Acta Mechanica

, Volume 230, Issue 7, pp 2339–2361 | Cite as

Reduced-order modeling and usefulness of non-uniform beams for flexoelectric energy harvesting applications

  • S. Faroughi
  • E. F. Rojas
  • A. AbdelkefiEmail author
  • Y. H. Park
Original Paper


Energy harvesting at micro- and nanoscales has recently seen a renewed interest that the flexoelectric effect can counter the inability of piezoelectric energy harvesters to generate enough energy at small scales. Almost all small-scale energy harvesters use uniform rectangular geometries, whereas at the macroscale energy harvesters use a wide array of geometries including tapered rectangular geometries. The incorporation of non-uniform effects into a piezoelectric system considering the flexoelectric effect should give insight into how these systems can benefit from different geometries. A non-uniform flexoelectric Euler–Bernoulli cantilever energy harvester is modeled using classical continuum theories and is examined at the microscale. The non-uniformity of the energy harvester is governed by linear and nonlinear tapering effects, with the nonlinearities represented by high-order polynomials. The system is assumed to be linear, only undergoing harmonic base excitation. The varied tapering ratios and powers of the geometric tapering, considering that only the thickness and the width of the beam are tapered, are compared with uniform systems. The results show that non-uniform beams exhibit more harvested power than their uniform counterparts and also increase the range of resonant frequencies where significant power can be generated. Nonlinear tapering increases the amount of power that could be harvested compared to linear tapering; however, the nonlinearity of the tapering effects is limited to cubic and quadratic forms. It is demonstrated that higher-order tapering effects reduce the amount of harvested power compared to the linear taper counterpart. Non-uniform beams prove to be more effective than their rectangular counterparts within a linear system, whereas optimal resistive loads decrease as the tapering effects increase.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Muralt, P., Muralt, P.: Ferroelectric thin films for micro-sensors and actuators: a review. J. Micromechanics Microengineering 10, 136 (2000)CrossRefGoogle Scholar
  2. 2.
    Kim, H.S., Kim, J.H., Kim, J.: A review of piezoelectric energy harvesting based on vibration. Int. J. Precis. Eng. Manuf. 12(6), 1129–1141 (2011)CrossRefGoogle Scholar
  3. 3.
    Abdelkefi, A.: Aeroelastic energy harvesting: a review. Int. J. Eng. Sci. 100, 112–135 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ke, L.L., Wang, Y.S., Wang, Z.D.: Nonlinear vibration of the piezoelectric beams based on the nonlocal theory. Compos. Struct. 94(6), 2038–2047 (2012)CrossRefGoogle Scholar
  5. 5.
    Abdelkefi, A., Barsallo, N.: Comparative modeling of low-frequency piezomagnetoelastic energy harvesters. J. Intell. Mater. Syst. Struct. 25(14), 1771–1785 (2014)CrossRefGoogle Scholar
  6. 6.
    Kapuria, S., Kumari, P., Nath, J.K.: Efficient modeling of smart piezoelectric composite laminates: a review. Acta Mech. 214(1–2), 31–48 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Raja, S., Rao, K.V., Gowda, T.M.: Improved finite element modeling of piezoelectric beam with edge debonded actuator for actuation authority and vibration behaviour. Int. J. Mech. Mater. Des 13(1), 25–41 (2017)CrossRefGoogle Scholar
  8. 8.
    Rao, K.V., Raja, S., Gowda, T.M.: Finite element modelling and vibration control study of active plate with debonded piezoelectric actuators. Acta Mech. 225(10), 2923–2942 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Akbar, M., Curiel-Sosa, J.L.: Evaluation of piezoelectric energy harvester under dynamic bending by means of hybrid mathematical/isogeometric analysis. Int. J. Mech. Mater. Des. 14(4), 647–667 (2018)CrossRefGoogle Scholar
  10. 10.
    Lumentut, M.F., Howard, I.M.: Electromechanical analysis of an adaptive piezoelectric energy harvester controlled by two segmented electrodes with shunt circuit networks. Acta Mech. 228(4), 1321–1341 (2017)CrossRefzbMATHGoogle Scholar
  11. 11.
    Tang, L., Wang, J.: Modeling and analysis of cantilever piezoelectric energy harvester with a new-type dynamic magnifier. Acta Mech. 229(11), 4643–4664 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pasharavesh, A., Ahmadian, M.T., Zohoor, H.: Electromechanical modeling and analytical investigation of nonlinearities in energy harvesting piezoelectric beams. Int. J. Mech. Mater. Des 13(4), 499–514 (2017)CrossRefzbMATHGoogle Scholar
