Research Papers

Modeling and Analysis of Piezoelectric Energy Harvesting Beams Using the Dynamic Stiffness and Analytical Modal Analysis Methods

[+] Author and Article Information
P. Bonello

School of Mechanical, Aerospace, and Civil Engineering, University of Manchester, Pariser Building, Sackville Street, Manchester M13 9PL, United Kingdomphilip.bonello@manchester.ac.uk

S. Rafique

School of Mechanical, Aerospace, and Civil Engineering, University of Manchester, Pariser Building, Sackville Street, Manchester M13 9PL, United Kingdom

J. Vib. Acoust 133(1), 011009 (Dec 17, 2010) (11 pages) doi:10.1115/1.4002931 History: Received March 15, 2010; Revised October 07, 2010; Published December 17, 2010; Online December 17, 2010

The modeling and analysis of base-excited piezoelectric energy harvesting beams have attracted many researchers with the aim of predicting the electrical output for a given base motion input. Despite this, it is only recently that an accurate model based on the analytical modal analysis method (AMAM) has been developed. Moreover, single-degree-of-freedom models are still being used despite the proven potential for significant error. One major disadvantage of the AMAM is that it is restricted to simple cantilevered uniform-section beams. This paper presents two alternative modeling techniques for energy harvesting beams and uses these techniques in a theoretical study of a bimorph. One of the methods is a novel application of the dynamic stiffness method (DSM) to the modeling of energy harvesting beams. This method is based on the exact solution of the wave equation and so obviates the need for modal transformation. The dynamic stiffness matrix of a uniform-section beam could be used in the modeling of beams with arbitrary boundary conditions or assemblies of beams of different cross sections. The other method is a much-needed reformulation of the AMAM that condenses the analysis to encompass all previously analyzed systems. The Euler–Bernoulli model with piezoelectric coupling is used and the external electrical load is represented by generic linear impedance. Simulations verify that, with a sufficient number of modes included, the AMAM result converges to the DSM result. A theoretical study of a bimorph investigates the effect of the impedance and quantifies the tuning range of the resonance frequencies under variable impedance. The neutralizing effect of a tuned harvester on the vibration at its base is investigated using the DSM. The findings suggest the potential of the novel concept of a variable capacitance adaptive vibration neutralizer that doubles as an adaptive energy harvester. The application of the DSM to more complex systems is illustrated. For the case studied, a significant increase in the power generated was achieved for a given working frequency through the application of a tip rotational restraint, the use of segmented electrodes, and a resized tip mass.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

Base-excited piezoelectric energy harvesting beams (series-connected bimorph, upper figure, parallel-connected bimorph, middle figure, and unimorph, lower figure)

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Figure 2

Free-body diagram (ambient damping forces not shown): (a) overall system and (b) exploded view

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Figure 3

Voltage FRF by the DSM and AMAM

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Figure 4

Voltage FRF for increasing values of resistive loads R in ohms

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Figure 5

Tip-to-base transmissibility for increasing values of resistive loads R in ohms

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Figure 6

Tip-to-base transmissibility peaks for different capacitive loads C=nCp

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Figure 7

Effect of mechanical damping at the first resonance frequency: (a) voltage and (b) mean specific power

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Figure 8

Direction-fixed base/free tip system

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Figure 11

Assembly of piezoelectric beam segments with external loads Z1,Z2

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Figure 12

Direction-fixed base/direction-fixed tip harvester with segmented electrodes

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Figure 13

Comparison of systems in Figs.  812 over a range of resistive loads (Z): (a) voltage amplitude at first resonance, (b) mean specific power at first resonance, and (c) first resonant frequency

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Figure 9

Force-referenced FRFs for a capacitive load C=Cp: (a) base displacement per unit base force ũ0/F̃0 and (b) voltage per unit base force ṽ/F̃0

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Figure 10

Base displacement per unit base force ũ0/F̃0 for different values of resistive loads R in ohms




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