Research Papers

Study the Effect of Tapering on the Nonlinear Behavior of an Exponentially Varying Width Piezoelectric Energy Harvester

[+] Author and Article Information
H. Salmani

Department of Aerospace Engineering,
Tarbiat Modares University,
P.O. Box 14115-111,
Tehran, Iran
e-mail: hamed.salmani@modares.ac.ir

G. H. Rahimi

Department of Mechanical Engineering,
Tarbiat Modares University,
Jalal Ale Ahmad Exp. Way,
P.O. Box 14115-111,
Tehran, Iran
e-mail: rahimi_gh@modares.ac.ir

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 16, 2017; final manuscript received March 3, 2018; published online May 7, 2018. Assoc. Editor: Alper Erturk.

J. Vib. Acoust 140(6), 061004 (May 07, 2018) (11 pages) Paper No: VIB-17-1543; doi: 10.1115/1.4039932 History: Received December 16, 2017; Revised March 03, 2018

It has been shown that exponentially tapering the width of a vibration-based piezoelectric energy harvester will result in increasing electric power per mass in a specified frequency. In this paper, a nonlinear solution of an exponentially decreasing width piezoelectric energy harvester is presented. Piezoelectric, inertial, and geometric nonlinearities are included in the presented model, while the exponentially tapered piezoelectric beam's mass normalized mode shapes are utilized in Galerkin discretization. The developed nonlinear coupled equations of motion are solved using method of multiple scales (MMS), and the steady states results are verified by experiment in high amplitude excitation. Finally, the exponentially tapering parameter effect is studied, and it is concluded that the voltage per mass of the energy harvester is improved by tapering at high exciting acceleration amplitudes.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Baker, J. , Roundy, S. , and Wright, P. , 2005, “Alternative Geometries for Increasing Power Density in Vibration Energy Scavenging for Wireless Sensor Networks,” AIAA Paper No. 2005-5617.
Mateu, L. , and Moll, F. , 2005, “Optimum Piezoelectric Bending Beam Structures for Energy Harvesting Using Shoe Inserts,” J. Intell. Mater. Syst. Struct., 16(10), pp. 835–845. [CrossRef]
Benasciutti, D. , Moro, L. , Zelenika, S. , and Brusa, E. , 2010, “Vibration Energy Scavenging Via Piezoelectric Bimorphs of Optimized Shapes,” Microsyst. Technol., 16(5), pp. 657–668. [CrossRef]
Matova, S. P. , Renaud, M. , Jambunathan, M. , Goedbloed, M. , and Van Schaijk, R. , 2013, “Effect of Length/Width Ratio of Tapered Beams on the Performance of Piezoelectric Energy Harvesters,” Smart Mater. Struct., 22(7), p. 75015. [CrossRef]
Salmani, H. , Rahimi, G. H. , and Hosseini Kordkheili, S. A. , 2015, “An Exact Analytical Solution to Exponentially Tapered Piezoelectric Energy Harvester,” Shock Vib., 2015, pp. 1–13. [CrossRef]
Erturk, A. , and Inman, D. J. , 2008, “A Distributed Parameter Electromechanical Model for Cantilevered Piezoelectric Energy Harvesters,” ASME J. Vib. Acoust., 130(4), p. 041002. [CrossRef]
Erturk, A. , and Inman, D. J. , 2008, “On Mechanical Modeling of Cantilevered Piezoelectric Vibration Energy Harvesters,” J. Intell. Mater. Syst. Struct., 19(11), pp. 1311–1325. [CrossRef]
Erturk, A. , and Inman, D. J. , 2009, “An Experimentally Validated Bimorph Cantilever Model for Piezoelectric Energy Harvesting From Base Excitations,” Smart Mater. Struct., 18(2), p. 25009. [CrossRef]
Rosa, M. , and De Marqui Junior, C. , 2014, “Modeling and Analysis of a Piezoelectric Energy Harvester With Varying Cross-Sectional Area,” Shock Vib., 2014, p. 930503.
Daqaq, M. F. , Stabler, C. , Qaroush, Y. , and Seuaciuc-Osorio, T. , 2008, “Investigation of Power Harvesting Via Parametric Excitations,” J. Intell. Mater. Syst. Struct., 20(5), pp. 545–557. [CrossRef]
