Piezoelectric energy harvester under parametric excitation: A theoretical and experimental investigation

Abstract

In the present work, both theoretical and experimental investigation of a vertical cantilever beam–based piezoelectric energy harvester are carried out under principal parametric resonance condition. A piezoelectric patch is attached near the fixed end of the cantilever beam along with an attached mass positioned at an arbitrary location. The extended Hamilton’s principle is used to derive the spatio-temporal equation of motion, and generalized Galerkin’s approximation is used to obtain the temporal nonlinear electromechanical governing equation of motion. The method of multiple scales is used to find the reduced modulation equations. Due to large transverse deflection and effect of rotary inertia of the attached mass, the system exhibits cubic and inertial nonlinearities. An experimental setup with slider crank mechanism–based shaker and a harvester consisting of a cantilever beam with piezoelectric patch and attached mass is designed and developed. The challenges posed by parametric resonance in crack development in the PZT and in the beam are reported. The theoretical and experimental output voltage and the power obtained are found to be in good agreement. Furthermore, a qualitative and quantitative comparative study of 17 energy harvesters has been carried out, and the normalized power densities have been compared.

Keywords

Cantilever beam piezoelectric energy harvester (PEH)principal parametric resonance nonlinear electromechanical method of multiple scales (MMS)

1. Introduction

The recent advancements in the field of sensors and actuators drastically reduce the power consumption of micro-electronic devices (Mahmoudi and Iniewski, 2016; Meindl, 1995). The next generation devices need to be self-reliant in the energy front. In this regard, these devices must be equipped with certain energy-harvesting unit, which should be capable of extracting energy from ambient sources. Available ambient energy in the form of solar, wind, flowing water, and vibration is sufficient to power such devices. Several transduction mechanisms (Beeby and White, 2010; Priya and Inman, 2009)—namely, electromagnetic, electrostatic, piezoelectric, and electrochemical—are exploited to convert these available energy into electricity. Among them, the piezoelectric-based transduction mechanism has advantage over others because of its high power density (Anton and Sodano, 2007) that makes it suitable for micro-scale applications (Poulin et al., 2004).

PZT materials are employed extensively in sensing, actuation, and energy harvesting purpose. Vibrational energy is converted into electrical energy by structures (i.e. beams, plates, shells, and combination of basic structures) embedded with PZT patches (in $d_{31}$ mode). A beam structure (Erturk and Inman, 2008; Friswell et al., 2012) or the combination of several beam-based PEH systems (Erturk et al., 2009; Harne et al., 2016; Leadenham and Erturk, 2015; Ramírez et al., 2018; Vijayan et al., 2015) is generally used for energy harvesting purpose due to their simplicity and size. Apart from the traditional PZT materials, wide ranges of flexible piezoelectric materials (Dagdeviren et al., 2016) are also available which can be integrated with the wearable devices for sensing purpose. In recent times, the linear and nonlinear energy harvesting systems under direct and base excitations are extensively explored. The system may be forced excited, parametrically excited, or combination of both. Although a large number of literatures are available on forced excitation of the system, very few studies focus on the investigation of the energy harvesting through parametric excitation. While in direct or forced excitation, the direction of force and the displacement are in the same direction; in case of parametric excitation, these directions are orthogonal to each other. The resonance in directly excited systems occurs when the external excitation frequency matches with any of the natural frequencies of the system; in case of parametric excitation, the resonance happens when excitation frequency is twice that of the natural frequencies which is known as principal parametric resonance of the system. One may also observe combination parametric resonance of sum and difference type (Nayfeh and Mook, 2008). Faraday (1831) observed the parametric excitation while Mathieu (1868) treated the phenomenon mathematically (Kovacic and Cartmell, 2012). The existence of a time varying coefficient in the differential equation is the characteristic of parametrically excited systems (Cartmell, 1990). There are several dynamical systems where parametric resonance occurs (Fossen and Nijmeijer, 2012) and cause danger to the system. Mechanical amplification (Rhoads et al., 2008) is also one of the characteristics of such systems. Existence of such phenomenon in systems such as ships rolling (Liaw and Bishop, 1995), pipes conveying fluids (Semler and Païdoussis, 1996), rail corrugation (Wu, 2008), and rail vibration and noise (Nordborg, 1998) causes instabilities. In some cases such as flutter suppression (Ibrahim and Castravete, 2006), quenching of self-excited vibration (Fatimah and Verhulst, 2003; Tondl and Ecker, 2003), enhanced damping properties (Dohnal et al., 2008), and vibration suppression in drive systems (Ecker and Pumhössel, 2012), it acts as stabilizers as well. It also helps in improving the sensitivity of sensors (Rugar and Grütter, 1991; Zhang and Turner, 2005). By using lumped mass approach, Daqaq et al. (2009) studied the parametrically excited nonlinear PEH system consists of a cantilever beam with a tip mass. Here the effect of electromechanical coupling and load resistance on output voltage, power, and critical excitation level is analyzed. Experiments are performed to validate the results where the beam is kept in horizontal position. Similar system is analyzed by Abdelkefi et al. (2012) by considering distributed parameter model. The effect of piezoelectric coefficients and quadratic air damping on harvested power is investigated. Jia et al. (2014) investigated the feasibility of a parametrically excited electromagnetic-based energy harvester system where a significant improvement in power and frequency bandwidth is achieved. In an investigation of the dynamics of a cubic nonlinear Mathieu-type equation, Alevras et al. (2017) suggested that the parametrically excited systems as an energy harvester can be very robust one for broad frequency range. The nonlinear and nonplanar cantilever beam when subjected to parametric excitation exhibits complex response behaviors (Arafat et al., 1998). Parametrically excited system caused by time varying damper (Scapolan et al., 2016) enhances bandwidth and power of the harvester as compared to the conventional directly excited systems. Yildirim et al. (2017) investigated an array of parametrically excited beam system, and here a significant increase in the bandwidth is reported. The system is suited for low-frequency operations, and each beam provides output power in mW range.

Autoparametric vibration phenomenon is also utilized for energy transduction purpose (Jia and Seshia, 2014) where parametric and autoparametric base excitation shows the superiority over direct excitation. Yan and Hajj (2017) also exploited the autoparametric base excitation to develop a cantilever-based combined absorber and harvester system. Higher order parametric excitation can also be explored as energy harvesters in micro-electromechanical systems (MEMS) at high frequencies (Jia et al., 2016). Bobryk and Yurchenko (2016) reported a significantly improved performance of energy harvester when subjected to the random parametric excitation. Li et al. (2018) studied the coherence resonance of buckled PEH under the stochastic parametric excitation. The nonlinear dynamics of cantilever beam with parametric excitation (Anderson et al., 1996; Cartmell and Roberts, 1987; Dugundji and Mukhopadhyay, 1973; Dwivedy and Kar, 2001; Zavodney and Nayfeh, 1989) is analyzed extensively, where rich nonlinear behaviors—that is, secondary resonances, bifurcations, and chaos—has been observed. In the previous work, Garg and Dwivedy (2019) analytically analyzed a similar PEH system with 1:3 internal resonance between the first two modes. The parametric instability regions along with time and frequency responses are determined, and turning point, pitchfork and Hopf bifurcations are observed. The present study brings light on few challenges pose by the PEH system under parametric excitation.

