical integration, which has been applied so far. Hence, it is necessary to obtain the approximate analytical response that allows for the analysis of the parameter space. To achieve this, an approximate expression of the restoring force of the QZS mechanism has to be obtained.

Figure 4.

Variations of the: (a) transverse displacement with reduced flow velocity, (b) harvested power for $0\le \overline{U}\le 10$ , and (c) harvested power for $7\le \overline{U}\le 10$ , for different combinations of $\gamma$ and $\beta$ that ensure QZS behaviour of the harvester, using the parameters of the system listed in Table 1.

Approximation of the restoring force of the QZS mechanism

Approximate mathematical model

The relationship between the nondimensional force and nondimensional displacement given by equation (2) is similar to that of a cubic function. Subsequent dynamic analysis of the QZS GPEH would be considerably simplified if its stiffness could be represented by a polynomial. The nondimensional force,

$${\overline{f}}_n=\overline{x}+2\beta \overline{x}[1-\frac{1}{\sqrt{{\left(\frac{\overline{x}}{{\overline{L}}_o}\right)}^2+{\gamma }^2}}],$$

(14)

can be expressed as a power series using Taylor series expansion and ignoring higher order terms, that is,

$${\overline{f}}_n=[1+2\beta (1-\frac{1}{\gamma })]\overline{x}+\frac{\beta }{{\overline{L}}_o^2{\gamma }^3}{\overline{x}}^3,$$

(15)

which can be written as

$${\overline{f}}_n={\overline{K}}_1\overline{x}+{\overline{K}}_3{\overline{x}}^3,$$

(16)

where

${\overline{K}}_1=[1+2\beta (1-\frac{1}{\gamma })]$ , and

$${\overline{K}}_3=\frac{\beta }{{\overline{L}}_o^2{\gamma }^3}.$$

From equation (16), the restoring force is QZS when ${\overline{K}}_1=0$ and ${\overline{K}}_3>0$ .

Introducing equation (16) into equation (8a), the approximate governing equations are

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{x}}^{″}+2{\zeta }_m{\overline{x}}^{\prime }+{\overline{K}}_1\overline{x}+{\overline{K}}_3{\overline{x}}^3-\frac{1}{2}\frac{\overline{U}{s}_1}{\overline{m}}{\overline{x}}^{\prime }}\\ \multicolumn{1}{c}{+\frac{s_3}{2\overline{m}\overline{U}}{\left({\overline{x}}^{\prime }\right)}^3-\kappa \overline{V}=0}\end{array}$$

(17a)

$${\overline{V}}^{\prime }+\frac{\overline{V}}{\alpha }+{\overline{x}}^{\prime }=0$$

(17b)

Figure 5 shows a plot of the cubic approximation and the exact expression of the force versus the displacement, for $\beta =0.5,\gamma =0.5$ , and $L_o=1$ . The error between the approximate and exact force increases as the system moves further from its static equilibrium position. Within the range of $-0.25\le \overline{\mathit{x}}\le 0.25$ , the approximate equation closely follows the exact force-deflection curve, demonstrating good accuracy.

Figure 5.

Force-displacement characteristics of the QZS mechanism for $\beta =0.5,\gamma =0.5$ , and $L_o=1$ .

Approximate analytical solution

The approximate analytical solutions of equations (17a) and (17b) are derived using the HBM. The HBM is an effective method for obtaining approximate analytical solutions of strongly nonlinear oscillations (Krack and Gross, 2019; Mickens, 1984, 2010). Assuming the appropriate solutions have the following form and ignoring higher harmonics,

$$\overline{x}={\overline{a}}_1\sin \mathrm{\Omega }\tau +{\overline{b}}_1\cos \mathrm{\Omega }\tau$$

(18a)

$$\overline{V}={\overline{a}}_2\sin \mathrm{\Omega }\tau +{\overline{b}}_2\cos \mathrm{\Omega }\tau$$

(18b)

where ${\overline{a}}_i$ and ${\overline{b}}_i$ (for $i=1,2)$ , are the sought Fourier coefficients, $\mathrm{\Omega }=\omega /{\omega }_n$ is the nondimensional frequency of the dynamic responses and $\omega$ is the dimensional frequency of the dynamic responses. Substituting equations (18a) and (18b) into equation (17a), setting ${\overline{K}}_1=0$ to achieve the QZS nonlinearity and neglecting terms with frequency $3\mathrm{\Omega }$ gives

$$\begin{array}{c}\multicolumn{1}{c}{[-{\mathrm{\Omega }}^2{\overline{a}}_1-2{\zeta }_m\mathrm{\Omega }{\overline{b}}_1+\frac{3}{4}{\overline{K}}_3({\overline{a}}_1^3+{\overline{a}}_1{\overline{b}}_1^2)}\\ \multicolumn{1}{c}{+\frac{\overline{U}{s}_1}{2\overline{m}}\mathrm{\Omega }{\overline{b}}_1-\frac{3{\mathrm{\Omega }}^3{s}_3}{8\overline{m}\overline{U}}({\overline{a}}_1^2{\overline{b}}_1+{\overline{b}}_1^3)-\kappa {\overline{a}}_2]\sin \mathrm{\Omega }\tau }\\ \multicolumn{1}{c}{+[-{\mathrm{\Omega }}^2{\overline{b}}_1+2{\zeta }_m\mathrm{\Omega }{\overline{a}}_1+\frac{3}{4}{\overline{K}}_3({\overline{a}}_1^2{\overline{b}}_1+{\overline{b}}_1^3)-\frac{\overline{U}{s}_1}{2\overline{m}}\mathrm{\Omega }{\overline{a}}_1}\\ \multicolumn{1}{c}{+\frac{3{\mathrm{\Omega }}^3{s}_3}{8\overline{m}\overline{U}}({\overline{a}}_1^3+{\overline{a}}_1{\overline{b}}_1^2)-\kappa {\overline{b}}_2]\cos \mathrm{\Omega }\tau =0.}\end{array}$$

(19)

From equation (19), the considered terms are balanced individually by setting the associated (Fourier) coefficients to zero. This leads to,

$$\begin{array}{c}\multicolumn{1}{c}{[-{\mathrm{\Omega }}^2+\frac{3}{4}{\overline{K}}_3({\overline{a}}_1^2+{\overline{b}}_1^2)]{\overline{a}}_1}\\ \multicolumn{1}{c}{-[2{\zeta }_m\mathrm{\Omega }-\frac{\overline{U}{s}_1}{2\overline{m}}\mathrm{\Omega }+\frac{3{\mathrm{\Omega }}^3{s}_3}{8\overline{m}\overline{U}}({\overline{a}}_1^2+{\overline{b}}_1^2)]{\overline{b}}_1-\kappa {\overline{a}}_2=0,}\end{array}$$

(20a)

$$\begin{array}{c}\multicolumn{1}{c}{[2{\zeta }_m\mathrm{\Omega }-\frac{\overline{U}{s}_1}{2\overline{m}}\mathrm{\Omega }+\frac{3{\mathrm{\Omega }}^3{s}_3}{8\overline{m}\overline{U}}({\overline{a}}_1^2+{\overline{b}}_1^2)]{\overline{a}}_1}\\ \multicolumn{1}{c}{+[-{\mathrm{\Omega }}^2+\frac{3}{4}{\overline{K}}_3({\overline{a}}_1^2+{\overline{b}}_1^2)]{\overline{b}}_1-\kappa {\overline{b}}_2=0.}\end{array}$$

(20b)

