Modelling of the circular edge-clamped interface of a hydraulic pressure energy harvester to determine power, efficiency and bandwidth

doi:10.1016/j.ymssp.2020.107013

Mechanical Systems and Signal Processing

Volume 146, 1 January 2021, 107013

https://doi.org/10.1016/j.ymssp.2020.107013 Get rights and content

Highlights

•
Generalized model for harvesting pressure ripples using the piezoelectric effect is developed.
•
Force-deflection relationship of the system is determined as a Duffing hardening response.
•
Expressions for input energy, output electrical energy and energy efficiency are developed.
•
Good agreement observed between model and experiment for output voltage and power.

Abstract

There is an increasing desire to monitor and control hydraulic systems in an autonomous and battery-free manner. One solution to this challenge is to harvest hydraulic pressure ripples and noise by exploiting the piezoelectric effect. This paper develops a new generalized model of a hydraulic piezoelectric harvester based on a circular edge-clamped flat plate interface with a central piezoelectric stack. Such a model allows the relationships between harvesting performance and structure to be assessed in detail. It is demonstrated that the force-deflection relationship of a circular edge-clamped plate with a central lumped mass follows a cubic hardening Duffing equation. A single degree of freedom (SDOF) lumped-parameter model of the system is established where the nonlinear frequency response resulting from hardening nonlinearities are explored. The input mechanical energy and the output electrical energy both exhibit a quadratic nonlinear relationship with vibration amplitude. The maximum output occurs at the jump-down frequency and the overall energy conversion efficiency of the system is determined. The optimum resistance load for maximum output energy and energy efficiency are obtained as non-dimensional excitation frequency. Experimental validation is performed and good agreement is observed between model results and experimental measurements. The developed model provides important insights into the optimization of power output and response of future hydraulic energy harvesting devices.

Graphical abstract

Introduction

Energy harvesting technologies are a promising alternative to conventional battery technologies since they are able to provide sustainable electric power for low-power electronic devices; these include autonomous wireless sensors and portable electronic devices [1], [2], [3]. Within hydraulic systems, there is an increasing need for continuous monitoring, such as pressure, temperature and vibration, and a desire to reduce cost of battery replacement and management. As a result, the ability to exploit the piezoelectric effect and harvest acoustic pressures from hydraulic pressure ripples has received increasing attention, for applications such as automotive suspension systems [4], [5], [6], industrial pumps [7] and energy-harvesting shock absorbers [8], [9]. Acoustic noise in hydraulics is often an undesirable consequence of the operation of pumps and actuators, and is generated in almost all pressurized fluid systems [10]. Generally, this undesired energy is attenuated using noise filter devices. However, it has been recently shown that by employing hydraulic pressure energy harvesters, the acoustic noise in the fluid can be converted into electrical energy to enable battery-free operation of low-power and self-powered sensor nodes in hydraulic systems [9], [11].

Hydraulic pressure energy harvesters (HPEH) convert acoustic pressure within hydraulic systems into electric energy. Energy conversion is often achieved via axial excitation of a piezoelectric stack through a fluid to mechanical interface. The generated voltage and electrical power of the HPEH is dependent on the amplitude and frequency of the pressure ripple, the dynamic fluid-structure interaction of the interface and the properties of the piezoelectric material. Generally, the dominant frequency components of pressure ripples are relatively low, namely in the range of hundreds of hertz [12], [13], [14], which is typically well below the fundamental resonance frequency of the piezoelectric element, namely in the order of tens of kHz. Therefore, under such ‘off-resonance’ conditions a Helmholtz resonator has been employed in an effort to improve energy harvesting efficiency. The Helmholtz resonator is a commonly used device that amplifies pressures over a narrowband and has been incorporated into acoustic energy harvesting devices to amplify pressure fluctuations and thereby increase the force applied to the piezoelectric stack [15], [16]. However, a traditional Helmholtz resonator needs to be large or the neck needs to be narrow to operate at sufficiently low frequencies. This remains a challenge to accommodate within any energy harvesting system with a small volume, and the need for a narrow neck leads to poor gain. To date, significant effort has been expended to improve the performance of traditional Helmholtz resonator for applications with pressure fluctuations through a variety of approaches, which include the design of the cavity shape [17], [18], modifying the architecture of the resonator neck [19], [20], and adding acoustic materials within the cavity [21].

The fluid to mechanical interface is another important component of a HPEH. The interface acts to convert the acoustic pressure into a mechanical force to deform the piezoelectric stack, while protecting the piezoelectric material from the fluid. The force-coupling behavior and deformation mechanisms at the interface are important factors that determine the output energy, bandwidth and conversion efficiency of the HPEH. Compared to the relatively large number of studies on the structural design of the Helmholtz resonator to improve the energy harvesting efficiency, the fluid to mechanical interface has yet to be studied in detail. The publications to date in this area are restricted to Aranda et al. [22], who analyzed the force transmission efficiency of flat-plate-interfaces for a range of dimensional configurations and pressure loads using finite element simulations. They provided an investigation of the force transmission ratio of two different types of fluid to mechanical interfaces for a flat metal plate and a conventional hydraulic piston, using an experimental setup with tunable ripple frequencies and amplitudes [23], [24]. In their studies, the force transmission efficiency was defined as the ratio of the normal force on the piezoelectric stack to the force acting on the exposed interface area of the harvester. Although the force transmission characteristics of hydraulic pressure energy harvesters, such as frequency and amplitude of the pressure ripple, have been presented for a range of system parameters, the results to date have been restricted to specific experimental HPEH devices. As a result, a general theoretical model and the interface mechanisms of the fluid to mechanical interface are essential to understand the output performance of future HPEH devices and therefore optimize their operation.

