Abstract
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 The chapter ends with the modelling of a piezoelectric truss where one or several bars have been replaced by active struts consisting of a collocated linear piezoelectric actuator and a force sensor.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.
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Notes
- 1.
Another observation is that a small linear shift appears in the phase diagram, due to the fact that these results have been obtained digitally (the sampling is responsible for a small delay in the system).
- 2.
The presence of \(\omega _i\) instead of \(\varOmega _i\) at the denominator of (4.77) is insignificant in practice.
- 3.
The piezoelectric layer contributes to A, B, and D with the stiffness properties under constant electric field.
- 4.
Piezoelectric transformers were introduced by Rosen in 1956; they have been very successful for low power applications such as power supply of laptop computers. Due to the high energy density of piezoelectric materials, the high electromechanical coupling factors, and the high-quality factor of the mechanical resonance (low damping), they tend to be lighter and more efficient than wire wound transformers whose efficiency tends to decrease rapidly as the size is reduced. Besides, they are free from electromagnetic interference and the solid-state nature of piezoelectric transformers is the key to mass production.
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Preumont, A. (2018). Piezoelectric Beam, Plate and Truss. In: Vibration Control of Active Structures. Solid Mechanics and Its Applications, vol 246. Springer, Cham. https://doi.org/10.1007/978-3-319-72296-2_4
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