Similarly, from equation (17b) the following equations are obtained

$$\mathrm{\Omega }{\overline{a}}_1+\mathrm{\Omega }{\overline{a}}_2+\frac{{\overline{b}}_2}{\alpha }=0,$$

(21a)

$$\mathrm{\Omega }{\overline{b}}_1-\frac{{\overline{a}}_2}{\alpha }+\mathrm{\Omega }{\overline{b}}_2=0.$$

(21b)

To obtain ${\overline{a}}_2$ and ${\overline{b}}_2$ , equations (21a) and (21b) are solved simultaneously. Therefore,

$${\overline{a}}_2=\frac{\alpha \mathrm{\Omega }}{{\left(\alpha \mathrm{\Omega }\right)}^2+1}(-\alpha \mathrm{\Omega }{\overline{a}}_1+{\overline{b}}_1),$$

(22a)

and

$${\overline{b}}_2=\frac{\alpha \mathrm{\Omega }}{{\left(\alpha \mathrm{\Omega }\right)}^2+1}(-{\overline{a}}_1-\alpha \mathrm{\Omega }{\overline{b}}_1).$$

(22b)

Substituting equation (22a) into equation (20a), the resulting equation is

$$\begin{array}{c}\multicolumn{1}{c}{[-{\mathrm{\Omega }}^2+\frac{3}{4}{\overline{K}}_3({\overline{a}}_1^2+{\overline{b}}_1^2)+\frac{\kappa {\left(\alpha \mathrm{\Omega }\right)}^2}{1+{\left(\alpha \mathrm{\Omega }\right)}^2}]{\overline{a}}_1-}\\ \multicolumn{1}{c}{[2{\zeta }_m\mathrm{\Omega }-\frac{\overline{U}{s}_1}{2\overline{m}}\mathrm{\Omega }+\frac{3{\mathrm{\Omega }}^3{s}_3}{8\overline{m}\overline{U}}({\overline{a}}_1^2+{\overline{b}}_1^2)+\frac{\kappa \alpha \mathrm{\Omega }}{1+{\left(\alpha \mathrm{\Omega }\right)}^2}]{\overline{b}}_1=0.}\end{array}$$

(23a)

Similarly, substituting equations (22b) into equation (20b) gives

$$\begin{array}{c}\multicolumn{1}{c}{[2{\zeta }_m\mathrm{\Omega }-\frac{\overline{U}{s}_1}{2\overline{m}}\mathrm{\Omega }+\frac{3{\mathrm{\Omega }}^3{s}_3}{8\overline{m}\overline{U}}({\overline{a}}_1^2+{\overline{b}}_1^2)+\frac{\kappa \alpha \mathrm{\Omega }}{1+{\left(\alpha \mathrm{\Omega }\right)}^2}]{\overline{a}}_1}\\ \multicolumn{1}{c}{+[-{\mathrm{\Omega }}^2+\frac{3}{4}{\overline{K}}_3({\overline{a}}_1^2+{\overline{b}}_1^2)+\frac{\kappa {\left(\alpha \mathrm{\Omega }\right)}^2}{1+{\left(\alpha \mathrm{\Omega }\right)}^2}]{\overline{b}}_1=0.}\end{array}$$

(23b)

Equations (23a) and (23b) can further be simplified into:

$${\mathrm{\Omega }}^2-\frac{3}{4}{\overline{K}}_3{\overline{A}}^2-\frac{\kappa {\left(\alpha \mathrm{\Omega }\right)}^2}{1+{\left(\alpha \mathrm{\Omega }\right)}^2}=0,$$

(24a)

$$2{\zeta }_m-\frac{\overline{U}{s}_1}{2\overline{m}}+\frac{3{\mathrm{\Omega }}^2{s}_3}{8\overline{m}\overline{U}}{\overline{A}}^2+\frac{\kappa \alpha }{1+{\left(\alpha \mathrm{\Omega }\right)}^2}=0,$$

(24b)

where $\overline{A}=\sqrt{{\overline{a}}_1^2+{\overline{b}}_1^2}$ , is the amplitude of the nondimensional transverse displacement.

The third term of equation (24a) is the nondimensional circuit-induced stiffness, that is,

$${\overline{k}}_e=\frac{\kappa {\left(\alpha \mathrm{\Omega }\right)}^2}{1+{\left(\alpha \mathrm{\Omega }\right)}^2},$$

(25a)

while the fourth term of equation (24b) is the nondimensional circuit-induced damping, that is,

$${\overline{c}}_e=\frac{\kappa \alpha }{1+{\left(\alpha \mathrm{\Omega }\right)}^2}.$$

(25b)

By solving equation (24a) using Mathematica^® the frequency of dynamic response is obtained as,

$$\mathrm{\Omega }=\frac{1}{2\sqrt{2}}\sqrt{4\kappa -\frac{4}{{\alpha }^2}+3{\overline{K}}_3{\overline{A}}^2+\frac{\sqrt{48{\alpha }^2{\overline{K}}_3{\overline{A}}^2+{(4-4{\alpha }^2\kappa -3{\alpha }^2{\overline{K}}_3{\overline{A}}^2)}^2}}{{\alpha }^2}},$$

(26)

and solving equation (24b) the amplitude of the nondimensional displacement is obtained as,

$$\overline{A}=\sqrt{\frac{8\overline{m}\overline{U}}{3{\mathrm{\Omega }}^2{s}_3}(\frac{\overline{U}{s}_1}{2\overline{m}}-2{\zeta }_m-{\overline{c}}_e)}.$$

(27)

Also, setting $\overline{A}=0$ in equations (27) and (26), the cut-in reduced velocity which signifies the onset of galloping is obtained as

$${\overline{U}}_g=\frac{2\overline{m}}{s_1}[2{\zeta }_m+\frac{\kappa \alpha }{{\left(\alpha \mathrm{\Omega }\right)}^2+1}],$$

(28a)

where

$$\mathrm{\Omega }=\frac{1}{\sqrt{2}}\sqrt{}\frac{{\alpha }^2\kappa -1-\sqrt{}{({\alpha }^2\kappa -1)}^2}{{\alpha }^2}.$$

(28b)

It can be seen clearly from equation (28a) that the onset of galloping of the QZS nonlinear GPEH is independent of the nonlinear stiffness of the three-spring mechanism. However, it is a function of $\kappa$ which includes ${\omega }_n$ in its definition. Hence, the stiffness of the system would change the onset of galloping in practice.

The nondimensional voltage output, $\overline{V}$ can be determined using equations (18) and (22) as

$$\overline{V}=\sqrt{{\overline{a}}_2^2+{\overline{b}}_2^2}=\frac{\alpha \mathrm{\Omega }}{\sqrt{1+{\left(\alpha \mathrm{\Omega }\right)}^2}}\overline{A},$$

(29)

and the nondimensional power output can be expressed using equation (13) as

$$\overline{P}=\frac{{\overline{V}}^2}{\alpha }=\frac{\mathrm{\alpha }{\mathrm{\Omega }}^2}{1+{\left(\mathrm{\alpha }\mathrm{\Omega }\right)}^2}{\overline{A}}^2.$$

(30)

Similarly, for the linear GPEH (obtained by setting ${\overline{K}}_3=0$ and ${\overline{K}}_1=1$ , in equation (17a)) the dimensionless frequency of the system is given as

$${\mathrm{\Omega }}_L=\frac{1}{\sqrt{2}}\sqrt{1-\frac{1}{{\alpha }^2}+\kappa +\frac{\sqrt{4{\alpha }^2+{(1-{\alpha }^2-{\alpha }^2\kappa )}^2}}{{\alpha }^2}}.$$

