Optimal piezoelectric beam shape for single and broadband vibration energy harvesting: Modeling, simulation and experimental results

doi:10.1016/j.ymssp.2014.07.014

Mechanical Systems and Signal Processing

Volumes 54–55, March 2015, Pages 417-426

https://doi.org/10.1016/j.ymssp.2014.07.014 Get rights and content

Highlights

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Mathematical modeling for the estimation of voltage is developed.
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Simulation is done in COMSOL Multiphysics software.
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Two designs for comb-shaped energy harvester are proposed.

Abstract

Harvesting energy from the surroundings has become a new trend in saving our environment. Among the established ones are solar panels, wind turbines and hydroelectric generators which have successfully grown in meeting the world’s energy demand. However, for low powered electronic devices; especially when being placed in a remote area, micro scale energy harvesting is preferable. One of the popular methods is via vibration energy scavenging which converts mechanical energy (from vibration) to electrical energy by the effect of coupling between mechanical variables and electric or magnetic fields. As the voltage generated greatly depends on the geometry and size of the piezoelectric material, there is a need to define an optimum shape and configuration of the piezoelectric energy scavenger. In this research, mathematical derivations for unimorph piezoelectric energy harvester are presented. Simulation is done using MATLAB and COMSOL Multiphysics software to study the effect of varying the length and shape of the beam to the generated voltage. Experimental results comparing triangular and rectangular shaped piezoelectric beam are also presented.

Introduction

Energy harvesting has been around for decades. To feed the world’s needs for energy, macro scale energy harvesting technologies have successfully established. On the other hand, for low powered electronics devices, harvesting energy from the ambient vibrations seems to be an ideal solution due to the definite life span and high cost for replacement of the traditional batteries. Three mechanisms are available for vibration energy harvesting; using electrostatic devices, electromagnetic field and utilizing piezoelectric based materials [1].

The efficiency of an energy harvester depends on the types of materials used. Piezoelectric energy harvester is widely used in this area of research as it possesses large power densities [2]. Lead Zicronate Titanate (PZT) is preferable due to its abundant vibration accessibility and high piezoelectric constants. Other materials include Lead Magnesium Niobate‐Lead Titanate (PMN‐PT), Lead Zinc Niobate‐Lead Titanate (PZN‐PT), Zinc Oxide and Polyvinylidene Difluiride (PVDF).

Increasing width and thickness of the beam resulted to greater power output [3]. Patel et al. also added that for a piezoelectric material, the influence of its length is greatly dependent on the thickness of the piezoelectric layer [4]. However, in the case of fixed mass, a shorter length, larger width, and lower ratio of piezoelectric layer thickness to total beam thickness are preferred [5].

Truncated beam produces better output voltage as the beam experiences higher stress. Sameh et al. concluded that a trapezoidal cantilever beam will produce more power per unit area compared to a rectangular beam as the distribution of strain is uniform [6]. In addition, cantilever beam is still in favor as it possesses lower resonance frequencies and relatively higher strain for a given force input [7]. However, other non-classical shapes are also being studied in some applications. This includes shell structure [8], spiral [9] and zigzag [10] configurations. Hence, there are a lot of other possibilities that can be explored in order to optimize the power generation of this type of piezoelectric energy harvester.

The main focus of this paper is to study the optimal beam of piezoelectric energy harvester for both single harmonic and broadband vibration. The equations of motions are derived from the Euler‐Bernoulli beam theory, and simulation studies are carried out using both MATLAB and COMSOL Multiphysics software. Experimental results are also presented to validate the findings in simulation studies.

Section snippets

Free vibration analysis of cantilever beam

In this section, the estimations of natural frequencies for rectangular, trapezoidal and triangular beams are presented. For a uniform beam that undergoes undamped free vibration, the governing equation of motion can be obtained as [11], [12] $E I \frac{\partial^{4} z (x, t)}{\partial x^{4}} + m \frac{\partial^{2} z (x, t)}{\partial t^{2}} = f_{0} (x, t)$ where EI is the bending stiffness, m is the mass per unit length of the beam and z(x,t) is the transverse displacement of the neutral axis (at point x and time t) due to bending. In addition, z(x,t) can be expressed as $z (x, t)$

Forced vibration analysis of cantilever beam

The estimation of voltage response is done in forced vibration analysis. Under the influence of base excitation, the governing equation of motion can be written as [2] $\frac{\partial^{2} M (x, t)}{\partial x^{2}} + m \frac{\partial^{2} z_{r e l} (x, t)}{\partial t^{2}} = - m \frac{\partial^{2} z_{b} (x, t)}{\partial t^{2}}$

The stress–strain relations for both beam and piezoelectric layer can be expressed as $σ_{1}^{b} = E_{b} ε_{1}^{b}$ $σ_{1}^{p} = E_{p} (ε_{1}^{p} - d_{31} E_{3})$ where ε is the strain, σ is the stress, d₃₁ is the piezoelectric moduli, and $E_{3}$ is the electric field in the piezoelectric. Also, the electric field, E₃ can be expressed as $E_{3} (t) = - v$

Simulation studies to find the optimal shape of single beam

To study the influence of the beam shape, the ratio of both widths of the beam $(b / a)$ is utilized (Fig. 5). The ratio starts from 1 to represent a rectangle and ends at 0 which is a triangle. The model is then simulated in a frequency range of 0–800 Hz.

From Fig. 6, the peak of the plot, which denotes the output voltage of each shape at its fundamental frequency, is the highest at widths-ratio equals to 0 (triangular shape). As the beam becomes truncated, the voltage generated increases and it is

Experimental result

Three piezoelectric bimorph actuators from Steiner & Martins, Inc (STEMiNC) are used in this experiment. Each beam consists of a layer of copper sandwiched between layers of PZT-5H material. The beams are manually cut into triangular shape at three different effective lengths; i.e. 35 mm, 30 mm and 25 mm. A clamp is used to hold the piezoelectric patch and mounted on an electrodynamic shaker. Dynamic signal analyzer (DSA) is used to perform frequency response analysis. Swept sine wave excitation

Conclusion

This paper presents the analysis of optimal piezoelectric shape and configuration of vibration based energy scavenging system. The derivations are based on Euler–Bernoulli beam theory and the determination of natural frequencies for arbitrary shapes is established from the energy method. The harvester is assumed to experience lateral vibration from its base excitation. Findings from both simulation and experiment are also shown. It is seen that triangular shape exhibits the highest voltage

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View full text

Optimal piezoelectric beam shape for single and broadband vibration energy harvesting: Modeling, simulation and experimental results

Highlights

Abstract

Introduction

Section snippets

Free vibration analysis of cantilever beam

Forced vibration analysis of cantilever beam

Simulation studies to find the optimal shape of single beam

Experimental result

Conclusion

J. Mechatron.

Energy harvesting vibration sources for microsystems applications

Measur. Sci. Technol.

Piezoelectric Energy Harvesting

A geometric parameter study of piezoelectric coverage on a rectangular cantilever energy harvester

J. Mater. Syst. Struct.

IEEE Trans. Ultrason. Ferroelectr. Freq. Control