**2.1 Inverse piezoelectric effect of piezoelectric actuator**

The piezoelectric constitutive equation states that the inverse piezoelectric effect of piezoelectric actuators can convert electrical energy into mechanical energy, resulting in small displacement and force changes with high resolution.

*Active Vibration Suppression Based on Piezoelectric Actuator DOI: http://dx.doi.org/10.5772/intechopen.103725*

**Figure 1.** *Structure diagram of piezoelectric stack actuator.*

The piezoelectric stack actuator used in this chapters is formed by stacking many thin piezoelectric ceramic sheets, and its structure is shown in **Figure 1**. Each piezoelectric ceramic sheet is equipped with electrodes and is separated by an insulating layer. The elongation of the piezoelectric ceramic sheet along the polarization direction is mainly related to the applied electric field strength but has nothing to do with its thickness. With the layered stack structure, the elongation of the piezoelectric ceramic sheets can be accumulated, so that the piezoelectric stack actuator can still generate a relatively large displacement at a lower operating voltage.

According to the inverse piezoelectric effect of piezoelectric materials, only considering the longitudinal force and elongation of the piezoelectric ceramic sheet under the action of the driving voltage, and assuming that the strain is uniformly distributed in the longitudinal polarization direction, then the longitudinal stress of a single piezoelectric ceramic sheet can be expressed as:

$$
\sigma = E\_p(\varepsilon - d\_{33}E) \tag{1}
$$

where *σ* is the longitudinal stress; *Ep* is the initial elastic modulus; *ε* is the longitudinal strain; *d*<sup>33</sup> is the longitudinal piezoelectric strain constant; *E* is the longitudinal electric field strength.

Assuming that the force-bearing area of the piezoelectric ceramic sheet is *Ap*, the thickness is *h*, the voltage applied at both ends is *V* and the resulting thickness deformation is *δ*. Then the longitudinal strain*ε* and longitudinal electric field strength *E* in Eq. (1) can be expressed as:

$$
\varepsilon = \delta/h \tag{2}
$$

$$E = V/\hbar$$

The force on the piezoelectric ceramic sheet can be expressed as the stress *σ* multiplied by the force-bearing area *Ap*, and the relationship between the output force and the force on the piezoelectric ceramic sheet is force and reaction force. Combining Eqs. (1–3), the output force of the piezoelectric ceramic sheet is:

$$f = -\sigma A\_p = A\_p E\_p / h(d\_{33}V - \delta) \tag{4}$$

Let *k* ¼ *ApEp=h*, which is the inherent constant of the piezoelectric ceramic sheet, the output force expression of the piezoelectric ceramic sheet can be simplified as:

$$f = k(d\_{33}V - \delta) \tag{5}$$

According to the output force expression Eq. (5), the output displacement of the piezoelectric ceramic sheet can be obtained as:

$$
\delta = d\_{33}V - f/k \tag{6}
$$

Since the piezoelectric stack actuator cascades several piezoelectric ceramic sheets in voltage parallel and physical series, its output force is the same as that of a single piezoelectric ceramic sheet, and its output displacement is the sum of the output displacement of all piezoelectric ceramic sheets:

$$\begin{cases} \quad f\_x = f \\ \Delta A = n\delta \end{cases} \tag{7}$$

where *f <sup>z</sup>* is the output force of the piezoelectric actuator, Δ*A* is the output displacement of the piezoelectric actuator, and*n* is the number of piezoelectric ceramic sheets in the piezoelectric actuator.

Although the output displacement of the piezoelectric stack actuator is the sum of the output displacements of all piezoelectric ceramic sheets, its stroke is still in the order of microns. In practical applications, it is usually necessary to cooperate with a displacement amplifying mechanism [14–16]. Therefore, it is necessary to analyze the displacement amplification characteristics of the displacement amplifying mechanism.

## **2.2 Displacement amplifying mechanism of piezoelectric actuator**

The mechanical displacement amplifying mechanism adopted in this chapter is based on the principle of triangular amplification to mechanically amplify the displacement of the piezoelectric actuator, to make up for the shortcoming of its insufficient stroke and expand its effective stroke. Dimensions such as the coordinate direction and angle of the displacement amplifying mechanism are shown in **Figure 2a**. The piezoelectric stack actuator with length A is placed in the x-direction of the displacement amplifying mechanism, and the inclination angle formed by the horizontal direction and the hypotenuse of the amplifying mechanism is *α*.

The amplification principle of the mechanical displacement amplifying mechanism is shown in **Figure 2b**. When the displacement changes in the *x*-direction, the inverse change of the displacement occurs in the *y*-direction. The displacement magnification

**Figure 2.**

*Displacement amplifying mechanism: (a) dimension labeling diagram; (b) amplifying schematic diagram.*

is defined as the ratio of the displacement change in the *y*-direction to the displacement change in the *x*-direction, which can be expressed as:

$$
\gamma = \frac{\Delta B}{\Delta A} \tag{8}
$$

According to the geometric transformation relationship shown in **Figure 2b**, we can get:

$$\begin{aligned} \Delta A &= l \cos a - l \cos \left( a - \Delta a \right) \\ \Delta B &= l \sin a - l \sin \left( a - \Delta a \right) \end{aligned} \tag{9}$$

Then the displacement magnification can be organized as:

$$\gamma = \frac{\Delta B}{\Delta A} = \frac{l \sin a - l \sin \left(a - \Delta a\right)}{l \cos a - l \cos \left(a - \Delta a\right)} = \frac{\tan a (1 - \cos \left(\Delta a\right)) + \sin \left(\Delta a\right)}{1 - \cos \left(\Delta a\right) - \tan a \sin \left(\Delta a\right)}\tag{10}$$

Since the stroke of the piezoelectric actuator is in the order of microns, the variation in length *ΔA* and the variation in inclination angle *Δα* are quite small. According to the equivalent infinitesimal principle: sin ð Þ <sup>Δ</sup>*<sup>α</sup>* <sup>≈</sup>Δ*α*, 1 � cosð Þ <sup>Δ</sup>*<sup>α</sup>* <sup>≈</sup> <sup>1</sup> <sup>2</sup> ð Þ <sup>Δ</sup>*<sup>α</sup>* <sup>2</sup> , tan ð Þ Δ*α* ≈Δ*α*. The Eq. (10) can be further organized as:

$$\gamma = \frac{\Delta a \tan a + 2}{\Delta a - 2 \tan a} \approx -\frac{1}{\tan a} \tag{11}$$

It can be seen from Eq. (11) that the magnification of the triangular mechanical displacement amplifying mechanism is not related to the length but is only related to the size of the inclination angle *α*. The negative sign indicates that the displacement changes in the *y*- and *x*-directions are opposite. The relationship between magnification and inclination angle is described in **Figure 3**.

It can be seen from **Figure 3** that the smaller the inclination angle, the larger the magnification (regardless of positive and negative). As the inclination angle increases, the change in magnification becomes insignificant. When the inclination angle reaches 45 degrees, the magnification is close to 1.

**Figure 3.** *Relationship between magnification and inclination angle.*