  13. 13.
    Goldschmidtboeing, F., Woias, P.: Characterization of different beam shapes for piezoelectric energy harvesting. J. Micromechanics Microengineering 18(10), 104013 (2008)CrossRefGoogle Scholar
  14. 14.
    Benasciutti, D., Moro, L., Zelenika, S., Brusa, E.: Vibration energy scavenging via piezoelectric bimorphs of optimized shapes. Microsyst Technol. 16(5), 657–668 (2010)CrossRefGoogle Scholar
  15. 15.
    Hosseini, R., Hamedi, M.: An investigation into resonant frequency of trapezoidal V-shaped cantilever piezoelectric energy harvester. Microsyst Technol. 22(5), 1127–1134 (2016)CrossRefGoogle Scholar
  16. 16.
    Siddiqui, N.A., Kim, D.J., Overfelt, R.A., Prorok, B.C.: Electromechanical coupling effects in tapered piezoelectric bimorphs for vibration energy harvesting. Microsyst Technol. 23(5), 1537–1551 (2017)CrossRefGoogle Scholar
  17. 17.
    Ben Ayed, S., Abdelkefi, A., Najar, F., Hajj, M.R.: Design and performance of variable-shaped piezoelectric energy harvesters. J. Intell. Mater. Syst. Struct. 25(2), 174–186 (2014)CrossRefGoogle Scholar
  18. 18.
    Erturk, A., Inman, D.J.: Issues in mathematical modeling of piezoelectric energy harvesters. Smart Mater. Struct. 17(6), 065016 (2008)CrossRefGoogle Scholar
  19. 19.
    Majdoub, M.S., Sharma, P., Cagin, T.: Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Phys. Rev. B Condens. Matter. Mater. Phys. 77(12), 125424 (2008)CrossRefGoogle Scholar
  20. 20.
    Curie, J., Curie, P.: Development by pressure of polar electricity in hemihedral crystals with inclined faces. Bull. Soc. Miner. Crystallogr. Fr. 3(1), 90 (1880)Google Scholar
  21. 21.
    Yan, Z., Jiang, L.: Modified continuum mechanics modeling on size-dependent properties of piezoelectric nanomaterials: a review. Nanomaterials 7(2), 27 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yudin, P.V., Tagantsev, A.K.: Fundamentals of flexoelectricity in solids. Nanotechnology 24(43), 432001 (2013)CrossRefGoogle Scholar
  23. 23.
    Zubko, P., Catalan, G., Tagantsev, A.K.: Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43(1), 387–421 (2013)CrossRefGoogle Scholar
  24. 24.
    Tadigadapa, S., Mateti, K.: Piezoelectric MEMS sensors: state-of-the-art and perspectives. Meas. Sci. Technol. 20(9), 092001 (2009)CrossRefGoogle Scholar
  25. 25.
    Nguyen, T.D., Mao, S., Yeh, Y.W., Purohit, P.K., McAlpine, M.C.: Nanoscale flexoelectricity. Adv. Mater. 25(7), 946–974 (2013)CrossRefGoogle Scholar
  26. 26.
    Heywang, W., Lubitz, H., Wersing, W.: Piezoelectricity: Evolution and Future of a Technology, vol. 114. Springer, Berlin (2008)CrossRefGoogle Scholar
  27. 27.
    Jiang, X., Huang, W., Zhang, S.: Flexoelectric nano-generator: materials, structures and devices. Nano Energy 2(6), 1079–1092 (2013)CrossRefGoogle Scholar
  28. 28.
    Hu, S., Shen, S.: Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Sci. China Phys. Mech. Astron. 53(8), 1497–1504 (2010)CrossRefGoogle Scholar
  29. 29.
    Majdoub, M.S., Sharma, P., Çagin, T.: Dramatic enhancement in energy harvesting for a narrow range of dimensions in piezoelectric nanostructures. Phys. Rev. B Condens. Matter Mater. Phys. 78(12), 121407 (2008)CrossRefGoogle Scholar
  30. 30.
    Bhaskar, U.K., et al.: A flexoelectric microelectromechanical system on silicon. Nat. Nanotechnol. 11(3), 263–266 (2016)CrossRefGoogle Scholar
  31. 31.
    Baroudi, S., Jemai, A., Najar, F.: Modeling and parametric analysis of a piezoelectric flexoelectric nanoactuator. Springer Proc. Phys. 199, 85–101 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang, K.F., Wang, B.L.: An analytical model for nanoscale unimorph piezoelectric energy harvesters with flexoelectric effect. Compos. Struct. 153, 253–261 (2016)CrossRefGoogle Scholar
  33. 33.
    Deng, Q., Kammoun, M., Erturk, A., Sharma, P.: Nanoscale flexoelectric energy harvesting. Int. J. Solids Struct. 51(18), 3218–3225 (2014)CrossRefGoogle Scholar
  34. 34.
    Moura, A., Erturk, A.: A distributed-parameter flexoelectric energy harvester model accounting for two-way coupling and size effects. In: Proceedings of the ASME 2016 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, pp. 1–10, (2016)Google Scholar
  35. 35.