(31)

The nondimensional amplitude of displacement is

$${\overline{A}}_L=\sqrt{\frac{8\overline{m}\overline{U}}{3{\mathrm{\Omega }}_L^2{s}_3}(\frac{\overline{U}{s}_1}{2\overline{m}}-2{\zeta }_m-\frac{\kappa \alpha }{{\left(\alpha {\mathrm{\Omega }}_L\right)}^2+1})},$$

(32a)

the cut-in reduced velocity is given as

$${\overline{U}}_c=\frac{2\overline{m}}{s_1}[2{\zeta }_m+\frac{\kappa \alpha }{{\left(\alpha {\mathrm{\Omega }}_L\right)}^2+1}],$$

(32b)

the nondimensional voltage output, ${\overline{V}}_L$ can be written as

$${\overline{V}}_L=\frac{\alpha {\mathrm{\Omega }}_L}{\sqrt{1+{\left(\alpha {\mathrm{\Omega }}_L\right)}^2}}{\overline{A}}_L,$$

(33)

while the nondimensional power output can be expressed as

$${\overline{P}}_L=\frac{\alpha {\mathrm{\Omega }}_L^2}{1+{\left(\mathrm{\alpha }{\mathrm{\Omega }}_L\right)}^2}{\overline{A}}_L^2.$$

(34)

Comparison of the approximate analytical solution with numerical solutions

The analytical solutions obtained using HBM are compared in Figure 6 with those obtained by numerical integration (using MATLAB’s ode45 function) using the parameters which are listed in Table 1. For the linear GPEH, as can be seen in Figure 6, the difference in the results obtained by numerical integration and the HBM is negligible. This implies that the HBM accurately predicts the mechanical response and the electrical response of the linear GPEH.

Figure 6.

Comparison between the analytical predictions and the numerical integration using the parameters of the system listed in Table 1: (a) amplitude of nondimensional displacement versus reduced velocity and (b) amplitude of nondimensional voltage versus reduced velocity. Results are for the linear GPEH.

Figure 7 shows the comparison between the approximate analytical solutions obtained using the HBM with those obtained by numerical integration, for the QZS nonlinear GPEH. Two cases are considered, the first case, referred to as Case I for which ${\overline{K}}_3=1$ , and the second case, referred to as Case II with ${\overline{K}}_3=20$ . The results for both Case I and Case II show that the HBM predicts a Hopf bifurcation point in the amplitude of nondimensional displacement versus reduced flow velocity parameter’s space, which is different from that obtained by numerical integration (Figure 7(a) and (b)). In the nondimensional voltage versus reduced flow velocity parameter space, the two solution methods predict the same Hopf bifurcation point, as shown in Figure 7(c) and (d). For a small value of ${\overline{K}}_3$ , the HBM using only the fundamental harmonic fails to accurately predict quantitatively the magnitude of the mechanical (displacement) but fairly predicts that of the electrical (voltage) responses of the harvester. For a large value of ${\overline{K}}_3$ , the difference in the magnitude of the mechanical response obtained using the HBM and numerical simulations becomes more obvious but remains within a reasonable range. For the output voltage, as shown in Figure 7(d), the difference remains negligible in the reduced velocity range $2.7\le \overline{U}\le 7$ , but starts increasing from $\overline{U}\ge 7$ . However, the HBM predicts with reasonable accuracy the qualitative behaviour, both mechanical and electrical of the harvester. In summary, the analytical solution obtained with HBM considering only the fundamental harmonic can be used to study the quantitative and qualitative behaviour of the electrical (voltage) responses of the QZS nonlinear GPEH with both low and large values of the cubic stiffness.

Figure 7.

Comparison between the analytical predictions and the numerical integration using the parameters of the system listed in Table 1: amplitude of nondimensional displacement versus reduced velocity (a) Case I ( ${\overline{K}}_3=1)$ and (b) Case II ( ${\overline{K}}_3=20)$ and amplitude of nondimensional voltage vs reduced velocity (c) Case I ( ${\overline{K}}_3=1)$ and (d) Case II ( ${\overline{K}}_3=20)$ .

Relative performance study

In this section, the performance of the QZS GPEH is compared to those of the conventional linear GPEH and GPEHs with different nonlinear restoring forces. For a fair comparison, the comparative study will be performed at the optimum values of the electrical to mechanical time-constant ratio ${\alpha }_{\mathrm{opt}}$ , using the properties of the harvester listed in Table 1. The optimum value of the electrical to mechanical time-constant ratio ${\alpha }_{\mathrm{opt}}$ , is the value of $\alpha$ which results in the maximum power output, while keeping other parameters constant.

Comparative study of QZS GPEH and linear GPEH

In this section, the potential advantages of the QZS nonlinear GPEH and the conventional linear GPEH are analysed and compared in terms of the mechanical response and the electrical response. For the QZS GPEH, the maximum nondimensional power output is obtained at ${\alpha }_{\mathrm{opt}}$ and ${\alpha }_{\mathrm{opt}}$ is obtained by differentiating equation (30) with respect to $\alpha$ and setting the result to zero, that is,

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d\overline{P}}{d\alpha }=}\\ \multicolumn{1}{c}{\frac{4\overline{U}\mathrm{\Omega }[4\overline{m}({\alpha }^2{\mathrm{\Omega }}^2-1)({\zeta }_m+\alpha \kappa +{\zeta }_m{\alpha }^2{\mathrm{\Omega }}^2)+s_1(\overline{U}-\overline{U}{\alpha }^4{\mathrm{\Omega }}^4)]}{3s_3{(1+{\alpha }^2{\mathrm{\Omega }}^2)}^3}=0,}\end{array}$$

(35)

which can be solved for $\alpha$ to obtain

$${\alpha }_{\mathrm{opt}1}=\frac{1}{\mathrm{\Omega }},$$

(36)

or

$$\begin{array}{c}\multicolumn{1}{c}{{\alpha }_{\mathrm{opt}2,3}=}\\ \multicolumn{1}{c}{\frac{-4\overline{m}\kappa \pm \sqrt{16{\overline{m}}^2{\kappa }^2-4(4\overline{m}{\zeta }_m-\overline{U}{s}_1)(4\overline{m}{\zeta }_m{\mathrm{\Omega }}^2-\overline{U}{\mathrm{\Omega }}^2{s}_1)}}{2(4\overline{m}{\zeta }_m{\mathrm{\Omega }}^2-\overline{U}{\mathrm{\Omega }}^2{s}_1)}.}\end{array}$$

(37)

When $\alpha ={\alpha }_{\mathrm{opt}1}=1/\mathrm{\Omega }$ , the frequency of the optimised QZS GPEH becomes

$$\mathrm{\Omega }=\frac{\sqrt{2\kappa +3{\overline{K}}_3{\overline{A}}^2}}{2}.$$

(38)

Hence the maximum nondimensional power of the QZS GPEH at ${\alpha }_{\mathrm{opt}1}=1/\mathrm{\Omega }$ from equation (30) is written as

$${\overline{P}}_{\max }=\frac{2\overline{U}}{3s_3}(\overline{U}{s}_1-\frac{2\overline{m}\kappa }{\sqrt{2\kappa +3{\overline{K}}_3{\overline{A}}^2}}-4\overline{m}{\zeta }_m).$$

(39)

When $\alpha ={\alpha }_{\mathrm{opt}2,3}$ as given by equation (37), the maximum nondimensional power is the same for both values of ${\alpha }_{\mathrm{opt}2,3}$ (each simply referred to as ${\alpha }_{\mathrm{opt}2}$ ) and is written as

$${\overline{P}}_{\max }=\frac{\overline{U}\mathrm{\Omega }{(\overline{U}{s}_1-4\overline{m}{\zeta }_m)}^2}{6s_3\overline{m}\kappa },$$