In the present study, a vertical cantilever-based PEH system is investigated under the harmonic excitation of parametric type. A mass is attached at an arbitrary position along with a piezoelectric patch (PZT-5H). Hamilton’s principle is used to find the governing equations, which involves cubic and inertial nonlinear terms along with electromechanical coupling terms. The method of multiple scales is applied to reduce the second-order nonlinear equation to the first-order form. An in-house shaker is developed to provide harmonic excitation for experimental investigations on the PEH system. The frequency response from analytical and experimental work is compared and analyzed. The parametric excitation causes very large and violent deflection of the cantilever beam system, which brings the challenge of dynamic stability of the system and the durability of the piezoelectric patch attached to it. Large deflection may cause crack in the beam system, which not only breaks the beam from the base (which is also observed by Zavodney and Nayfeh (1989) and by present experimental work) but also severely damages the PZT patch. How these challenges affect the voltage response of the system is investigated in the present work.

2. Design and development of the experimental setup

The objective of this work is to develop a low-cost shaker to provide parametric excitation to the energy harvester. The requirement of the frequency of excitation, amplitude (stroke length), specimen dimensions, and its weight are the basic parameters on which the design and development of the harvester is carried out. The basic specimen considered in this work is a vertical steel cantilever beam having dimensions of 300 mm × 25.5 mm × 5.1 mm attached with a unimorph piezoelectric patch (50 mm × 25 mm × 0.5 mm) near to its fixed end. A mass (m) is attached at an arbitrary distance d from the fixed end. The neodymium magnets (rectangular plate form, 25 mm × 10 mm × 1 mm, weight of 4.5-g each) are used as the attached mass here so that the mass and its position are adjustable along the beam. The PZT patch (PZT-5H) by Sparkler Ceramics Pvt. Ltd. is attached near the fixed end of the cantilever beam. The PZT patch is glued on to the beam surface by using Araldite epoxy-resin (Grade AV138M/HV 998) with a 60-40 ratio.

3. Shaker design

A slider-crank mechanism (Figure 1(a)) is used to convert the rotational motion of DC motor to translational motion of the slider. A base plate is taken to mount the DC motor with the slider-crank mechanism and the electronic components. The CAD model of the setup developed using ProE is shown in Figure 1(b). To design the components, initially kinematic and dynamic analysis of the slider-crank mechanism has been carried out.

Figure 1.

(a) Schematic diagram of slider-crank mechanism and (b) CAD model of the piezoelectric energy harvester (PEH) with different parts.

4. Kinematic analysis

The schematic diagram of the mechanism is shown in Figure 1(a). The mechanism is assumed to have stroke lengths $l_{s}$ between 13 and 22 mm with a maximum frequency of 25 Hz (1500 rpm) so as to excite the harvester near twice the principal parametric resonance condition that is twice the natural frequency of the system. As the stroke length is considered to be in the range of 13–22 mm, a crank is made out of a circular disk having radius 12 mm and having slots to connect the connecting rod at 6.5, 8, 9, and 11 mm. The crank is attached to the shaft of the DC motor. In Figure 1(a), OP is the radius of the crank of length r and PQ is the connecting rod of length $l_{c}$ . A guided extension rod (QR) of length $l_{r}$ is bolted to the connecting rod at point Q in vertical position. A holder is placed over the rod QR to attach a cantilever beam, and it can accommodate a beam up to 35-mm width. As it is required to move the point Q in a straight line, a rod QR is attached to the connecting rod and it is made to move inside a guide supported by ball bearings. The crank radius r decides the stroke length (displacement of point R, $l_{s}$ ) by the relation $l_{s} = 2 r$ . The length QR is decided based on kinematic constraints.

5. Dynamic analysis

A dynamic analysis is performed to decide the required motor torque to develop the PEH system. The components are made up of mild steel. The components, which are supported by the DC motor with their weights, are mentioned in Table 1. The total load of the basic supporting structure is 251 g.

Table 1.

Components supported by DC motor with weights.

Components	Weight (g)
Crank with connecting rod	48
Slider with holder	129
Specimen	62
Others (screws)	12
Total	251

The required torque is calculated using the following expression (Uicker et al., 2011)

\begin{matrix} T = (m_{s} r^{2} ω^{2} (\cos θ + \frac{\cos 2 θ}{n}) + m_{s} gr) \times \\ (\sin θ + \frac{\sin 2 θ}{2 \sqrt{n^{2} - \sin^{2} θ}}) \end{matrix}

(1)

Taking $m_{s} = 0.225 kg$ , $r = 0.0065 m$ , $l_{c} = 0.05 m$ , $n = l_{c} / r$ , $N = 1500 rpm$ , and $g = 9.8 m / s^{2}$ , the variation of the torque for the basic structure with angular position for few cycles is shown in Figure 2. Based on the torque calculation, a 24-V DC motor is selected which is capable of producing 6000 rpm. However, for the present setup a 12-V supply is used, so the motor can provide maximum speed of 3000 rpm (25 Hz). The operating range of motor speed is decided based on the natural frequency of the cantilever beam system which is found to be 3.6 Hz (Figure 5(b)) and 4.3 Hz (Figure 11(b)) for beam specimens with substrates S1 and S3, respectively (as mentioned in Table 2). The experimentally obtained acceleration characteristic of the holder (without beam system attachment) is shown in Figure 2(b). The acceleration time response (after filtering) is obtain at a particular speed of DC motor by using the accelerometer.

Figure 2.

(a) Variation of torque with crank angle and (b) experimental acceleration time response (after filtering) of the holder without beam system attachment at a particular speed of DC motor by using the accelerometer.

Table 2.

Material and geometric properties of the substrate and piezoelectric patch (PZT-5H).

