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TABLE 20.5 Resonant Frequency Expressions for
Various Piezoelements
Expression (L, length; D,
Operating Mode diameter; T, width)
Transverse mode, thin bar N 1 = f r × L
Radial mode, disc N p = f r × D
Thickness mode, disc N t = f r × T
Length mode, cylinder N 3 = f r × L
Shear mode, plate N s = f r × T

E

L
PZT

Steel (PZT)
(a) (b)
FIGURE 20.82

depends on the thickness of the layer of piezomaterial. Maximum possible relative change in length is up
to 0.13%. The shortest time of expansion is one third of the period at resonant frequency oscillations of
the mechanical system containing piezoelement. Piezoelement resonant properties are described by fre-
quency constant N i , which is the resonance frequency f r multiplied by the linear dimension governing the
resonance. Table 20.5 shows the expressions for various operating modes.
Voltage being changed (V remains constant after the change), piezomaterial continues to expand/
contract in the same direction, decaying exponentially towards stability. This drift, known as creep, can
be estimated by
∆Lt() = ∆L 1 + γ lg0.1) (20.35)
(
where ∆L is 0.1 s expansion after the positioning process, γ is the drift factor. It depends on the design and
mechanical load and lies between 0.01 and 0.02. Hysteresis is common in the piezomaterials as well. PZT
hysteresis is a fairly constant fraction of the stroke and the width of the hysteresis curve for it can be as
large as 20% of the stroke. Due to compensation strategies hysteresis errors decrease up to 3%. Piezoelement
with electrodes laminated on to it is electrical capacitor. Because of extremely high piezomaterial internal
resistance (more than 100 MΩ) only a small discharge current flows if piezomaterial remains static in the
expanded state. Thus, the piezoelement being separated from the source of high voltage, its expansion is
decreasing slowly. This, in turn, causes a change in the charge, which results in a current:

dV
------- =
i = dQ C------ (20.36)
dt dt
where Q is the charge, C is the capacity, and V is the voltage.
Table 20.6 shows different piezoelement sensing/actuating possibilities for some shape cases. Basic
types of piezoactuators used are stacked and of laminar design. Laminar design actuators consist of
piezoelectric strips with electrodes bonded onto them. Stacked (multilayer) actuators consist of some
thin wafers of piezoactive material between metallic electrodes in parallel connection (Fig. 20.82(a)).

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