(40)

where $\mathrm{\Omega }$ (the frequency of the optimised solution) is given as

$$\mathrm{\Omega }=\sqrt{f_1+\frac{1}{32}\sqrt{f_2+{\left(f_3\right)}^2}},$$

(41)

and

$$\{\begin{array}{c}{f}_1=\frac{\kappa }{2}+\frac{3{\overline{K}}_3{\overline{A}}^2}{4}-\frac{{\zeta }_m^2}{2}+\frac{\overline{U}{\zeta }_m{s}_1}{4\overline{m}}-\frac{{\overline{U}}^2{s}_1^2}{32{\overline{m}}^2}\hfill \\ \multicolumn{1}{c}{f_2=-192(4\kappa {\overline{K}}_3{\overline{A}}^2+3{\overline{K}}_3^2{\overline{A}}^4)}\\ \multicolumn{1}{c}{f_3=16({\zeta }_m^2-\kappa -\frac{3{\overline{K}}_3{\overline{A}}^2}{2})-\frac{8\overline{U}{\zeta }_m{s}_1}{\overline{m}}+\frac{{\overline{U}}^2{s}_1^2}{{\overline{m}}^2}}\end{array}.$$

Substituting equation (41) into (40), the maximum nondimensional power is obtained as

$${\overline{P}}_{\max }=p_1\sqrt{p_2+\sqrt{p_3}},$$

(42)

where

$$\{\begin{array}{c}{p}_1=\frac{\overline{U}{(\overline{U}{s}_1-4\overline{m}{\zeta }_m)}^2}{24\sqrt{2}\overline{m}\kappa s_3}\hfill \\ \multicolumn{1}{c}{p_2=3{\overline{K}}_3{\overline{A}}^2-16{\zeta }_m^2+16\kappa +\frac{8\overline{U}\overline{m}{\zeta }_m{s}_1-{\overline{U}}^2{s}_1^2}{{\overline{m}}^2}}\\ \multicolumn{1}{c}{p_3=\frac{-48{\overline{m}}^2{\overline{K}}_3{\overline{A}}^2{(-4\overline{m}{\zeta }_m+\overline{U}{s}_1)}^2+{[16{\overline{m}}^2({\zeta }_m^2-\kappa -2)+\overline{U}{s}_1(\overline{U}{s}_1-8\overline{m}{\zeta }_m)]}^2}{{\overline{m}}^4}}\end{array}$$

Similarly, for the linear GPEH, when $\alpha ={\alpha }_{\mathrm{opt}4}=1/{\mathrm{\Omega }}_L$ , the nondimensional frequency of the optimised linear GPEH becomes

$${\mathrm{\Omega }}_L=\frac{\sqrt{2+\kappa }}{\sqrt{2}}.$$

(43)

For $\kappa <<1,{\mathrm{\Omega }}_L\approx 1$ . This implies that the galloping frequency is approximately equal to the natural frequency of the harvester and is consistent with results obtained from experimental (Ewere et al., 2014; Zhao et al., 2012; Zhao and Yang, 2015) and analytical (Abdelkefi et al., 2012; Bibo and Daqaq, 2014; Sirohi and Mahadik, 2012) studies of SDOF linear GPEHs.

By substituting equation (43) into equation (34), the maximum nondimensional power of the linear GPEH at ${\alpha }_{\mathrm{opt}4}=1/{\mathrm{\Omega }}_L$ can be written as

$${\overline{P}}_{\mathrm{Lmax}}=\frac{2\overline{U}(\overline{U}{s}_1-\frac{\sqrt{2}\overline{m}\kappa }{\sqrt{2+\kappa }}-4\overline{m}{\zeta }_m)}{3s_3}.$$

(44)

When

$$\begin{array}{c}\multicolumn{1}{c}{{\alpha }_{\mathrm{opt}5,6}=}\\ \multicolumn{1}{c}{\frac{-4\overline{m}\kappa \pm \sqrt{16{\overline{m}}^2{\kappa }^2-4(4\overline{m}{\zeta }_m-\overline{U}{s}_1)(4\overline{m}{\zeta }_m{\mathrm{\Omega }}_L^2-\overline{U}{\mathrm{\Omega }}_L^2{s}_1)}}{2(4\overline{m}{\zeta }_m{\mathrm{\Omega }}_L^2-\overline{U}{\mathrm{\Omega }}_L^2{s}_1)},}\end{array}$$

(45)

the maximum nondimensional power is the same for both values of ${\alpha }_{\mathrm{opt}5,6}$ (each simply referred to as ${\alpha }_{\mathrm{opt}5}$ ) and is written as

$${\overline{P}}_{\mathrm{Lmax}}=\frac{\overline{U}{\mathrm{\Omega }}_L{(\overline{U}{s}_1-4\overline{m}{\zeta }_m)}^2}{6s_3\overline{m}\kappa },$$

(46)

where ${\mathrm{\Omega }}_L$ (the frequency of the optimised solution) is given as

$${\mathrm{\Omega }}_L=\sqrt{1-f_4+\frac{1}{32}\sqrt{f_5+{\left(f_6\right)}^2}},$$

(47)

and

$$\{\begin{array}{c}{f}_4=\frac{{\zeta }_m^2}{2}+\frac{\kappa }{2}+\frac{\overline{U}{\zeta }_m{s}_1}{4\overline{m}}-\frac{{\overline{U}}^2{s}_1^2}{32{\overline{m}}^2}\hfill \\ \multicolumn{1}{c}{f_5=1024(\kappa -1)}\\ \multicolumn{1}{c}{f_6=16({\zeta }_m^2-\kappa -2)-\frac{8\overline{U}{\zeta }_m{s}_1}{\overline{m}}+\frac{{\overline{U}}^2{s}_1^2}{{\overline{m}}^2}}\end{array}.$$

Substituting equation (47) into (46), the maximum nondimensional power is obtained as

$${\overline{P}}_{\mathrm{Lmax}}=p_1\sqrt{p_4+\sqrt{p_5}},$$

(48)

where

$$\{\begin{array}{c}{p}_4=32-16{\zeta }_m^2+16\kappa +\frac{8\overline{U}\overline{m}{\zeta }_m{s}_1-{\overline{U}}^2{s}_1^2}{{\overline{m}}^2}\hfill \\ \multicolumn{1}{c}{p_5=1024(\kappa -1)+\frac{{[16{\overline{m}}^2({\zeta }_m^2-\kappa -2)+\overline{U}{s}_1(\overline{U}{s}_1-8\overline{m}{\zeta }_m)]}^2}{{\overline{m}}^4}}\end{array}.$$

As stated above, both values of ${\alpha }_{\mathrm{opt}2,3}$ given by equation (37) and both values of ${\alpha }_{\mathrm{opt}5,6}$ given by equation (45), yield the same value of the maximum output power. A sensitivity analysis is performed to provide more insight into this behaviour by varying the nondimensional electromechanical coupling coefficient $\kappa$ while keeping other parameters of the harvester listed in Table 1 constant. Figure 8 shows the variation of ${\alpha }_{\mathrm{opt}}$ with respect to $\kappa$ , for different values of $\overline{U}$ . It can be observed that in both cases considered, ${\alpha }_{\mathrm{opt}4}$ is constant (i.e. ${\alpha }_{\mathrm{opt}4}\approx 1$ ) and is the result of linear impedance matching (Bibo and Daqaq, 2014). ${\alpha }_{\mathrm{opt}5}$ and ${\alpha }_{\mathrm{opt}6}$ have the same value up to the point when ${\alpha }_{\mathrm{opt}}=0.9$ . Beyond this point, ${\alpha }_{\mathrm{opt}5}$ decreases with $\kappa$ , while ${\alpha }_{\mathrm{opt}6}$ increases with $\kappa$ , and the values of these two optimal electric loads result in the same maximum output power. It should be noted that $\alpha$ is optimal when the external electric load, $R$ embedded within $\alpha$ is optimal, that is, ${\alpha }_{\mathrm{opt}}={\omega }_n{R}_{\mathrm{opt}}{C}_p$ . Since both ${\alpha }_{\mathrm{opt}5}$ and ${\alpha }_{\mathrm{opt}6}$ produce equal maximum output power, the energy harvesting circuit can be designed to operate in two different modes: the high current/low voltage mode with low optimal electric load or the high voltage/low current mode with large optimal electric load.