Property	PZT patch	Substrate S1	Substrate S2	Substrate S3
Young’s Moduli, GPa	$E_{p} = 66.7$	$E_{b} = 190$	$E_{b} = 190$	$E_{b} = 190$
Density, kg m^–3	$ρ_{p} = 7500$	$ρ_{b} = 7800$	$ρ_{b} = 7800$	$ρ_{b} = 7800$
Length, m	$L_{p} = 5 \times 10^{- 2}$	$L_{b} = 30 \times 10^{- 2}$	$L_{b} = 29.1 \times 10^{- 2}$	$L_{b} = 30 \times 10^{- 2}$
Width, m	$b_{p} = 2.5 \times 10^{- 2}$	$b_{b} = 2.55 \times 10^{- 2}$	$b_{b} = 2.55 \times 10^{- 2}$	$b_{b} = 2.55 \times 10^{- 2}$
Height, m	$t_{p} = 5 \times 10^{- 4}$	$t_{b} = 5.1 \times 10^{- 4}$	$t_{b} = 5.1 \times 10^{- 4}$	$t_{b} = 6.5 \times 10^{- 4}$
Capacitance, nF	$C_{p} = 71.30$	–	−	−
Piezoconstant, CN^–1	$d_{31} = - 285 \times 10^{- 12}$	−	−	−
Mass, g	$m_{p} = 5$	$m_{b} = 39$	$m_{b} = 35$	$m_{b} = 44$

The photograph of the fabricated setup is shown in Figure 3. The encoder is connected to the DC motor to read the rotational speed in cycles per minute. Microcontroller (mbed NXP LPC1768) is programmed in such a way that one can regulate the rotational speed of the DC motor by just rotating the knob and display the output in display unit (16 × 2 LCD). The microcontroller required 5-V DC power, which is provided by USB cable. Power supply converts AC into DC and is connected to the motor driver. The motor driver provides 12 or 24-V power to the motor.

Figure 3.

(a) Experimental setup of PEH system with electronic circuit and oscilloscope and (b) the harvester with PZT patch.

6. Analytical model

The schematic diagram of parametrically excited PEH system consists of a cantilever beam of length L, mass m at a distance d from its fixed end, and a PZT patch of length $L_{p}$ near its base is shown in Figure 4. The electrodes of PZT patch are connected to an external load resistance through a circuit to find out the power output of the harvester. The longitudinal and transverse motion of the beam is denoted by $u (ξ, t)$ and $v (ξ, t)$ respectively, where $ξ$ is the curvilinear coordinate along the beam length.

Figure 4.

Schematic diagram of piezoelectric energy harvester (PEH) system with parametric excitation.

The nondimensional nonlinear coupled electromechanical temporal governing equations of motion of the system are expressed in the following equations. The detailed derivation can be found in Appendix 1

\begin{matrix} \overset{\cdot\cdot}{q} + 2 ϵ ζ \overset{\cdot}{q} + q + ϵ α q^{3} + ϵ β q {\overset{\cdot}{q}}^{2} + ϵ γ q^{2} \overset{\cdot\cdot}{q} + ϵ η V = \\ ϵ q \hat{f} \cos (ϕ τ) \end{matrix}

(2)

\overset{\cdot}{V} + χ V + Θ \overset{\cdot}{q} = 0

(3)

where,

\begin{matrix} x = \frac{s}{L}, β = \frac{d}{L}, τ = ω t, λ = \frac{r}{L}, μ = \frac{m}{ρ L}, Γ = \frac{z_{0}}{z_{r}}, \\ J = \frac{J_{0}}{ρ L r^{2}}, ϕ = \frac{Ω}{ω}, ζ = ζ^{*} \frac{ν}{ϵ}, χ = \frac{1}{C_{p} ω R_{l}}, Θ = \frac{\bar{K} r}{C_{p} r_{v}}, \\ η = \frac{- \bar{η} r_{v}}{ρ ℏ ω^{2} r ϵ L^{2}} \int_{0}^{1} {δ_{x} (x) - δ_{x} (x - 1)} ψ (x) dx, \\ \bar{K} = \frac{e_{31} t_{pc} b_{p}}{L} \int_{0}^{1} ψ_{xx} (x) dx, t_{pc} = \frac{t_{p} + t_{s}}{2} - \bar{y} \\ r_{v} = \frac{\bar{V}}{V} = \frac{\bar{K} r}{C_{p}} \end{matrix}

where $\overset{•}{()} = d () / d τ$ . Here the damping, geometric and inertial nonlinearities, electromechanical coupling, and excitation terms are small, and hence considered of the order of $ϵ$ (bookkeeping parameter). Here $α, β, γ, and η$ are the coefficients of cubic, inertial $(q {\overset{\cdot}{q}}^{2}, q^{2} \overset{\cdot\cdot}{q})$ , and coupling terms, respectively. The coefficients appear in equations (2) and (3) are defined in Appendix 2. By observing the right-hand side (RHS) term of equation (2) the time-varying forcing term appears as a parameter, which is a typical characteristic of parametrically excited systems that provides energy input to the system. In the absence of PZT patch, the equation (2) reduced to that of Zavodney and Nayfeh (1989).

7. Perturbation analysis

A uniform first-order approximate analytical solution of equations (2) and (3) is obtained by using the standard method of multiple scales. In the present work, the nonlinear electromechanical system is explored as an energy harvester for the principal parametric resonance condition. The time dependence is expressed into multiple time scales as $T_{n} = ϵ^{n} τ; n = 0, 1, 2, \dots$ , and the time derivatives are expressed as follows: $d / d τ = D_{0} + ϵ D_{1} + O (ϵ^{2})$ and $d^{2} / d τ^{2} = D_{0}^{2} + 2 ϵ D_{0} D_{1} + O (ϵ^{2})$ , where $D_{i} = \partial / \partial T_{i}$ . The system solution $(q, V)$ expanded in a series form as

q (τ; ϵ) = q_{0} (T_{0}, T_{1}) + ϵ q_{1} (T_{0}, T_{1}) + O (ϵ^{2})

(4)

V (τ; ϵ) = V_{0} (T_{0}, T_{1}) + ϵ V_{1} (T_{0}, T_{1}) + O (ϵ^{2})

(5)

Substituting equations (4) and (5) into equations (2) and (3) and further equating the coefficients involving the terms of $ϵ^{0}$ and $ϵ^{1}$ to zero yields the set of equations in different order of $ϵ$

D_{0}^{2} q_{0} + q_{0} = 0

(6a)

D_{0} V_{0} + χ V_{0} = - D_{0} q_{0}

(6b)

\begin{matrix} D_{0}^{2} q_{1} + q_{1} = - 2 ζ D_{0} q_{0} - 2 D_{0} D_{1} q_{0} + q_{0} \hat{f} \cos ϕ τ \\ - η V_{0} - α q_{0}^{3} - β q_{0} {(D_{0} q_{0})}^{2} - γ q_{0}^{2} D_{0}^{2} q_{0} \end{matrix}

(7a)

D_{0} V_{1} + χ V_{1} = - D_{0} V_{0} - (D_{0} q_{1} + D_{1} q_{0})

(7b)