Stanton, S. C. , Erturk, A. , Mann, B. P. , and Inman, D. J. , 2010, “Resonant Manifestation of Intrinsic Nonlinearity Within Electroelastic Micropower Generators,” Appl. Phys. Lett., 97(25), p. 254101. [CrossRef]
Stanton, S. C. , Erturk, A. , Mann, B. P. , and Inman, D. J. , 2010, “Nonlinear Piezoelectricity in Electroelastic Energy Harvesters: Modeling and Experimental Identification,” J. Appl. Phys., 108(7), pp. 1–9. [CrossRef]
Stanton, S. C. , Erturk, A. , Mann, B. P. , Dowell, E. H. , and Inman, D. J. , 2012, “Nonlinear Nonconservative Behavior and Modeling of Piezoelectric Energy Harvesters Including Proof Mass Effects,” J. Intell. Mater. Syst. Struct., 23(2), pp. 183–199. [CrossRef]
Masana, R. , and Daqaq, M. F. , 2011, “Electromechanical Modeling and Nonlinear Analysis of Axially Loaded Energy Harvesters,” ASME J. Vib. Acoust., 133(1), p. 011007. [CrossRef]
Goldschmidtboeing, F. , Eichhorn, C. , Wischke, M. , Kroener, M. , and Woias, P. , 2011, “The Influence of Ferroelastic Hysteresis on Mechanically Excited PZT Cantilever Beams,” 11th International Workshop Micro Nanotechnology Power Generation Energy Conversion Application (PowerMEMS), Seoul, Korea, Nov. 15–18, pp. 114–117.
Abdelkefi, A. , Nayfeh, A. H. , and Hajj, M. R. , 2012, “Global Nonlinear Distributed-Parameter Model of Parametrically Excited Piezoelectric Energy Harvesters,” Nonlinear Dyn., 67(2), pp. 1147–1160. [CrossRef]
Abdelkefi, A. , Nayfeh, A. H. , and Hajj, M. R. , 2012, “Effects of Nonlinear Piezoelectric Coupling on Energy Harvesters Under Direct Excitation,” Nonlinear Dyn., 67(2), pp. 1221–1232. [CrossRef]
Leadenham, S. , and Erturk, A. , 2014, “Unified Nonlinear Electroelastic Dynamics of a Bimorph Piezoelectric Cantilever for Energy Harvesting, Sensing, and Actuation,” Nonlinear Dyn., 79(3), pp. 1727–1743. [CrossRef]
Garg, A. , and Dwivedy, S. K. , 2016, “Nonlinear Dynamics of Axially Loaded Piezoelectric Energy Harvester,” Procedia Eng., 144, pp. 592–599. [CrossRef]
Silva, C. J. , and Daqaq, M. F. , 2016, “Nonlinear Flexural Response of a Slender Cantilever Beam of Constant Thickness and Linearly-Varying Width to a Primary Resonance Excitation,” J. Sound Vib., 389, pp. 438–453.
Meirovitch, L. , 2001, Fundamentals of Vibrations, Mc Graw-Hill, New York, Chap. 6. [PubMed] [PubMed]
Meitzler, D. B. A. H. , Tiersten, H. F. , and Warner, A. W. , 1988, “IEEE Standard on Piezoelectricity,” Institute of Electrical and Electronics Engineers, Piscataway, NJ, Standard No. ANSI/IEEE Std 176-1987.
Nayfeh, A. H. , and Pai, P. F. , 2004, Linear and Nonlinear Structural Mechanics, Wiley, Hoboken, NJ, Chap. 4. [CrossRef]
Arafa, M. , and Baz, A. , 2004, “On the Nonlinear Behavior of Piezoelectric Actuators,” J. Vib. Control, 10(3), pp. 387–398.


Grahic Jump Location
Fig. 2

Piezoelectric beam cross section

Grahic Jump Location
Fig. 3

Experimental setup

Grahic Jump Location
Fig. 4

Experimental setup structural elements

Grahic Jump Location
Fig. 5

Analytical and experimental output voltage

Grahic Jump Location
Fig. 6

Experimental model output voltage for 9 g exciting acceleration amplitude

Grahic Jump Location
Fig. 7

Output voltage: (a) c = 0, (b) c = 5, (c) c = 10, (d) c = 15, and (e) c = 20

Grahic Jump Location
Fig. 8

Output voltage backbone curve

Grahic Jump Location
Fig. 9

Output voltage backbone curve for models with the same natural frequencies

Grahic Jump Location
Fig. 1

Piezoelectric energy harvester

Grahic Jump Location
Fig. 10

Comparing linear and nonlinear results—voltage per mass

Grahic Jump Location
Fig. 11

Comparing linear and nonlinear results—power per mass




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In