Theoretical and experimental investigation of parametrically excited piezoelectric energy harvester

In the present work, a cantilever beam based piezoelectric energy harvester is investigated both theoretically and experimentally. The harvester is consists of a harmonically base excited vertical cantilever beam with a piezoelectric patch at the fixed end and a mass attached at an arbitrary position. The Euler-Bernoulli beam theory is applied considering the cantilever beam to be slender. The temporal nonlinear electromechanical governing equation of motion is obtained by using generalized Galerkin’s method considering two-mode approximation. Here for principal parametric resonance condition the steady state response of the voltage is obtained by using the method of multiple scales. The results are validated by developing an experimental setup of the harvester. For the harvester having a dimension of 295 mm 24 mm 7.6 mm,   a maximum voltage of 40 V is obtained for a base motion of 9 mm with a frequency of 10.07 Hz when 15 gm mass is attached at a distance of 140 mm from the fixed end.


Introduction
Development in smart devices used for sensing, human and structural health monitoring and actuation requires low power to operate. These devices are operated by batteries which in addition, are having a limited life span, environmentally unfriendly and requires a regular replacement. This dependency of smart devices on power supply is not really smart. So to overcome this drawback researchers are exploring other nontraditional means to power these devices by extracting untapped ambient energy from sources such as light, wind, temperature and potential gradient, noise, sound and vibration etc. This energy can be transformed by three basic transduction mechanisms namely electromagnetic, electrostatic and piezoelectric. Out of three the piezoelectric transduction mechanism is getting the most attention due to its high power density and ease of application [10].
In general, linear vibration based piezoelectric energy harvester systems work over a short range of bandwidth near the resonance frequency. Energy transduction reduces sharply for mistuned (away from resonance) linear harvesting systems. To address this issue researchers are focusing on tapping the rich dynamics which is outcome of inherent or induced nonlinearity. Nonlinear systems display behaviours such as bifurcation, chaos, MATEC Web of Conferences 211, 02009 (2018) https://doi.org/10.1051/matecconf/201821102009 VETOMAC XIV internal resonance that linear systems cannot. The multi-branched fixed point response of a nonlinear system may be periodic, quasi-periodic and chaotic in nature. Energy harvesters based on nonlinear vibration have potential to extract energy due to multiple resonance conditions which may lead to enhanced frequency band width. The source of nonlinearity can be of the geometric or material in nature. It can also arise due to external excitation and constraints such as impact, friction, backlash and freeplay [4,5]. The frequency bandwidth over which the higher transduction of mechanical energy to electrical energy takes place can be increased by exploiting the inherent or induced nonlinearity [6].
A large response can be produced by small parametric excitation [7] even when the frequency of excitation is away from the fundamental frequency of the system. Daqaq et al., [3] investigated a nonlinear parametrically excited cantilever beam based harvester. A critical amplitude of excitation is necessary to be maintained always to get nontrivial system response. In a similar work where parametric excitation is considered Abdelkefi et al., [9] developed a distributed parameter model of parametrically excited nonlinear PEH system having a cantilever beam and a tip mass.
In the present work a parametrically excited cantilever beam with piezoelectric patch and an attached mass is considered for dynamic analysis. The similar system is analyzed by Zavodney and Nayfeh [1] considering single mode approximation and without piezoelectric element. Also by considering two mode approximation similar model is analysed analytically by Kar and Dwivedy [2] and Dwivedy and Kar [8] without piezoelectric patch.
Here the Generalized Galerkin's method is used to discretize the spatio-temporal equation of motion to its temporal form. Steady state response and output voltage is obtained by using perturbation method namely the Method of multiple scales (MMS) to reduce the equations of motion into first order differential equations. In house experimental setup is developed in order to verify the results obtained analytically.

Mathematical modeling
The schematic of parametrically excited (  The inextensibility condition defines the longitudinal motion in terms of lateral. The circuit equation [10] for series connection under the external load resistance, l R becomes The boundary conditions are (0, ) 0, (0, ) 0, ( , ) 0, Here , , , p s t C Here we assume that the terms

Perturbation analysis
To obtained analytical expression for transverse displacement, generated voltage and power, a uniform first order approximate analytical solution of Eq. (4) and (5) is obtained. Method of multiple scales (MMS) is used to meet this end which describe the dynamics of the nonlinear system. The system dynamics under parametric resonance case is studied. In order to implement MMS the time dependence is discretize into multiple time scales as ; 0,1, 2,...
Later substituting these expansions of solution (Eq. (6)) in Eq. (4) and comparing the coefficients involving terms 0  and 1  , yields the following set of differential equations The solution of Eq. (7) expressed in the following form Here   1 n A T is a slowly varying complex valued function and cc is an abbreviation for complex conjugate term. Further expanding Eq. (8) by utilizing Eq. (9) the secular and near secular terms are obtained for first two modes (

1, 2 n 
). These obtained secular terms should be eliminated so that finite amplitude response exist.

Case : Principal parametric excitation (
Considering only single mode approximation ( 1 n  ) and when excitation frequency (  ) is near to twice of first natural frequency of the system that is

 
The variation of voltage with nondimensional frequency of excitation is shown in Fig. 2 for two different parameter of damping. For higher damping the bandwidth increases slightly. The experimental findings (see Fig. 2) matches with analytical analysis. A voltage range from 30 V to 50 V is obtained when the system excited near to parametric excitation. The voltage follows the amplitude topologically due to proportionality.

Experimental setup
In house experimental setup consists of mbed NXP LPC1768 Microcontroller, encoder, display unit, power supply, PM DC motor (24 V, 3000 rpm), energy harvester etc., as shown in Fig. (3). Output voltage is obtained using the oscilloscope (InfiniiVision DSO-X-3024A  Table 1 and Table 2. The developed shaker is used to excite the beam near twice its first natural frequency which is termed as parametric excitation. The frequency range of the shaker is 25 Hz and stroke length is 9 mm.

Conclusion
A parametrically excited harvesting system consists of a cantilever beam with piezoelectric patch and attached mass is analysed. Analytical expressions are developed to measure the steady-state amplitude and voltage for parametric excitation case. Method of multiple scales is used to obtain reduced expressions which are compared with experimental findings and found to be in good agreement. High voltage of approximately 40 V is obtained. One can vary the mass and its position along the beam to adjust the frequency of the system accordingly to the frequency of excitation. This analysis is limited to parametric excitation but one may analyse the system by considering multimode dynamics with internal resonance.