The solution of differential equations (equations (6a) and (6b)) of the order $O (ϵ^{0})$ is expressed as

q_{0} = A (T_{1}) e^{i T_{0}} + cc; V_{0} = - ZA (T_{1}) e^{i T_{0}} + cc

(8)

where cc is the complex conjugate of the preceding terms and $A (T_{1})$ is an unknown complex function to be determined later. Using equation (8) into equation (7a), the following secular and near secular terms are obtained which should be eliminated for finite amplitude response of the system

\begin{matrix} \frac{\hat{f}}{2} \bar{A} e^{i T_{0} ϕ - i T_{0}} - (3 α + β - 3 γ) \bar{A} A^{2} e^{i T_{0}} \\ - 2 i (A' + ζ A) e^{i T_{0}} - η ZA e^{i T_{0}} = 0 \end{matrix}

(9)

One may obtain similar expression for single-mode approximation without considering 1:3 internal resonance case from Garg and Dwivedy (2019). To express the nearness of $ϕ$ to that of 2 (principal parametric resonance), the detuning parameter $σ$ is introduced as $ϕ = 2 + ε σ$ . The complex function $A (T_{1})$ is expressed into polar form as $A (T_{1}) = \frac{1}{2} a (T_{1}) e^{i ϑ (T_{1})}$ where amplitude $a (T_{1})$ and phase $ϑ (T_{1})$ are slowly varying function of time scale $T_{1}$ . Now by using this polar form in equation (9) and eliminating the secular term (the coefficient of $e^{i T_{0}}$ ) to find the bounded solution, one obtains the following reduced modulation equations

a' = ζ_{eff} a + a F_{eff} \sin (δ)

(10)

\frac{1}{2} a δ' = \frac{1}{2} σ_{eff} a - N_{eff} a^{3} + a F_{eff} \cos (δ)

(11)

where

\begin{matrix} N_{eff} = \frac{3 α}{8} + \frac{β}{8} - \frac{3 γ}{8}; ζ_{eff} = - ζ + ζ_{e} - ζ_{a}; \\ σ_{eff} = σ + σ_{e}; F_{eff} = \frac{\hat{f}}{4} ζ_{e} = \frac{η χ}{2 (χ^{2} + 1)}; \\ ζ_{a} = \frac{4 μ_{a} a}{3 π}; σ_{e} = \frac{η}{(χ^{2} + 1)}; δ = σ T_{1} - 2 ϑ; \\ \tan δ_{0} = \frac{- ζ_{eff}}{- 0.5 σ_{eff} + N_{eff} a^{2}} \end{matrix}

where $ζ, ζ_{e}, ζ_{a}, and σ_{e}$ represent the internal material damping, the damping term caused by electric circuit, aerodynamic damping (Abdelkefi et al., 2012; Baker et al., 1967; Daqaq et al., 2009), and the shift in systems frequency due to electromechanical coupling respectively, which depends on parameters $η and ξ$ . Also, $N_{eff}$ denotes the total effect due to cubic $(α)$ and inertial $(β, γ)$ nonlinear terms in the equation of motion of the beam. The nontrivial $(a \neq 0)$ frequency response of the reduced equations can be calculated by the following frequency response expression

σ = 2 N_{eff} a^{2} - σ_{e} \pm 2 \sqrt{F_{eff}^{2} - ζ_{eff}^{2}}

(12)

The above expression is parabolic in nature with respect to the nondimensional amplitude a. Steady-state displacement, output voltage, and power are obtained from the following expressions

v (s, t) = r ψ (s) a \cos (ω t + ϑ)

(13)

V (t) = r_{v} \frac{1}{\sqrt{χ^{2} + 1}} a \cos (ω t + ϑ + φ)

(14)

P = \frac{a^{2}}{R_{l}} {(r_{v} \frac{1}{\sqrt{χ^{2} + 1}})}^{2}

(15)

here $φ = \overset{- 1}{\tan} (χ)$

8. Results and discussion

For result and discussion purpose, the geometric and material properties of the specimen (cantilever beam and piezoelectric patch of PZT-5H) as shown in Figure 1(b) are given in Table 2.

It may be noted that a mass of 27 g is attached at a distance of 213 mm from the base of the system. Furthermore, the PZT is placed at a distance of 5.7 mm from the fixed base. Initially, the free vibration response of the system is found out to obtain the natural frequency and the damping ratio of the system. Figure 5(a) shows the voltage–time response of the harvester, which is obtained experimentally by simply tapping the end of the harvester in open circuit condition. As the voltage is proportional to the displacement of the system, using the logarithmic decrement method, the damping ratio of the system is found to be 0.0134.

Figure 5.

(a) Open circuit voltage–time response of the system when just tapping the top edge of the beam for damping coefficient calculation by logarithmic decrement method and (b) corresponding FFT of the voltage–time signal.

The corresponding FFT of the voltage–time signal is plotted in Figure 5(b), where the fundamental frequency of the PEH system is found to be 3.6 Hz. By using the expression given in Appendix 2, the natural frequency of the system is found to be 3.53 Hz. Figure 6(a) shows the frequency response plot obtained by using equation (12). The external load resistance is kept at $100 k Ω$ . It may be noted that by sweeping up the frequency up to 6.76 Hz, the beam has a rigid-body motion and does not undergo transverse vibration, which is one of the characteristics of the parametrically excited systems.

Therefore, the piezoelectric patch does not undergo any strain and hence, no voltage is produced (Figure 6(b)). Furthermore, sweeping up this frequency, after this critical value where it undergoes supercritical pitchfork bifurcation ( $P F_{L}$ ), the system has a nontrivial response and the corresponding output voltage starts to increase with increase in the external frequency. This continues until the saddle-node bifurcation (marked SN in Figure 6(b)), and a further increase in frequency leads to jump down phenomenon where the beam response may again become trivial. Here the beam may develop cracks near the fixed end. On sweeping down the frequency, the jump up phenomenon occurs near the subcritical pitchfork bifurcation point ( $P F_{R}$ ) and the trivial response jumps to the nontrivial branch. The acceleration level for parametric onset is found to be ( $| \overset{\cdot\cdot}{z} | = z_{0} (2 π f)^{2} = 11.7 m / s^{2}$ , where $z_{0} = 6.5 mm$ and $f = 6.76 Hz$ ).

Figure 6.

(a) Amplitude response with frequency of excitation at $R_{l} = 100 k Ω$ , (b) comparison of analytical and experimental voltage response, (c) comparison of analytical and experimentally obtained power with frequency of excitation, (d) time response of the output voltage at a particular frequency of excitation of $f = 7.18 Hz$ (with piezo-patch with hairline cracks), (e) FFT of the voltage-time response shown in Figure 6(d), and (f) input acceleration time plot with FFT of the filtered signal.

To compare the analytical result with experimental result, the system is excited near to twice its fundamental frequency ( $2 ω$ ) to find out the voltage response by sweeping up the frequency. Figure 6(b) shows the plot for the variation of the voltage with frequency. Here both theoretical and experimental results are plotted, and both results are found to be almost in good agreement up to 7.33 Hz.

