Piezoelectric Beam, Plate and Truss

  • André Preumont
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 246)


This chapter begins with a discussion of the constitutive equations of a general piezoelectric material and of the various actuation and sensing modes. Next, Hamilton’s principle is applied to the piezoelectric Euler–Bernoulli beam, leading to the definition of the piezoelectric loads associated with the various electrode shapes when used as actuator (voltage driven) and a discussion of the laminar sensors (when associated with a charge amplifier); the duality is highlighted, and various modal filters are discussed. The model is applied to a collocated piezoelectric beam, leading to alternating poles and zeros, and a special attention is drawn on the influence of modal truncation on the location of the zeros. Next, the attention is given to the two-dimensional constitutive equations of a piezoelectric laminate and the Kirchhoff plate theory (this work is implemented in a finite element code SAMCEF); the equivalent piezoelectric loads and sensor output are defined, and the duality is pointed out. The beam theory and the plate theory are compared, and the limitations of the beam model for a collocated structure are explained. The chapter ends with the modelling of a piezoelectric truss where one or several bars have been replaced by active struts consisting of a collocated linear piezoelectric actuator and a force sensor. The chapter concludes with a short list of references and a set of problems, including Rosen’s piezoelectric transformer.


Piezoelectric material Constitutive equations Piezoelectric beam Laminar sensor Modal sensor Duality Collocated beam Modal truncation Kirchhoff plate theory Piezoelectric laminate Equivalent piezoelectric loads Finite element Piezoelectric truss Rosen’s piezoelectric transformer 


  1. 1.
    Agarwal BD, Broutman LJ (1990) Analysis and performance of fiber composites, 2nd edn. Wiley, New YorkGoogle Scholar
  2. 2.
    Allik H, Hughes TJR (1970) Finite element method for piezoelectric vibration. Int J Numer Methods Eng 2:151–157CrossRefGoogle Scholar
  3. 3.
    Benjeddou A (2000) Advances in piezoelectric finite element modeling of adaptive structural element: a survey. Comput Struct 76:347–363CrossRefGoogle Scholar
  4. 4.
    Burke SE, Hubbard JE (1987) Active vibration control of a simply supported beam using spatially distributed actuator. IEEE Control Syst Mag 7:25–30CrossRefGoogle Scholar
  5. 5.
    Cady WG (1946) Piezoelectricity: an introduction to the theory and applications of electromechanical phenomena in crystals. McGrawHill, New YorkGoogle Scholar
  6. 6.
    Crawley EF, Lazarus KB (1991) Induced strain actuation of isotropic and anisotropic plates. AIAA J 29(6):944–951CrossRefGoogle Scholar
  7. 7.
    de Marneffe B (2007) Active and passive vibration isolation and damping via shunted transducers. Ph.D. thesis, Université Libre de Bruxelles, Active Structures LaboratoryGoogle Scholar
  8. 8.
    Dimitriadis EK, Fuller CR, Rogers CA (1991) Piezoelectric actuators for distributed vibration excitation of thin plates. Trans ASME J Vib Acoust 113:100–107CrossRefGoogle Scholar
  9. 9.
    Eer Nisse EP (1967) Variational method for electrostatic vibration analysis. IEEE Trans Sonics Ultrason SU-14(4):153–160Google Scholar
  10. 10.
    Garcia Lage R, Mota Soares CM, Mota Soares CA, Reddy JN (2004) Layerwise partial mixed finite element analysis of magneto-electro-elastic plates. Comput Struct 82:1293–1301CrossRefGoogle Scholar
  11. 11.
    Heyliger P, Pei KC, Saravanos D (1996) Layerwise mechanics and finite element model for laminated piezoelectric shells. AIAA J 34(11):2353–2360CrossRefzbMATHGoogle Scholar
  12. 12.
    Holterman J, Groen P (2012) Piezoelectric materials and components, Stichting Applied PiezoGoogle Scholar
  13. 13.
    Hwang W-S, Park HC (1993) Finite element modeling of piezoelectric sensors and actuators. AIAA J 31(5):930–937CrossRefGoogle Scholar
  14. 14.
    Lee C-K (1990) Theory of laminated piezoelectric plates for the design of distributed sensors/actuators - Part I: governing equations and reciprocal relationships. J Acoust Soc Am 87(3):1144–1158CrossRefGoogle Scholar
  15. 15.
    Lee C-K, Moon FC (1990) Modal sensors/actuators. Trans ASME J Appl Mech 57:434–441CrossRefGoogle Scholar
  16. 16.
    Lee C-K, Chiang W-W, O’Sullivan TC (1991) Piezoelectric modal sensor/actuator pairs for critical active damping vibration control. J Acoust Soc Am 90(1):374–384CrossRefGoogle Scholar
  17. 17.
    Lerch R (1990) Simulation of piezoelectric devices by two and three dimensional finite elements. IEEE Trans Ultrason Ferroelectr Freq Control 7(3):233–247CrossRefGoogle Scholar
  18. 18.
    Piefort V (2001) Finite element modeling of piezoelectric active structures. Ph.D. thesis, Université Libre de Bruxelles, Active Structures LaboratoryGoogle Scholar
  19. 19.
    Preumont A (2006) Mechatronics, dynamics of electromechanical and piezoelectric systems. Springer, BerlinzbMATHGoogle Scholar
  20. 20.
    Preumont A, François A, de Man P, Piefort V (2003) Spatial filters in structural control. J Sound Vib 265:61–79MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rosen CA (1956) Ceramic transformers and filters. In: Proceedings of the electronic component symposium, pp 205–211Google Scholar
  22. 22.
    Tiersten HF (1967) Hamilton’s principle for linear piezoelectric media. In: Proceedings of the IEEE, pp 1523–1524Google Scholar
  23. 23.
    Uchino K (2000) Ferroelectric devices. Marcel Dekker, New YorkGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Active Structures LaboratoryUniversité Libre de BruxellesBrusselsBelgium

Personalised recommendations