Figure 8.

Variation of the optimum mechanical to electrical time-constant ratio ( ${\alpha }_{\mathrm{opt}}$ ) with nondimensional electromechanical coupling coefficient ( $\kappa$ ) for different values of the reduced velocity of the linear GPEH: (a) $\overline{U}=8$ , and (b) $\overline{U}=10$ . Dash-dotted line – magenta represents ${\alpha }_{\mathrm{opt}4}$ ; solid line – blue represents ${\alpha }_{\mathrm{opt}5}$ ; and dotted line – red represents ${\alpha }_{\mathrm{opt}6}$ . The other parameters are the same as listed in Table 1.

To determine which of these optimisation results ( ${\alpha }_{\mathrm{opt}1}$ or ${\alpha }_{\mathrm{opt}2}$ ) provides the highest nondimensional power for the QZS GPEH, and ${\alpha }_{\mathrm{opt}4}$ or ${\alpha }_{\mathrm{opt}5}$ for the linear GPEH, a comparison is performed by setting ${\overline{K}}_3=1$ . The results are depicted in Figure 9. For the QZS GPEH, it is observed from Figure 9(a) that the maximum nondimensional power obtained based on ${\alpha }_{\mathrm{opt}2}$ is greater than the maximum nondimensional power obtained based on ${\alpha }_{\mathrm{opt}1}$ . For the linear GPEH, it is clearly seen that for $\overline{U}>5$ , the maximum nondimensional power obtained based on ${\alpha }_{\mathrm{opt}5}$ becomes greater than the maximum nondimensional power obtained based on ${\alpha }_{\mathrm{opt}4}$ . This result shows that to optimise the energy harvested by a linear GPEH, ${\alpha }_{\mathrm{opt}5}$ should be used rather than ${\alpha }_{\mathrm{opt}4}$ . Hence, to compare the performance of the QZS nonlinear GPEH with that of the linear GPEH, the nondimensional maximum power will be based on the optimisation using ${\alpha }_{\mathrm{opt}2}$ for the QZS GPEH and ${\alpha }_{\mathrm{opt}5}$ for the linear GPEH.

Figure 9.

Variation of the maximum nondimensional power with reduced velocity for the different values of the optimum mechanical to electrical time-constant ratio: (a) QZS GPEH ( ${\overline{K}}_3=1$ ) and (b) linear GPEH. Dashed line – black represents ${\alpha }_{\mathrm{opt}1}$ and blue represents ${\alpha }_{\mathrm{opt}4}$ while the solid line – magenta represents ${\alpha }_{\mathrm{opt}2}$ and red ${\alpha }_{\mathrm{opt}5}$ . The other parameters are those of the harvester listed in Table 1.

Table 2 shows a set of aerodynamic coefficients that characterise square-sectioned bluff bodies and the mechanical damping ratio of the energy harvesting beams used in Bibo et al. (2015), Chen et al. (2022), Ewere and Wang (2013) and Zhao et al. (2016). It can be seen in Table 2, that the percentage difference between the highest and lowest values of $s_1$ is relatively small, while the percentage differences between the highest and lowest values of ${\zeta }_m$ and $s_3$ are high. To investigate the effect of these parameters on the performance of the GPEHs, two different parameter sets, referred to as Set I and Set II, are defined to examine both low and high values reported in the literature. Set I parameters are $s_1=2.5$ , and $s_3=10$ (low value of $s_3$ ), and ${\zeta }_m=0.002$ (low damping); while Set II parameters are $s_1=2.5$ , and $s_3=168.4$ (high value of $s_3$ ), and ${\zeta }_m=0.011$ (high damping). ${\alpha }_{\mathrm{opt}4}$ (equation (44)) is commonly used in the literature to obtain the maximum power and is used here to obtain the nondimensional power for the linear GPEH as functions of $\overline{m}{\zeta }_m$ and $\overline{m}\kappa$ for given values of $\overline{U}$ , $s_1$ , and $s_3$ , as shown in Figure 10.

Table 2.

Aerodynamic coefficients and mechanical damping ratios of different galloping piezoelectric energy harvesting systems.

Parameter (symbol)	Value
	Ewere and Wang (2013)	Chen et al. (2022)	Bibo et al. (2015)	Zhao et al. (2016)
Mechanical damping ratio $({\zeta }_m)$	0.002	0.01	0.003	0.011
Linear aerodynamic coefficient $(s_1)$	2.69	2.3	2.5	2.69
Cubic aerodynamic coefficient $(s_3)$	93.33	10	130	168.4

Figure 10.

Isolines of maximum nondimensional power as a function of $\overline{m}{\zeta }_m$ and $\overline{m}\kappa$ for a linear GPEH at $\overline{U}=10$ : (a) Set I and (b) Set II.

It can be seen in Figure 10 that Set I produces high power, and Set II produces low power. To compare GPEHs that produce both high and low power, two points, one for each set, which are physically obtainable, are randomly selected in Figure 10. The other parameters of the system are based on these two points. For Set I, considering point $P_1$ ( $\overline{m}{\zeta }_m=4;\overline{m}\kappa =3;$ and ${\overline{P}}_{\mathrm{Lmax}}=3.9$ ) of Figure 10(a), the other parameters of the system are $\overline{m}=2000$ , and $\kappa =0.0015$ . Similarly, for Set II, considering point $P_2$ ( $\overline{m}{\zeta }_m=2;\overline{m}\kappa =1.5;$ and ${\overline{P}}_{\mathrm{Lmax}}=0.6$ ) of Figure 10(b), the other parameters of the system are $\overline{m}=182$ and $\kappa =0.0083$ .

The result of the comparison between the QZS nonlinear GPEH and linear GPEH is shown in Figure 11. It can be seen from Figure 11(a; Set I) and Figure 11(b; Set II), that the QZS nonlinear GPEH outperforms the linear GPEH. Also, from Figure 11, it is observed that the onset of galloping is the same for both the QZS and the linear system. This can be explained using equations (28a) and (31). From both equations, it is observed that, for given values of $\alpha$ and $\kappa$ , the real positive values of the frequency of dynamic response are always the same for both the linear and QZS nonlinear systems. Furthermore, it can be seen from Figure 11 that the onset of galloping using Set I parameters is higher compared to that using Set II parameters. This is because for both the QZS nonlinear GPEH and the linear GPEH, higher values of the mass parameter, $\overline{m}$ , result in higher values of the cut-in reduced velocities (see equations (28a) and (32b)).

Figure 11.