The difference between experimental and analytical findings may be due to the parameter taken for theoretical analysis and minor imperfection in the experimental setup. The theoretical prediction of critical threshold frequency and the highest output voltage closely resembles the experimental findings. It may be noted that the error is found to be around 2% near the supercritical pitchfork bifurcation point and saddle-node bifurcation point. In between these points the error is coming to be around 10–15%. The error is calculated as the ratio of the total deviation in output voltage by total theoretical voltage. Also one may note that the analytically predicted response consists of stable (solid blue lines) and unstable (solid red lines) nontrivial branches of fixed points, which coincides with experiment findings.

The steady-state power is then calculated as shown in Figure 6(c). One may achieve a maximum power of nearly 22.7 mW around a frequency of excitation of 7.25 Hz for external load resistance of $R_{l} = 100 k Ω$ . Figure 6(d) shows the experimental voltage–time response for a particular frequency of excitation $f = 7.18$ and load resistance of $R_{l} = 100 k Ω$ where the average voltage is found to be $V = 40.21 V$ . The FFT of the output voltage versus time response shown in Figure 6(d) is plotted in Figure 6(e), and the time response of the input acceleration along with FFT of the filtered signal is shown in Figure 6(f).

When the harvester system is excited near the parametric resonance condition, few hairline cracks are observed transverse to the length of the piezo-patch (Figure 7(c)). This is because the patch is placed near to the base (spanned from 5.7 to 10.7 mm length from the base) where the bending moment is high and large amplitude of vibration causes PZT patch to break transversely and visible hairline cracks appear due to the brittle nature (low Young’s modulus) of patches. Due to these cracks, the time response (shown in Figure 6(d)) differs from well-expected continuous periodic profile and sharp peaks rather than smooth variations are visible. Here, near the SN bifurcation point the beam may develop cracks close to the fixed end due to high transverse deformation, which produces maximum bending moment at the base, and after few cycles the beam may break as shown in Figure 7(a). The corresponding close-up figure is shown in Figure 7(b), where the direction of the crack is transverse to the length of the beam. This could be further quantified in terms of the amplitude of the base acceleration, which is responsible for the onset of a crack in the beam. From the expression of the base excitation ( $z = z_{0} \cos (Ω t)$ ), one can find the amplitude of the base acceleration ( $| \overset{\cdot\cdot}{z} | = z_{0} (2 π f)^{2} = 13.78 m / s^{2}$ , where $z_{0} = 6.5 mm$ and $f = 7.33 Hz$ ) at which the crack in the beam is possible due to high amplitude transverse displacement.

Figure 7.

(a) Cracked beam with piezo-patch, (b) cracked beam (zoomed view), and (c) piezoelectric patch (PZT-5H) with visible hairline cracks in transverse direction and disintegration near the electrode separation.

The length of the beam after disintegration (Figure 7(a)) from near its fixed end is reduced to $L_{b} = 291 mm$ , and it is used as next specimen (S2). Figure 8 shows the frequency and time response of output voltage for external load resistance of $25 k Ω$ . The natural frequency of the system increases due to the reduced length of the harvester; hence to trigger the parametric excitation, the frequency of excitation shifts toward higher frequency side as shown in Figure 8(a).

Figure 8.

(a) Experimental voltage response with frequency of excitation at $R_{l} = 25 k Ω$ and (b) voltage–time response at a particular frequency of excitation of $f = 9.63 Hz$ (when its length reduces to $L_{b} = 291 mm$ from previous length of $L_{b} = 300 mm$ ).

The voltage–time response corresponds to $f = 9.63$ is shown in Figure 8(b) where an average voltage of 11.82 V is achievable. The amplitude and voltage response grows as frequency of excitation f and the right side bend of the response shows the hardening behavior, which observed earlier as well. In Figure 9(a) the time response of the voltage is shown for $R_{l} = 25 k Ω$ ; one can achieve an average voltage of 17 V in this case for the frequency of excitation $f = 9.85$ . The corresponding point is from voltage response in Figure 8(a). Figure 9(b) captures the voltage–time response of pre and post-failure of piezoelectric patch attached to the beam due to high rate of deformation (a violent transverse movement of the beam) when external frequency (f) is nearly $> 9.85$ . The time response becomes trivial from nontrivial as time progresses and some high peaks are observed which shows that the electric circuit sometimes gets completed, depend on the nature of the disintegration of the PZT patch.

Figure 9.

Experimental time response of output voltage at $R_{l} = 25 k Ω$ and frequency of excitation (a) f = 9.85 Hz.(b) f > 9.85 Hz.

Since the PZT patch disintegrated in the previous experiment, a new specimen is fabricated for further analysis. The new specimen (with substrate S3) is having all the material and geometric properties similar to previous specimens with substrates S1 and S2 (as mentioned in Table 2), except the thickness. The beam thickness ( $t_{b} = 0.65 mm$ ) of the new specimen is more than that of previous ones, and hence its natural frequency is slightly higher than the previous specimen. Figure 10 shows the variation of voltage and power with external load resistance when the frequency of excitation kept at a constant value of $f = 8.28 Hz$ . Here few readings of output voltages are taken at a particular value of load resistance to plot the variation. This is achieved by keeping the rpm of DC motor at a constant level. The quadratic curve fit depicts the average variation of voltage and power. It is observed that as the load resistance increases, the voltage and corresponding power (Figure 10(b)) also increases and then stabilizes for certain resistance values. This shows that the load resistance influences the power output and at a certain value of $R_{l}$ , the output power gets maximum. One may obtain the optimal resistance by performing power limit analysis as given in Liao and Liang (2018). The voltage–time response at a particular load resistance $R_{l} = 800 k Ω$ is shown in Figure 11(a), and corresponding FFT is plotted in Figure 11(b). The voltage–time response is amplitude-modulated response with frequency of oscillation around 4.3 Hz.

Figure 10.

Experimental voltage and power output with variation of load resistance $R_{l}$ (for beam system with substrate S3) with parameters μ=0.7, β = 0.71 and f = 8.28 Hz.

Figure 11.

Experimental (a) voltage–time response for load resistance $R_{l} = 800 k Ω$ and (b) FFT of the voltage–time response.

The continuous and smooth variation of the voltage–time response means the PZT patch is working well, and there is no evident hairline crack is developed until now, at this frequency level. The average value of voltage amplitude is nearly 54.72 V. The participation of other higher harmonics of fundamental frequency is shown in an insight zoomed figure in Figure 11(b) where their participation is weak as compared to the fundamental harmonic. The amplitude modulation may be due to certain imbalances, which causes minor irregularities in energy transfer.

