Shock and Vibration

Volume 2015 (2015), Article ID 426876, 13 pages

http://dx.doi.org/10.1155/2015/426876

## An Exact Analytical Solution to Exponentially Tapered Piezoelectric Energy Harvester

^{1}Department of Aerospace Engineering, Tarbiat Modares University, Jalal Ale Ahmad Highway, P.O. Box 14115-111, Tehran, Iran^{2}Department of Mechanical Engineering, Tarbiat Modares University, Jalal Ale Ahmad Highway, P.O. Box 14115-111, Tehran, Iran^{3}Department of Aerospace Engineering, Sharif University of Technology, Azadi Street, P.O. Box 11365-11155, Tehran, Iran

Received 26 January 2015; Accepted 20 April 2015

Academic Editor: Chao Tao

Copyright © 2015 H. Salmani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It has been proven that tapering the piezoelectric beam through its length optimizes the power extracted from vibration based energy harvesting. This phenomenon has been investigated by some researchers using semianalytical, finite element and experimental methods. In this paper, an exact analytical solution is presented to calculate the power generated from vibration of exponentially tapered unimorph and bimorph with series and parallel connections. The mass normalized mode shapes of the exponentially tapered piezoelectric beam with tip mass are implemented to transfer the proposed electromechanical coupled equations into modal coordinates. The steady states harmonic solution results are verified both numerically and experimentally. Results show that there exist values for tapering parameter and electric resistance in a way that the output power per mass of the energy harvester will be maximized. Moreover it is concluded that the electric resistance must be higher than a specified value for gaining more power by tapering the beam.

#### 1. Introduction

The dream of making sensor network nodes to monitor mechanical systems’ behavior is being achieved by applying low power, light weight, and self-sustained wireless sensors, which are capable of gaining required energy from environment. Converting light, thermal, and mechanical energies into electrical energy has been the most common methods to supply sensors’ power. Electromagnetic, electrostatic, and piezoelectric transducers are three usual conversion methods of mechanical energy into electrical energy [1]. Electromagnetic convertors have been widely used in order to generate electricity out of mechanical energy since the early 1930s [2]. During the last decade, implementing piezoelectric materials in energy harvesting devices has been extensively investigated by researchers, because of their light weight, direct implementation, and wide range of frequencies applications. Piezoelectric energy harvester is usually constructed of piezoelectric layers attached to a substrate, and its fundamental frequency is adjusted by employing tuning mass, while it is mounted on a vibrating system.

As the highest performance of the piezoelectric energy harvesters occurs at their fundamental frequencies, at the early stage of the development of these devices, an equivalent single degree of freedom (SDOF) model is employed to estimate generated power [3, 4]. Later on, Erturk and Inman [5] improved SDOF model by adding separately treated strain rate and air damping terms and introducing a correction factor for varying tip mass to beam mass ratio. Although SDOF model gives a simple expression to study the energy harvester’s behavior, it lacks several important aspects of the device such as dynamic mode shapes and accurate strain distribution. These lacks have been managed through introducing a correction factor for an improved lumped parameter model by Wang and Lu [6]. Knowing that SDOF model is a simple method to assess the harvester at a limited frequency range around the fundamental frequency of the beam, Sodano et al. [7] proposed a Rayleigh-Ritz method to calculate the power harvested from piezoelectric material including damping effect in higher modes. Erturk and Inman [8, 9] also employed eigenfunction expansion method using mass normalized mode shapes of unimorph and bimorph beams with tip mass to solve the coupled electromechanical formulation of energy harvesting devices.

The need for more efficient devices has encouraged researchers to present electrical and mechanical solutions in order to increase the power generation of piezoelectric vibration based energy harvesters. The most common approach used to increase the performance of energy harvesters is their geometry modification, which creates uniform strain distribution through the beam’s length. In this situation the electric charge is distributed uniformly and average power is higher than varying charge piezoelectric layer [10]. Baker et al. [11] observed that a triangular beam can harvest 50% more energy than the rectangular one, while the local overstrain occurrence is vanished by uniform strain distribution. Accepting the effects of uniform strain distribution on the harvested energy from the beam, researchers have started to examine the effects of different parameters on the harvester’s performance either numerically [12–18] or analytically [19–23]. Although numerical methods are valuable for easy implementation as well as doing parameters study, such as geometry [13–17] and tip mass [12, 18], they are time consuming and do not lead to an exact solution for the problem. On the other hand, deriving analytical solution is an intricate or almost impossible task if the governing differential equation does not belong to those familiar ones with closed form solution.

Due to the existence of exact solution for eigenmodes of a uniform beam problem, a closed form expression can be obtained to present the harvester’s behavior [8, 9]. However, solving trapezoidal beams dictates taking into account assumed modes as space dependent variables. Goldschmidtboeing and Woias [19] used rectangular beam eigenmodes to form mass, stiffness, coupling, and capacitance matrices of a triangular beam which were derived by Rayleigh-Ritz method. Dietl and Garcia [20] also applied admissible mode shapes of slender prismatic beam with tip mass to study the performance of a rectangular, linear taper and reverse taper energy harvester. Rosa and de Marqui Jr. [21] implemented Rayleigh-Ritz method and Euler-Bernoulli assumptions to investigate the behavior of linear and reverse tapered energy harvesters. The Differential Quadrature Method (DQM) is another way that was used to extract mode shapes of a beam with variable width in order to form the reduced order model of electromechanical equations [23]. In addition to tapering with just varying width, the piezoelectric energy harvesters can be also modified by changing the beam thickness [22].

According to this literature survey the approximated mode shapes have been employed to solve the nonuniform cross section piezoelectric energy harvesters’ governing equations. In this paper, exact normalized mode shapes of exponential beam with tip mass [24] are utilized to convert the governing electromechanical equations into modal space to calculate the power generated from piezoelectric energy harvesters. The strain rate and air damping terms are also considered to define the damping effect more accurately. After verifying the proposed formulations numerically and experimentally, parametric study is performed on beam’s length, tapering parameter, and electric resistance to investigate their effects on the energy harvesting device’s performance. In this study, we have considered unimorph and bimorph with series connection as well as bimorph with parallel connection for all the cases.

#### 2. Analytical Solution

##### 2.1. Electromechanical Formulation

Piezoelectric energy harvester, as shown in Figure 1, is composed of a beam embedded by piezoelectric layers and is excited at its base. It is assumed that the piezoelectric layer is perfectly bonded to the substrate layer, while the integrated beam’s width is varying exponentially through the length by , where is tapering parameter. A proof mass is also attached on the tapered beam’s tip to regulate the fundamental frequency.