Variation of the nondimensional output power with the reduced velocity: (a) Set I (low value of $s_3$ and ${\zeta }_m$ ) and (b) Set II (high value of $s_3$ and ${\zeta }_m$ ). Linear (dashed – blue) represent linear GPEH, and QZS (solid – red) represent QZS nonlinear GPEH ( ${\overline{K}}_3$  = 1).

Comparative study of SDOF GPEHs with different types of nonlinear restoring forces

In this section, the relative performance of a SDOF GPEH with different types of nonlinear restoring forces is investigated numerically. The system shown in Figure 2 becomes bi-stable when

$$\beta >\frac{\gamma }{2(1-\gamma )},$$

(49a)

or

$$\gamma <\frac{2\beta }{1+2\beta },$$

(49b)

in this configuration, ${\overline{K}}_1<0$ , and ${\overline{K}}_3>0$ . Similarly, the system shown in Figure 2 becomes hardening when

$$\beta <\frac{\gamma }{2(1-\gamma )},$$

(50a)

or

$$\gamma >\frac{2\beta }{1+2\beta },$$

(50b)

in this configuration, ${\overline{K}}_1>0$ and ${\overline{K}}_3>0$ . Finally, to achieve the softening nonlinearity, the linear oblique springs used in the system shown in Figure 2 are replaced with nonlinear oblique springs which are softening, with linear stiffness and cubic softening nonlinear stiffness (Kovacic et al., 2008). In such a configuration, the equivalent linear stiffness ( ${\overline{K}}_{1\mathrm{s}})$ is positive, while the equivalent cubic stiffness ( ${\overline{K}}_{3\mathrm{s}})$ is negative. The nondimensional potential energy (PE) of the three-linear spring mechanism can be obtained by integrating the force given by equation (16) with respect to $\overline{x}$ to give

$$\overline{\mathcal{V}}=\frac{1}{2}{\overline{K}}_1{\overline{x}}^2+\frac{1}{4}{\overline{K}}_3{\overline{x}}^4.$$

(51)

The nondimensional PE of the softening nonlinearity is estimated by replacing ${\overline{K}}_1$ with ${\overline{K}}_{1\mathrm{s}}$ , and ${\overline{K}}_3$ with ${\overline{K}}_{3\mathrm{s}}$ , respectively, in equation (51). The comparisons here will be based on equal potential energy or equal magnitude of the restoring force due to a unit displacement of the system. This criterion is chosen because if the comparison is based on equal magnitudes of effective linear stiffness and effective cubic stiffness for the hardening, softening, and bi-stable as in Bibo et al. (2015), it will be impossible to have a QZS nonlinearity with the same magnitude of effective linear stiffness as those of a hardening, softening, or bi-stable nonlinearity. The parameters listed in Table 1 are used to perform the simulations. Figure 12(a) shows the nondimensional PE diagram due to a unit deflection, for the hardening ( ${\overline{K}}_1=5$ and ${\overline{K}}_3=10)$ , softening ( ${\overline{K}}_{1\mathrm{s}}=20$ and ${\overline{K}}_{3\mathrm{s}}=-20)$ , bi-stable ( ${\overline{K}}_1=-10$ and ${\overline{K}}_3=40)$ , and QZS ( ${\overline{K}}_1=0$ and ${\overline{K}}_3=20)$ nonlinearities. Figure 12(b) shows the nondimensional restoring force diagram due to a unit deflection, for the hardening ( ${\overline{K}}_1=10$ and ${\overline{K}}_3=10)$ , softening ( ${\overline{K}}_{1\mathrm{s}}=30$ and ${\overline{K}}_{3\mathrm{s}}=-10)$ , bi-stable ( ${\overline{K}}_1=-20$ and ${\overline{K}}_3=40)$ , and QZS ( ${\overline{K}}_1=0$ and ${\overline{K}}_3=20)$ nonlinearities.

Figure 12.

(a) Nondimensional potential function versus nondimensional displacement for the hardening ( ${\overline{K}}_1=5$ and ${\overline{K}}_3=10)$ , softening ( ${\overline{K}}_{1s}=20$ and ${\overline{K}}_{3s}=-20)$ , bi-stable ( ${\overline{K}}_1=-10$ and ${\overline{K}}_3=40)$ , and QZS ( ${\overline{K}}_1=0$ and ${\overline{K}}_3=20)$ nonlinearities, and (b) nondimensional restoring force vs nondimensional displacement, for the hardening ( ${\overline{K}}_1=10$ and ${\overline{K}}_3=10)$ , softening ( ${\overline{K}}_{1\mathrm{s}}=30$ and ${\overline{K}}_{3\mathrm{s}}=-10)$ , bi-stable ( ${\overline{K}}_1=-20$ and ${\overline{K}}_3=40)$ , and QZS ( ${\overline{K}}_1=0$ and ${\overline{K}}_3=20)$ nonlinearities. Dash-dotted line – red represents hardening; dotted line – black represents softening; solid line – magenta represents bi-stable; and dashed line – blue represents QZS.

For a fair comparison of the performance of each SDOF GPEH with the different types of nonlinear restoring force elements considered namely: hardening, softening, bi-stable, and QZS, the results are obtained at the individual optimal mechanical to electrical time-constant ratio, ${\alpha }_{\mathrm{opt}}$ that results in the power peaks.

Comparison based on potential energy ( $\overline{\mathcal{V}}$ )

Figure 13 depicts the electrical and dynamic characteristics when the nondimensional PE due to a unit nondimensional displacement is the same for the four configurations. Two cases are considered based on the value of the nondimensional PE, namely: Case III when the nondimensional PE, $\overline{\mathcal{V}}$ is equal to 5, and Case IV when the nondimensional PE, $\overline{\mathcal{V}}$ is equal to 10. Table 3 lists the values of the equivalent linear and cubic stiffness of the four nonlinear GPEHs for Cases III and IV. The variation of the nondimensional power output with the mechanical to electrical time-constant ratio for the different types of nonlinear GPEHs being compared at $\overline{U}=8$ which is shown in Figure 13(a; Case III) and Figure 13(b; Case IV). Figure 13(c; Case III) and Figure 13(d; Case IV) show the variation of the nondimensional power output with the reduced flow velocity for the different types of nonlinear GPEHs being compared, while Figure 13(d) and (e) shows the phase portrait of Case III for the different types of nonlinear GPEHs being compared at $\overline{U}=7.5$ and $\overline{U}=7.8$ , respectively.

Figure 13.

The characteristics of the four nonlinear GPEHs when the nondimensional PE due to a unit nondimensional displacement is equal for all four nonlinear elements: (a and b) nondimensional power for varying mechanical to electrical time-constant ratio at $\overline{U}=8$ , (a) Case III ( $\overline{\mathcal{V}}=5$ ) and (b) Case IV ( $\overline{\mathcal{V}}=10$ ); (c and d) nondimensional power for varying reduced flow velocities at ${\alpha }_{\mathrm{op}t}$ , (c) Case III and (d) Case IV; (e and f) phase portrait for Case III, (e) $\overline{U}=7.5$ and (f) $\overline{U}=7.8$ . Case III: H ( ${\overline{K}}_1=5$ and ${\overline{K}}_3=10)$ , S ( ${\overline{K}}_{1\mathrm{s}}=20$ and ${\overline{K}}_{3\mathrm{s}}=-20)$ , B ( ${\overline{K}}_1=-10$ and ${\overline{K}}_3=40)$ , and Q ( ${\overline{K}}_1=0$ and ${\overline{K}}_3=20)$ . Case IV: H ( ${\overline{K}}_1=5$ and ${\overline{K}}_3=10)$ , S ( ${\overline{K}}_{1\mathrm{s}}=20$ and ${\overline{K}}_{3\mathrm{s}}=-20)$ , B ( ${\overline{K}}_1=-10$ and ${\overline{K}}_3=40)$ , and Q ( ${\overline{K}}_1=0$ , and ${\overline{K}}_3=20)$ . H represents hardening (dash-dotted line/hexagram marker – red); S for softening (dotted line/downward-pointing triangle marker – black); B for bi-stable (solid line/pentagram marker – magenta); and Q for QZS ( dashed line/square marker – blue) GPEH.