In Figure 12, the variation of output voltage with nondimensional distance ( $β$ ) and frequency of excitation (f) along with variation of f with $β$ is shown while keeping $m = 27 g$ and load resistance $R_{l} = 100 k Ω$ invariant. The average voltage is kept stable within a certain range by keeping the transverse displacement near the triggering onset frequency of excitation as $β$ and f increase. In Figure 12(c), as $β$ increases, the threshold frequency (at which beam starts the transverse vibration) reduces linearly. By a suitable position of the attached mass m, one may adjust the natural frequency of the system and critical threshold excitation value can be reduced further. Voltage–time response at specific point $P_{1}$ located in Figure 12(a) is shown in Figure 12(d) where at $β = 0.533$ the output voltage is found to be $V = 13.5799 V$ .

Figure 12.

Experimental findings of (a) voltage with nondimensional distance of mass from the fixed end of the beam ( $β$ ),(b) corresponding voltage response, (c) frequency of excitation with $β$ , and (d) voltage–time response at point $P_{1}$ located inFigure 11(a) while keeping attached mass $m = 27 g$ and $R_{l} = 100 k Ω$ constant.

The voltage–time response at point $P_{2}$ as mentioned in Figure 12(a) is shown in Figure 13(a). By observing Figure 13(a), it is found that the pattern of the response changes (ellipsoidal zone), which denotes the patch abnormalities caused by sudden high deformation. This pattern carries forward in the responses shown in Figure 13(b) to (d) for $β = 0.833$ . High voltage peaks are also observed in Figure 13(b) to (d), which further provides concrete proof that PZT patch has suffered some permanent structural damage. One conclusion can be drawn from here is that beyond a certain transverse deflection of beam system the PZT patch may get damaged. The energy extraction from a broken piezo-patch poses challenges related to the circuitry section.

Figure 13.

Experimental findings of voltage–time response, where mass $m = 27 g$ , $β = 0.833$ , and $R_{l} = 100 k Ω$ .

This is probably the first time when time response from a broken patch is analyzed. However, to avoid such kind of disruption in energy transduction two possible solutions are suggested. First, the large amplitude vibration may be restricted by using stoppers and second, the more flexible MFCs (micro-fiber composites) may be used instead of using brittle PZT patches. Apart from MFCs, a wide range of flexible piezoelectric materials (Dagdeviren et al., 2016) are available, which can be integrated into the structures to not only harvest energy but in wearable devices for sensing purpose as well. It also allows using post-buckle structures and curved surfaces. However, use of MFC slightly increases the cost of the system. Another challenge is to avoid the cracks near the fixed end of the cantilever beam system for long-lasting use and keeping the structural integrity intact. The best way to avoid this failure is by providing concave-shaped supporting structure of suitable radius (Kluger et al., 2015) to the base attachment element (holder) or using stoppers (Searle et al., 2018) to prevent large transverse displacement. The provision of mass at an arbitrary position and its value adjustment is helpful to tune the frequency of the harvester as per the available ambient frequency of excitation.

A qualitative comparison of several energy harvesters is presented in Table 3. Comparison has been made based on the type of used piezoelectric patch (viz. PZT, MFC, or PVDF), its configuration (unimorph or bimorph), type of external excitation (direct or parametric), orientation of the specimen (horizontal or vertical), nature of study (analytical or experimental), length covered by the piezoelectric patch (partial or full), and transduction mechanism utilized (piezoelectric or electromagnetic). One sample case (Daqaq et al., 2009) is described here, and the interpretation in a similar manner for the other cases can be drawn from Table 3. Daqaq et al. (2009) were the first to analyze energy harvesting via parametric excitation. In this work, a lumped parameter model of the cantilever beam with a tip mass and MFC patch is used. The beam is kept in the horizontal position in this experimental study. It may be noted that in the present work, the cantilever beam is placed in the vertical position where the effect of gravity is accommodated in the dynamics of the system. Also, the mass can be attached to any arbitrary position along the beam, and PZT-5H patch is used in the present work. Abdelkefi et al. (2012) studied a similar system where a nonlinear distributed parameter model is developed and the effect of nonlinear piezoelectric coupling coefficients on the output voltage and power are analyzed analytically. The unimorph piezoelectric patch spans the whole length of the beam, and any issues in using PZTs under parametric excitation are not reported. Few systems utilized the parametrically excited energy harvesters based on electromagnetic transduction mechanism as well (Jia et al., 2014; Kuang and Zhu, 2019; Yildirim et al., 2016). In such systems, the volume of the system is generally high as compared to the harvesters based on piezoelectric transduction. A combination of direct excitation along with parametric excitation (Mam et al., 2017; Searle et al., 2018; Xia et al., 2019) or autoparametric with parametric (Jia and Seshia, 2014) is also explored for performance enhancement. The emphasis is given on harvesting from low-frequency direct excitation and post-buckled beam system in the work of Friswell et al. (2012). Zhu et al. (2013) and Panyam et al. (2018) explore the effect of axial load in PEH system where the former used piezofilm. Zaghari et al. (2015) analyzed the PEH system with time varying stiffness. In the previous work of Garg and Dwivedy (2019), a system similar to the present one is studied where it undergoes parametric and 1:3 internal resonance. Here due to the internal resonance, the first two modes of the system interacts and the energy transfer takes place between them which helps in extracting more output power and enhances the bandwidth of the harvester. Without considering internal resonance and using unimorph PZT in the limited span instead of bimorph MFC, the system can be reduced to the present system.

Table 3.

Qualitative comparison of the present work with previous literature.

References	PZT/MFC	Type: U, B \| P, D, AP \| H, V	A/Exp	Length of patch covered	PE/EM
Daqaq et al. (2009)	MFC	U \| P \| H	A + Exp	Partial	PE
Abdelkefi et al. (2012)	PZT	U \| P \| H	A	Full	PE
Friswell et al. (2012)	MFC	U \| D \| V	A + Exp	Partial	PE
Zhu et al. (2013)	PVDF	U \| P \| V	A + Exp	Partial	PE
Jia et al. (2014)	–	–	A + Exp		EM
Jia and Seshia (2014)	PZT	B \| P − AP \| V	A + Exp	Partial	PE
Yildirim et al. (2016)	–	–	A + Exp		EM
Yan and Hajj (2017)	PZT	B \| P \| V	A	Full	PE
Yildirim et al. (2017)	MFC	U \| P \| H	A + Exp	Partial	PE
Mam et al. (2017)	PZT	B \| D − P \| V	A + Exp	Partial	PE
Panyam et al. (2018)	MFC	U \| P \| H	A + Exp	Partial	PE
Searle et al. (2018)	MFC	U \| D − P \| H	A + Exp	Partial	PE
Zaghari et al. (2015)	PZT	U \| P \| H	A + Exp	Partial	PE
Garg and Dwivedy (2019)	MFC	B \| P \| V	A	Full	PE
Kuang and Zhu (2019)	–	–	A + Exp		EM
Xia et al. (2019)	PZT	B \| D − P \| H	A	Full	PE
Present work	PZT	U \| P \| V	A + Exp	Partial	PE

MFC: micro-fiber-composite; U: unimorph; B: bimorph; P: parametric; D: direct; AP: autoparametric; H: horizontal; V: vertical; A: analytical; Exp: experimental; PE: piezoelectric; EM: electromagnetic.