In this paper, a new theoretical model of a circular edge-clamped flat plate, and its interface within a HPEH device, is established to study the output electrical energy and energy conversion efficiency of the system. Such a configuration is selected since it is the most commonly used fluid to mechanical interface in the hydraulic piezoelectric harvesters [18], [24]. The force-deflection relationship of a circular edge-clamped flat plate with a central lumped mass attachment is obtained and a cubic hardening nonlinear behavior is observed. The reaction force between the piezoelectric stack and housing is derived. A SDOF lumped-parameter model of the electromechanical coupling system subjected to harmonic excitation is established. The harmonic balance method is used to obtain the solutions. The nonlinear frequency responses resulting from hardening nonlinearities are presented and the input mechanical energy and the output electrical energy of the electromechanical coupled system of a flat plate interface-piezoelectric stack are studied in detail. The expression of energy conversion efficiency of the system, which is defined as the ratio between output electrical energy to input mechanical energy, is then derived. Ultimately, the analysis provides a new generalized model that enables optimization of HPEH output and efficiency.

Section snippets

Model description

The general structure of a hydraulic pressure energy harvester is shown in Fig. 1. A thin metallic flat circular plate, with thickness h and radius R, is used as the fluid to mechanical interface to isolate the fluid from the piezoelectric stack. The piezoelectric stack is located at the center of the circular plate, which acts as the mechanical to electrical transfer component. The hydraulic pressure, p(t), arising from the pressure ripple is applied on the plate, which bends the circular

Dynamic behavior of the electromechanical coupling system

The dynamic behavior of the system is examined in this section. On substituting Eq. (5) and Eq. (16) into Eq. (1), the dynamic equations of the electromechanical coupling system can be written as $m \ddot{z} + c_{d} \dot{z} + k_{1} z + k_{3} z^{3} + θ V = γ F cos (ω t)$ $C_{p} \dot{V} + \frac{1}{R_{r}} V = θ \dot{z}$ where the force coefficient γ, arising from the housing constraint is expressed as $γ = 1 - \frac{K_{s} R^{2}}{4 (K_{s} R^{2} - 16 π D)}$

Substituting Eq. (8) and Eq. (15) into Eq. (19), the force coefficient γ can be rewritten as $γ = 1 - \frac{1}{4} \frac{1}{1 - \frac{4 π}{3 (1 - ν^{2})} \frac{E}{E_{s}} \frac{h^{3} h_{s}}{R^{2} A_{s}}}$

For the HPEH, the thickness of the plate

Experimental verification

To validate the performance of the proposed model, experiments were conducted to the circular edge-clamped flat plate interface, as shown in Fig. 11. A thin stainless flat plate was clamped circularly and a piezoelectric stack was located at the center of the plate, whose radius was R = 20 mm. The thickness of the plate is an important parameter determining the system nonlinearity and the compression strain in the piezoelectric stack, as well as the output voltage and power. Accordingly, two

Conclusions

In this paper a new generalized model has been developed for the output electrical energy and energy conversion efficiency of a hydraulic piezoelectric energy harvester based on a flat plate interface with a centered piezoelectric stack. The force-deflection relationship of the electromechanical coupling system has been determined to be a cubic hardening Duffing equation. The reaction force between the piezoelectric stack and the housing has been obtained and is proportional to the pressure

CRediT authorship contribution statement

Huifang Xiao: Conceptualization, Methodology, Investigation, Writing - original draft. Haotang Qie: Formal analysis, Software, Validation. Chris R Bowen: Resources, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant number 51775037] and the China Scholarship Council [grant number 201906465020]. The authors would like to thank Nick Gathercole and Stephen Coombes in University of Bath for their help with the measurement setup and evaluation.

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Modelling of the circular edge-clamped interface of a hydraulic pressure energy harvester to determine power, efficiency and bandwidth

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Model description

Dynamic behavior of the electromechanical coupling system

Experimental verification

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Renew. Sustain. Energy Rev.

Mech. Syst. Sig. Process.

Appl. Energy

Mech. Syst. Sig. Process.

J. Adv. Res.

Mech. Syst. Sig. Process.

Mech. Syst. Sig. Process.

Mech. Syst. Sig. Process.

Nano Energy

Sens. Actuators, A

Aerosp. Sci. Technol.

Appl. Acoust.

Appl. Acoust.

J. Sound Vib.

Eur. J. Mech. A. Solids

Extreme Mech. Lett.

Mech. Syst. Sig. Process.

Eng. Struct.

Piezoelectric and ferroelectric materials and structures for energy harvesting applications

Energy Environ. Sci.