Table 3.

Values of the equivalent linear and cubic stiffness and their corresponding harvesting systems for Case III and Case IV.

Type of GPEH	Case III		Case IV
	${\overline{K}}_1$ , ${\overline{K}}_{1s}$	${\overline{K}}_3$ , ${\overline{K}}_{3s}$	${\overline{K}}_1$ , ${\overline{K}}_{1s}$	${\overline{K}}_3$ , ${\overline{K}}_{3s}$
Hardening GPEH	$5$	$10$	$10$	$20$
Softening GPEH	$20$	$-20$	$30$	$-20$
Bi-stable GPEH	$-10$	$40$	$-5$	$50$
QZS GPEH	$0$	$20$	$0$	$40$

From Figure 13(a) and (b), the optimum values of the mechanical to electrical time-constant ratio, ${\alpha }_{\mathrm{opt}}$ of the four nonlinear GPEHs being compared are obtained. The values of ${\alpha }_{\mathrm{opt}}$ for Case III are: hardening ( ${\alpha }_{\mathrm{opt}}=0.38)$ , softening ( ${\alpha }_{\mathrm{opt}}=0.19)$ , bi-stable ( ${\alpha }_{\mathrm{opt}}=$ 0.92), and QZS ( ${\alpha }_{\mathrm{opt}}=0.56)$ . For Case IV, the values of ${\alpha }_{\mathrm{opt}}$ are: hardening ( ${\alpha }_{\mathrm{opt}}=0.31)$ , softening ( ${\alpha }_{\mathrm{opt}}=0.19)$ , bi-stable ( ${\alpha }_{\mathrm{opt}}=$ 0.56), and QZS ( ${\alpha }_{\mathrm{opt}}=0.45)$ . It is seen from Figure 13(c) and (d) that the QZS harvester outperforms all the other configurations up to the point when the bi-stable harvester escapes the single potential well and goes into inter-well oscillations (as shown in Figure 13(d) and (e) for Case III), resulting in better power harvesting performance than the QZS and the other harvesters. Also, for the bi-stable GPEH, for Case III ( $\overline{\mathcal{V}}=5$ ) between $7.55<\overline{U}<7.75$ , there is a jump of the stable periodic orbits from intra-well oscillations to inter-well oscillations, while for Case IV ( $\overline{\mathcal{V}}=10$ ) the snap-through occurs between $4.3<\overline{U}<4.5$ .

It can be summarised that based on equal potential energy for the four restoring force elements, a GPEH designed with a QZS restoring force will outperform all other configurations, as long as the oscillations are confined in single potential wells, while a GPEH designed with a bistable restoring force has superior performance only when inter-well oscillations are activated. Furthermore, a bi-stable GPEH with a higher potential energy will snap-through from intra-well to inter-well oscillations at a lower reduced flow velocity, compared to one with a lower potential energy.

Comparison based on the magnitude of restoring force ( ${\overline{f}}_n$ )

The power harvesting potentials of the four configurations based on the equal magnitude of the restoring force due to a unit nondimensional displacement are investigated in this section. Figure 14 shows the performance and dynamic characteristics when the magnitude of the restoring force due to a unit nondimensional displacement is the same for the four configurations. Again, two cases are investigated: Case V, when the nondimensional restoring force ${\overline{f}}_n$ is equal to 20, and Case VI, when the nondimensional force ${\overline{f}}_n$ is equal to 40. Table 4 lists the values of the equivalent linear and cubic stiffness of the four nonlinear GPEHs for Cases V and VI.

Figure 14.

The behaviour of the four nonlinear GPEHs when the nondimensional restoring force due to a unit nondimensional displacement is equal for all four nonlinear elements: (a and b) nondimensional power for varying mechanical to electrical time-constant ratio at $\overline{U}=8$ , (a) Case V ( ${\overline{f}}_n=20$ ) and (b) Case VI ( ${\overline{f}}_n=40$ ); (c and d) nondimensional power for varying reduced flow velocities at ${\alpha }_{\mathrm{opt}}$ , (c) Case V and (d) Case VI; (e and f) phase portrait for Case VI (e) $\overline{U}=7$ and (f) $\overline{U}=7.2$ . Case V: H ( ${\overline{K}}_1=10$ and ${\overline{K}}_3=10)$ , S ( ${\overline{K}}_{1\mathrm{s}}=30$ and ${\overline{K}}_{3\mathrm{s}}=-10)$ , B ( ${\overline{K}}_1=-20$ and ${\overline{K}}_3=40)$ , and Q ( ${\overline{K}}_1=0$ and ${\overline{K}}_3=20)$ . Case VI: H ( ${\overline{K}}_1=10$ and ${\overline{K}}_3=30)$ , S ( ${\overline{K}}_{1\mathrm{s}}=50$ and ${\overline{K}}_{3\mathrm{s}}=-10)$ , B ( ${\overline{K}}_1=-10$ , and ${\overline{K}}_3=50)$ , and Q ( ${\overline{K}}_1=0$ and ${\overline{K}}_3=40)$ . H represents hardening (dash-dotted line/hexagram marker – red); S for softening (dotted line/downward-pointing triangle marker – black); B for bi-stable (solid line/pentagram marker – magenta); and Q for QZS (dashed line/square marker – blue) GPEH.

Table 4.

Values of the equivalent linear and cubic stiffness and their corresponding harvesting systems for Case V and Case VI.

Type of GPEH	Case V		Case VI
	${\overline{K}}_1$ , ${\overline{K}}_{1\mathrm{s}}$	${\overline{K}}_3$ , ${\overline{K}}_{3\mathrm{s}}$	${\overline{K}}_1$ , ${\overline{K}}_{1\mathrm{s}}$	${\overline{K}}_3$ , ${\overline{K}}_{3\mathrm{s}}$
Hardening GPEH	$10$	$10$	$10$	$30$
Softening GPEH	$30$	$-10$	$50$	$-10$
Bi-stable GPEH	$-20$	$40$	$-10$	$50$
QZS GPEH	$0$	$20$	$0$	$40$