The studies of PEH systems mentioned in Table 3 are extended in Table 4 for the quantitative comparison of the normalized power density (NPD). Here, the parameters considered for the characterization of the harvesting system are the input acceleration, volume, frequency, maximum power output, and NPD. NPD is determined using the expression $NPD = power / (volume \times acceleratio n^{2})$ , which is taken from the work of Beeby et al. (2007). The present system yields a maximum power output and NPD (in $μ W c m^{- 3} m^{- 2} s^{4}$ ) value of 22.7 mW and of 15.9 respectively. It may be observed from Table 4 that the maximum power obtained in the present case is better than most of the harvesters presented in the table. It is less than from the work of Abdelkefi et al. (2012) where full-length patch was taken. Also it is found to be less than that of Yan and Hajj (2017) where very high acceleration was applied and the volume is higher than the present case. Now comparing NPD, it is found to be better than most of the cases. It is less than that of Friswell et al. (2012) where the acceleration is very less, but it may be noted that the output power is higher in the present case than that of Friswell et al. (2012).

Table 4.

Quantitative comparison of the present work with previous literature.

References	Acceleration( $m s^{- 2}$ )	Volume( $c m^{3}$ )	Frequency(Hz)	Maximum poweroutput	≈ NPD( $μ W c m^{- 3} m^{- 2} s^{4}$ )
Daqaq et al. (2009)	3.14	9.51	5.37	0.06 mW	6.42E – 01
Abdelkefi et al. (2012)	12	11.7	–	80 mW	4.75E + 01
Friswell et al. (2012)	1.19	1.23	2.45	0.2 mW	1.15E + 02
Zhu et al. (2013)	6.37	2.66	9	–	3.30E – 03
Jia et al. (2014)	0.57	1800	3.57	171.5 mW	2.93E + 02
Jia and Seshia (2014)	20	1.21	20	0.15 mW	3.10E – 01
Yildirim et al. (2016)	9	–	54.4	–	4.14E – 04
Yan and Hajj (2017)	98.3	170.5	8.44	1.4 W	8.50E – 01
Yildirim et al. (2017)	20	1.12	12.1	2.15 mW	1.87E + 00
Mam et al. (2017)	57.13	10.92	29.8	0.96 mW	2.70E – 02
Panyam et al. (2018)	60	1.51	224	0.04 mW	1.66E – 02
Searle et al. (2018)	–	4.1	10.8	0.068 mW	–
Zaghari et al. (2015)	47	15.44	10.9	0.0087 mW	2.55E – 04
Garg and Dwivedy (2019)	36.9	3.94	15.3	9 W	1.67E + 03
Kuang and Zhu (2019)	4.9	–	18	3.6 mW	–
Xia et al. (2019)	–	0.6	–	0.36 W	–
Present work	13.38	7.98	7.22	22.7 mW	1.59E + 01

NPD: normalized power density.

In comparison with the previous studies where nonlinear (geometric and inertial type) parametric excited harvesters are analyzed, the present work differs mainly in placing the attached mass in arbitrary position along the beam length and keeping the beam in vertical position instead horizontal in order to let gravity assist in large transverse displacement of the beam system. Also, the present work is the first in reporting the challenges in using PZT and cracks in the beam system due to large deformations where previous works failed to do so. The future work involves the experimental investigation on how in such kind of parametrically excited systems one may bring more stability.

9. Conclusion

In the present analysis, the vertical cantilever beam–based piezoelectric energy harvester under parametric excitation is investigated theoretically and experimentally. The system consists of an arbitrary positioned mass and piezoelectric patch near its fixed end. The nonlinear coupled electromechanical governing equations of motions are derived by using Hamilton’s principle, and method of multiple scales is used to reduce the second-order nonlinear governing equations to a set of first-order reduced equations that are used for finding the steady-state response and the stability of the system. The results from these equations are compared with those available in the literature and by developing an experimental setup.

A shaker with piezoelectric energy harvester is developed to experimentally validate the theoretical findings of the vertical beam-based harvester. The limiting supercritical bifurcation point at which the nontrivial state is observed theoretically is found to be in good agreement from the experimental findings. Furthermore, the theoretical and experimental responses are found to be in good agreement. The maximum output voltage obtained is also found to be in close agreement. The output voltage and power are compared with those found in the literature. It is observed that the present vertical cantilever beam–based harvester nearly produced more output voltage and power than that of a horizontally placed cantilever beam with same configurations. Hence one may use the developed reduced equations for finding the response, voltage, and power for similar piezoelectric energy harvester. A broadband range of frequency between the subcritical and supercritical pitchfork bifurcation points may be used to harvest the energy. The amount of harvested voltage and power depends on the system parameter, which may suitably be adjusted by using the developed reduced equations to harvest more energy. The challenges of cracking in beam and breaking of piezo-patch during experiments are discussed in this work. In addition, some suggestion on how the challenges can be overcome is also discussed for system stability and durability. A qualitative and quantitative comparison of 17 energy harvesters have been carried out, and the present harvester is found to be better than most of the harvesters in terms of the NPD. The present experimental and theoretical investigation will help to overcome the challenges posed by the parametrically excited energy harvester.

Footnotes

Appendix 1

In the present work, the extended Hamilton’s principle $\int_{t_{1}}^{t_{2}} (δ T - δ U + δ W_{nc}) dt$ is used to derive the governing equation of motion of the system. The kinetic energy (T) and potential energy (U) of the harvester under parametric excitation are expressed by

(16)

\begin{matrix} T = \frac{1}{2} \int_{0}^{L} [ρ^{*}] (\overset{\cdot}{u} {(s, t)}^{2} + \overset{\cdot}{v} {(s, t)}^{2}) ds \\ + \frac{1}{2} J_{0} \overset{\cdot}{φ} {(d)}^{2} \end{matrix}

(17)

