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Research Papers

# A Distributed Parameter Electromechanical Model for Cantilevered Piezoelectric Energy Harvesters

[+] Author and Article Information
A. Erturk1

Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering and Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061erturk@vt.edu

D. J. Inman

Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061

Here, as a common practice, the abbreviation PZT is used for a generic piezoelectric ceramic, rather than a specific material.

If the excitation due to the external damping of the medium (which is generally air) is negligible when compared to the inertial excitation (i.e., $Nrc(t)≪Nrm(t)$), one can simply set it equal to zero $(Nrc(t)=0)$.

Yet, one can obtain the value of $c$ for nonzero initial conditions both in the mechanical domain (for the beam) and in the electrical domain (for the electrical circuit). Note that, for nonzero initial conditions in the mechanical domain, the Duhamel integral given by Eq. 33 for the modal mechanical response must be modified accordingly (i.e., initial displacement and velocity terms must be introduced).

Excitation due to air damping is ignored.

1

Corresponding author.

J. Vib. Acoust 130(4), 041002 (Jun 11, 2008) (15 pages) doi:10.1115/1.2890402 History: Received August 16, 2007; Revised December 07, 2007; Published June 11, 2008

## Abstract

Cantilevered beams with piezoceramic layers have been frequently used as piezoelectric vibration energy harvesters in the past five years. The literature includes several single degree-of-freedom models, a few approximate distributed parameter models and even some incorrect approaches for predicting the electromechanical behavior of these harvesters. In this paper, we present the exact analytical solution of a cantilevered piezoelectric energy harvester with Euler–Bernoulli beam assumptions. The excitation of the harvester is assumed to be due to its base motion in the form of translation in the transverse direction with small rotation, and it is not restricted to be harmonic in time. The resulting expressions for the coupled mechanical response and the electrical outputs are then reduced for the particular case of harmonic behavior in time and closed-form exact expressions are obtained. Simple expressions for the coupled mechanical response, voltage, current, and power outputs are also presented for excitations around the modal frequencies. Finally, the model proposed is used in a parametric case study for a unimorph harvester, and important characteristics of the coupled distributed parameter system, such as short circuit and open circuit behaviors, are investigated in detail. Modal electromechanical coupling and dependence of the electrical outputs on the locations of the electrodes are also discussed with examples.

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## Figures

Figure 1

Unimorph piezoelectric energy harvester under translational and small rotational base motions

Figure 2

(a) The unimorph piezoelectric energy harvester used for the parametric case study and (b) a detail from its cross section (dimensions are in millimeters)

Figure 3

Voltage FRF for five different values of load resistance (with the enlarged view of Mode 1 resonance showing the short circuit and open circuit behaviors)

Figure 4

Variation of voltage output with load resistance for base excitations at the short circuit and open circuit resonance frequencies of the first vibration mode

Figure 5

Current FRF for five different values of load resistance (with the enlarged view of Mode 1 resonance showing the short circuit and open circuit behaviors)

Figure 6

Variation of current output with load resistance for base excitations at the short circuit and open circuit resonance frequencies of the first vibration mode

Figure 7

Power output FRF for five different values of load resistance (with the enlarged views of Mode 1 and Mode 2 resonances showing the short circuit and open circuit behaviors)

Figure 8

Variation of power output with load resistance for base excitations at the short circuit and open circuit resonance frequencies of the first vibration mode

Figure 9

Relative tip motion FRF for the uncoupled system and for the coupled system with five different values of load resistance (with the enlarged views of Mode 1 and Mode 2 resonances showing the short circuit and open circuit behaviors)

Figure 10

Variation of relative tip displacement to base displacement ratio with load resistance for base excitations at the short circuit and open circuit resonance frequencies of the first vibration mode

Figure 11

Effect of the location of continuous electrode pair on the voltage FRF (focusing on the vibrations around the second mode)

Figure 12

Effect of the location of continuous electrode pair on the voltage FRF (focusing on the vibrations around the third mode)

Figure 13

(a) Cross section of a unimorph harvester and (b) the transformed cross section

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