Figure 14(a; Case V) and Figure 14(b; Case VI) show the variation of nondimensional power with the mechanical to electrical time-constant ratio at $\overline{U}=8$ , from which the respective ${\alpha }_{\mathrm{opt}}$ can be obtained for each of the four configurations. The values of ${\alpha }_{\mathrm{opt}}$ for Case V are: hardening ( ${\alpha }_{\mathrm{opt}}=0.31)$ , softening ( ${\alpha }_{\mathrm{opt}}=0.19)$ , bi-stable ( ${\alpha }_{\mathrm{opt}}=0.19$ ), and QZS ( ${\alpha }_{\mathrm{opt}}=0.56)$ . For Case VI, the values of ${\alpha }_{\mathrm{opt}}$ are: hardening ( ${\alpha }_{\mathrm{opt}}=0.31)$ , softening ( ${\alpha }_{\mathrm{opt}}=0.13)$ , bi-stable ( ${\alpha }_{\mathrm{opt}}=$ 0.62), and QZS ( ${\alpha }_{\mathrm{opt}}=0.44)$ . It is observed from Figure 14(c; Case V) that the QZS harvester can harvest more power than the other configurations, and the oscillations of the bistable system are confined to a single potential well for the range of reduced flow velocity considered. Different performance and dynamic characteristics of the four configurations are obtained when the magnitude of the restoring force is equally increased to 40 due to a unit nondimensional displacement (Case VI) as shown in Figure 14(d). It is clearly shown in Figure 14(d) that the harvester design with a QZS nonlinearity harvests more power than the other configurations for $3.5<\overline{U}<7$ . In this reduced flow velocity range, t the energy harvesting characteristics of the softening and bi-stable harvesters are similar. At $\overline{U}=7$ , the dynamic trajectories of the bi-stable harvester escape the single potential well resulting in cross-well oscillations. As a result, at reduced velocities greater than seven, $\overline{U}>7$ , the bi-stable harvester harvests more power than the other configurations. Figure 14(e) and (f) show the phase portrait of the four nonlinear systems for Case VI at $\overline{U}=7$ and $\overline{U}=7.2$ , respectively. It is observed that at $\overline{U}=7$ , the bi-stable harvester is confined in a single potential well, while at $\overline{U}=7.2$ , the bi-stable harvester is undergoing inter-well oscillations.

It can be summarised that based on an equal magnitude of restoring force for the four different restoring force elements, a GPEH designed with a QZS restoring force performs better than the other configurations, as long as the oscillations are confined in single potential wells, while a GPEH designed with a bi-stable restoring force has superior performance only when inter-well oscillations are activated. Also, a bi-stable SDOF GPEH with a higher magnitude of restoring force will snap-through from intra-well to inter-well oscillations at a lower reduced flow velocity, compared to one with a lower magnitude of restoring force.

Case study

In this subsection, the proposed QZS nonlinear GPEH is compared with conventional GPEHs from previous studies that the authors performed experimental validation. Two cases are considered, and the properties of the harvesters are listed in Table 5. In each case, the properties of the QZS nonlinear GPEH is the same as those of the harvester being compared, and ${\overline{K}}_3=1$ . Also, the comparison is performed at ${\alpha }_{\mathrm{opt}}$ as discussed in subsection 5.1. Figure 15 shows the results of the comparison. It can be observed from both cases, that the proposed QZS has the capacity to harvest more maximum power than the conventional GPEH.

Table 5.

Nondimensional properties of two galloping piezoelectric energy harvesting systems.

Parameter (symbol)	Value
	Ewere and Wang (2013)	Zhao et al. (2016)
Mechanical damping ratio $({\zeta }_m)$	0.002	0.011
Harvester to flow mass ratio $(\overline{m})$	172	61
Electrical to mechanical time-constant ratio $(\alpha )$	0.443	0.896
Nondimensional electromechanical coupling $(\kappa )$	0.0234	0.0114
Linear aerodynamic coefficient $(s_1)$	2.69	2.69
Cubic aerodynamic coefficient $(s_3)$	93.33	168.4

Figure 15.

Variation of the maximum nondimensional output power with the reduced velocity using the parameters of the systems listed in Table 5: (a) QZS versus linear (Ewere and Wang, 2013) and (b) QZS versus linear (Zhao et al., 2016). Linear (dashed – blue) represent linear GPEH (Ewere and Wang, 2013), Linear (dash-dotted – magenta) represent linear GPEH (Zhao et al., 2016), and QZS (solid – red) represent QZS nonlinear GPEH ( ${\overline{K}}_3$  = 1).

Discussions

One of the design requirements for good performance of a linear GPEH is for the harvester to have a low natural frequency, which implies that a large mass and a low stiffness spring element is required. However, having a large mass supported on a low stiffness spring may result in a large extension of the spring and may be impractical. To overcome this limitation, a QZS nonlinear GPEH, which can support the vibrating mass without a large static deformation while maintaining a low natural frequency is utilised in this study.

It is observed from Figure 4(c) that a QZS nonlinear GPEH designed with a QZS mechanism with a stiffness-displacement curve that is lower and flatter can harvest more power than one with a stiffness-displacement curve that is higher and steeper. Hence, it can be inferred that designing a QZS nonlinear GPEH with a different QZS mechanism whose stiffness-displacement curve is much lower and flatter over a wide range of displacement can significantly improve the energy harvesting performance of the GPEH.

Using the approximate analytical solutions of the approximate coupled analytical model of the QZS nonlinear GPEH obtained via HBM, an optimisation of the power with respect to the mechanical to electrical time-constant ratio ( $\alpha$ ) reveals that the optimisation of $\alpha$ given by equation (37) results in a higher level of output power compared to the optimisation of $\alpha$ by using the traditional linear impedance matching. This is also true for the linear GPEH.

A nonlinear GPEH designed with a QZS nonlinearity performs better than its counterpart designed with a hardening or softening nonlinearity. At low velocities, when the bi-stable harvester is trapped in intra-well oscillations, the QZS harvester produces more energy than the bi-stable harvester. A harvester with the QZS nonlinearity is inferior to its bi-stable counterpart only when inter-well oscillations are activated. It is observed from Figure 13(c) and (d), that when the potential energy of the bi-stable nonlinearity is increased, the cut-in reduced flow velocity at which snap-through occurs in the harvester is reduced. This is due to a reduction in the potential barrier, that is, the potential well becomes shallower while maintaining the same unstable equilibrium position resulting in easier jumps between potential wells. The same phenomenon is observed when the magnitude of the restoring force of the bi-stable system is increased (Figure 14(c) and (d)). Finally, for all the cases considered, the hardening GPEH performs better than the softening GPEH.

Conclusions

This paper investigated the concept of exploiting the QZS nonlinearity to improve the performance of GPEH. A coupled analytical aero-electro-mechanical model of the QZS nonlinear GPEH is developed in which the QZS nonlinearity is realised using a simple system comprising a vertical linear spring in parallel with two transverse linear springs, the galloping force is modelled based on the quasi-steady hypothesis, and piezoelectric transduction is assumed. The effects of the geometrical parameter and the stiffness ratio on the transverse displacement and output power are investigated numerically, and it was shown that a GPEH with a QZS mechanism whose stiffness-displacement curve is much lower, and flatter over a wide range of displacement can significantly improve the energy harvesting performance of the GPEH.

In order to perform an analysis of the parameter space and to optimise the output power of the QZS nonlinear GPEH, an approximation of the exact expression of the restoring force of the QZS mechanism is obtained. Analytical solutions of the approximate model are obtained using the HBM, which are fairly in agreement with numerical solutions. A power optimisation analysis performed for both the QZS nonlinear GPEH, and the linear GPEH reveals that three optimal values of the mechanical to electrical time-constant ratio exist for the two systems. A comparison of the power harvesting potentials of the QZS nonlinear harvester and the linear harvester shows that the QZS nonlinear harvester has the potential to harvest more energy than its linear counterpart.

The approximate form of the restoring force of the three-spring system is considered to model different types of nonlinear restoring forces, namely: hardening, softening, and bi-stable restoring force, whose influences on the performance of GPEH are compared with that of the QZS nonlinearity. It is observed that for similar design parameters and equal magnitude of potential energy or restoring force due to a unit nondimensional displacement, of the three-spring system, the harvester designed with a QZS nonlinearity performs better than the harvester designed with hardening or softening nonlinearity. Also, a harvester with a QZS nonlinearity outperforms the bi-stable harvester as long as the oscillations of the bi-stable harvester are within a single potential well.