    Moura, A.G., Erturk, A.: Electroelastodynamics of flexoelectric energy conversion and harvesting in elastic dielectrics. J. Appl. Phys. 121(6), 064110 (2017)CrossRefGoogle Scholar
  36. 36.
    Rupa, N.S., Ray, M.C.: Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity. Int. J. Mech. Mater. Des 13(3), 453–467 (2017)CrossRefGoogle Scholar
  37. 37.
    Kundalwal, S.I., Shingare, K.B., Rathi, A.: Effect of flexoelectricity on the electromechanical response of graphene nanocomposite beam. Int. J. Mech. Mater. Des. (2018).
  38. 38.
    Yang, W., Liang, X., Shen, S.: Electromechanical responses of piezoelectric nanoplates with flexoelectricity. Acta Mech. 226(9), 3097–3110 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sidhardh, S., Ray, M.C.: Exact solutions for flexoelectric response in elastic dielectric nanobeams considering generalized constitutive gradient theories. Int. J. Mecha. Mater. (2018).
  40. 40.
    Wang, K.F., Wang, B.L.: Non-linear flexoelectricity in energy harvesting. Int. J. Eng. Sci. 116, 88–103 (2017)CrossRefzbMATHGoogle Scholar
  41. 41.
    Liang, X., Zhang, R., Hu, S., Shen, S.: Flexoelectric energy harvesters based on Timoshenko laminated beam theory. J. Intell. Mater. Syst. Struct. 28(15), 2064–2073 (2017)CrossRefGoogle Scholar
  42. 42.
    Reddy, J.N., Pang, S.D.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103(2), 023511 (2008)CrossRefGoogle Scholar
  43. 43.
    Zhang, D.P., Lei, Y.J., Adhikari, S.: Flexoelectric effect on vibration responses of piezoelectric beams embedded in viscoelastic medium based on nonlocal elasticity theory. Acta Mech. 229(6), 2379–2392 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Liang, X., Hu, S., Shen, S.: Effects of surface and flexoelectricity on a piezoelectric beam. Smart Mater. Struct. 23(3), 035020 (2014)CrossRefGoogle Scholar
  45. 45.
    Yan, Z.: Modeling of a nanoscale flexoelectric energy harvester with surface effects. Phys. E Low Dimens. Syst. Nanostruct. 88, 125–132 (2017)CrossRefGoogle Scholar
  46. 46.
    Toupin, R.: The elastic dielectric. J. Ration Mech. Anal. 5(6), 849–915 (1956)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Erturk, A.: Assumed-modes modeling of piezoelectric energy harvesters: Euler–Bernoulli, Rayleigh, and Timoshenko models with axial deformations. Comput. Struct. 106–107, 214–227 (2012)CrossRefGoogle Scholar
  48. 48.
    Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley, London (2011)CrossRefGoogle Scholar
  49. 49.
    Meirovitch, L.: Fundamentals of Vibrations by Leonard Meirovitch (20.... McGraw-Hill, New York (2001)Google Scholar
  50. 50.
    Erturk, A., Inman, D.J.: An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations. Smart Mater. Struct. 18(2), 025009 (2009)CrossRefGoogle Scholar
  51. 51.
    Clough, R.W., Penzien, J.: Dynamics of Structures. Computers and Structures, Inc., Berkeley (2003)zbMATHGoogle Scholar
  52. 52.
    Trindade, M., Benjeddou, A.: Effective electromechanical coupling coefficients of piezoelectric adaptive structures: critical evaluation and optimization. Mech. Adv. Mater. Struct. 16(3), 210–223 (2009)CrossRefGoogle Scholar
  53. 53.
    Lesieutre, G.A., Davis, C.L.: Can a coupling coefficient of a piezoelectric device be higher than those of its active material? J. Intell. Mater. Syst. Struct. 8(10), 859–867 (1997)CrossRefGoogle Scholar
  54. 54.
    Chu, B., Salem, D.R.: Flexoelectricity in several thermoplastic and thermosetting polymers. Appl. Phys. Lett. 101(10), 103905 (2012)CrossRefGoogle Scholar
  55. 55.
    Guney, H.Y.: Elastic properties and mechanical relaxation behaviors of PVDF (poly(vinylidene fluoride)) at temperatures between \(-20\) and \(100^\circ \text{ C }\) and at 2 MHz ultrasonic frequency. J. Polym. Sci. Part B Polym. Phys. 43(20), 2862–2873 (2005)CrossRefGoogle Scholar
  56. 56.
    Murayama, N., Nakamura, K., Obara, H., Segawa, M.: The strong piezoelectricity in polyvinylidene fluroide (PVDF). Ultrasonics 14(1), 15–23 (1976)CrossRefGoogle Scholar
  57. 57.
    Akgöz, B., Civalek, Ö.: Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Compos. Struct. 98, 314–322 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  • S. Faroughi
    • 1
  • E. F. Rojas
    • 2
  • A. Abdelkefi
    • 2
    Email author
  • Y. H. Park
    • 2
  1. 1.Department of Mechanical EngineeringUrmia University of TechnologyUrmiaIran
  2. 2.Department of Mechanical and Aerospace EngineeringNew Mexico State UniversityLas CrucesUSA

Personalised recommendations