U = \frac{1}{2} \int_{0}^{L} EI φ' {(s)}^{2} ds - \int_{0}^{L} [ρ^{*}] g u (s, t) ds

where $ρ, δ, J_{0}, φ, EI, and g$ denote the unit mass density of beam system, Dirac delta function, rotary inertia of mass m, angle of rotation, the flexural rigidity of beam system, and gravity constant, respectively. The lateral and longitudinal dynamics is governed by inextensibility condition (Kar and Dwivedy, 1999) as $u (s, t) = \frac{1}{2} \int_{0}^{s} {v' (ξ, t)}^{2} d ξ + z (t)$ . Here $z (t)$ denotes the harmonic base excitation. On observing the deflected beam geometry, the relation between the angle of rotation $φ (s)$ and transverse deflection $v (s, t)$ is defined as $\sin φ = v'$ and $\cos φ = (1 - v'^{2})^{1 / 2}$ . As per the Euler–Bernoulli beam theory, the curvature $κ (s, t)$ is

(18)

\begin{matrix} κ (s, t) = φ' = \frac{\partial φ (s)}{\partial s} = \frac{v ″}{\cos φ} = \frac{v ″}{\sqrt{1 - {v'}^{2}}} \\ \approx v ″ (1 + \frac{1}{2} {v'}^{2}) \end{matrix}

The non-conservative virtual work done $W_{nc}$ involves the work done in dissipation of energy by viscous force effect and electrical forces in moving charges by piezoelectric elements (Erturk and Inman, 2008)

(19)

\begin{matrix} δ W_{nc} = - \int_{0}^{L} c \overset{\cdot}{v} (s, t) δ v ds \\ - \int_{0}^{L} [H (s)] M_{ele} (s, t) δ κ (s) ds \end{matrix}

The total moment due to piezoelectric patches $M_{ele}$ is expressed as follows (here the whole expression is valid for bimorph case and later half part is for unimorph case)

(20)

M_{ele} = \int_{- h_{2}}^{- h_{1}} d_{31} E_{p} b_{p} E_{e} y dy + \int_{h_{1}}^{h_{2}} d_{31} E_{p} b_{p} (- E_{e}) y dy

Here,

\begin{matrix} E_{e} = \frac{\bar{V} (t)}{t_{p}}; h_{1} = \frac{1}{2} t_{b} - \bar{y}; h_{2} = \frac{1}{2} t_{b} + t_{p} - \bar{y}; \\ \bar{y} = \frac{E_{p} t_{p} b_{p} (t_{p} + t_{b})}{2 (E_{p} t_{p} b_{p} + E_{b} t_{b} b_{b})} \end{matrix}

The governing nonlinear equation of motion becomes

(21)

\begin{matrix} ρ^{*} v_{tt} + c v_{t} + E I^{*} [v_{ssss} + {v_{s} {(v_{s} v_{ss})}_{s}}_{s}] \\ - {[J^{*} {(v_{s})}_{tt}]}_{s} - {(N v_{s})}_{s} - H_{ss} \bar{η} \bar{V} = 0 \end{matrix}

Here

\begin{matrix} ρ^{*} = ρ + m δ (s - d), ρ = ρ_{b} A_{b} + H ρ_{p} A_{p}, \\ E I^{*} = E_{b} I_{b} + H E_{p} I_{p}, J^{*} = J_{0} δ (s - d), \\ \bar{η} = \frac{e_{31} b_{p}}{t_{p}} [h_{2}^{2} - h_{1}^{2}], A_{b} = b_{b} t_{b}, A_{p} = b_{p} t_{p}, \\ H = H (s - L_{1}) - H (s - L_{2}), \\ e_{31} = d_{31} E_{p}, I_{p} = \frac{1}{3} b_{p} [{(t_{p} + \frac{1}{2} t_{b})}^{3} - \frac{1}{8} {t_{b}}^{3}], \\ I_{b} = \frac{1}{12} b_{b} {t_{b}}^{3}, L_{p} = L_{2} - L_{1} \\ N = \frac{1}{2} \int_{s}^{L} ρ^{*} \int_{0}^{ξ} {(v_{s}^{2})}_{tt} d η d ξ + m (z_{tt} - g) \int_{s}^{L} δ (ξ - d) d ξ \\ + ρ L (1 - \frac{s}{L}) (z_{tt} - g) \end{matrix}

where subscript “p” is used to denote terms related to PZT, and $()_{s}$ and $()_{t}$ denote the differentiation with respect to curvilinear coordinate “s,” that is, $d () / ds$ and time $d () / dt$ respectively. The term N denotes the longitudinal load at “s.” The electric circuit equation in unimorph configuration can be obtained by applying the Gauss’s law (Erturk and Inman, 2008)

(22)

\frac{dQ (t)}{dt} = \frac{d}{dt} \int_{A_{p}} D . n d A_{p}

Here $D$ and $n$ denote the electric displacement vector and outward normal to the surface. The electric displacement vector ( $D$ ) in terms of axial stress $(σ_{p})$ and generated electric field $(E_{e})$ in the present case is defined by constitutive relation $D_{3} = d_{31} σ_{p} + \hat{e} E_{e}$ , where $σ_{p} = E_{p} ϵ_{p} - e_{31} E_{e}$ and $ϵ_{p} = - y κ (s, t)$ . Using these expressions in equation (22), the circuit equation becomes

(23)

C_{p} {\bar{V}}_{t} + \frac{1}{2 R_{l}} \bar{V} - i (t) = 0

Here

i (t) = - e_{31} t_{pc} b_{p} \int_{L_{1}}^{L_{2}} {(v_{ss})}_{t} ds, C_{p} = \frac{\hat{e} b_{p} L_{p}}{t_{p}}

The four boundary conditions at fixed and free ends are represented as follows

(24)

v (0, t) = 0, v_{s} (0, t) = 0, v_{ss} (L, t) = 0, v_{sss} (L, t) = 0

At the fixed end, displacement and slope are zero and at the free end, the bending moment and shear forces are zero. By considering single-mode approximation in the Galerkin’s method, the transverse displacement $v (s, t)$ is represented by a scaling factor r, time modulation $q_{i} (t)$ , and shape function $ψ (s)$ (Zavodney and Nayfeh, 1989) of the beam system as $v (s, t) = r ψ (s) q (t)$ . After nondimensionalizing, the nonlinear coupled electromechanical temporal governing equations of motion for single mode are given as follows

(25)

\begin{matrix} \overset{\cdot\cdot}{q} + 2 ϵ ζ \overset{\cdot}{q} + q + ϵ α q^{3} + ϵ β q {\overset{\cdot}{q}}^{2} + ϵ γ q^{2} \overset{\cdot\cdot}{q} + ϵ η V \\ = ϵ q \hat{f} \cos (ϕ τ) \end{matrix}

(26)

\overset{\cdot}{V} + χ V + Θ \overset{\cdot}{q} = 0

Appendix 2 Acknowledgements

The authors are grateful to Mr. Arun Borgohain and Mr. Ayushman Gogoi of Yantrabot Pvt. Limited for their help to fabricate the experimental setup.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the financial support provided by the Department of Mechanical Engineering, Indian Institute of Technology Guwahati.

ORCID iD

Anshul Garg

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