**2.2 Piezoelectric stick-slip actuators using asymmetrical flexure hinges**

The trapezoid-type stick–slip actuator developed above can make the slider move by oblique displacement of the flexure hinge mechanism, but it is hard to realize a larger tangential displacement along the motion direction. Therefore, we proposed an idea of using a new four-bar mechanism with different thicknesses of both side flexure hinges, as shown in **Figure 3**. There are considerable improvements in output performance of the actuator based on this design idea.

A four-bar mechanism using asymmetrical flexure hinges is combined with a symmetrical indenter to generate controllable tangential displacement [11]. As shown in **Figure 3(a)**, the stator is mainly comprised of an asymmetrical flexure

#### **Figure 3.**

*Research on the piezoelectric stick–slip actuators using asymmetrical flexure hinges. (a) the basic structure of the stator. (b) and (c) simulation results of the asymmetrical flexure hinge mechanism. (d) Prototype and (e) experimental system of the proposed actuator. (f)–(h) The test results of the actuator [11]. (i) The basic structure of the rhombus-type flexure hinge mechanism. (j) and (k) the experimental results of the actuator [12].*

hinge driving mechanism, a preload screw, a tungsten chrome steel sheet and a piezoelectric stack. There are right-circle flexure hinges located in both side of the asymmetrical flexure hinge driving mechanism respectively. The thickness of two right-circle flexure hinges on the one side is *T*1, and the other side is *T*2 (*T*<sup>1</sup> ≠ *T*2). And the thickness between the stack and the indenter is *T*3. By changing the thickness *T*1, *T*2 and *T*3 of the actuators, the oblique movement required by the indenter can be realized.

Deformation of the actuator caused by the different minimum thicknesses of the flexure hinges results in a larger tangential displacement. In the simulation analysis stage, the initial thickness *T*1, *T*2 and *T*3 of the asymmetrical flexure hinge driving mechanism are set as 0.3 mm, 0.3 mm and 4.5 mm, respectively. The simulation results of the asymmetrical flexure hinge driving mechanism under the different ratio of *T*1/*T*2 and the different thickness of *T*3 is shown in **Figure 3(b)** and **(c)**. Based on the results, the mechanism can not only generate the tangential displacement but also change the normal force. Finally, the structure parameters are identified as ratio *T*1/*T*2 = 0.2 and thickness *T*3 = 1 mm.

The actuator mainly consists of a base, a slider, a stator and an adjusting stage, as shown in **Figure 3(d)**. The adjusting stage is used to adjust the preload force between the stator and the slider. The experimental system of the prototype is shown in **Figure 3(e)**. All the experimental equipment is placed on the vibration isolation optical platform.

The experimental results are shown in **Figure 3(f )**–**(h)**. First, when the voltage is 100 Vp-p, the velocity is parabolic with the increase of frequency. When the frequency is 470 Hz, and the locking force is 2 N respectively, the obtained maximum velocity of the actuator is 11.75 mm/s. When the locking forces are 3.5 N and 5 N, the obtained maximum velocities of the actuator under the frequency of 490 Hz are 13.72 and 15.04 mm/s, respectively. Besides, the minimum starting voltage is 20.25 Vp-p at the locking force of 3.5 N, and the displacement resolution is 65 nm. Since frictional force drives the slider to produce linear motion, the improvement of the load capacity may be attributed to the increase of frictional forces as the locking force increased. The maximum loads are 160 g, 300 g and 440 g under locking forces of 2 N, 3.5 N and 5 N, respectively. And the maximum efficiency of the actuator under the locking force of 2 N is 0.98% at the load of 80 g. The maximum efficiency of the actuator under the locking force of 3.5 N is 2.43% at the load of 200 g. The maximum efficiency of the actuator under the locking force of 5 N is 3.66% at the load of 280 g.

Furthermore, a rhombus-type flexure hinge mechanism with an asymmetrical structure is proposed [12], which is easy to produce the parasitic motion of the large stroke. The proposed rhombus-type flexure hinge mechanism can improve the velocity of the actuator with a compact stator. The experimental results show that the prototype achieves a maximum velocity of 13.08 mm/s at a frequency of 570 Hz under the voltage of 100 Vp-p; and the maximum efficiency is 1.26% with a load of 135 g.

#### **2.3 Mode conversion piezoelectric stick-slip actuator**

A novel stick–slip piezoelectric stick–slip actuator based on mode conversion flexure hinge is proposed which is mainly uses the three chutes of the mode conversion flexure hinge to make the driving foot produce lateral displacement [13], as illustrated in **Figure 4(a)** and **(b)**. Under the influence of the symmetrical waveform, the mode conversion flexure hinge can obtain a lateral displacement and push the slider to move a distance.

The mode conversion flexure hinge is comprised of a bracket, a driving foot, four circular notch joins, a beam and three chutes. The deformation of the *The Asymmetric Flexure Hinge Structures and the Hybrid Excitation Methods for Piezoelectric… DOI: http://dx.doi.org/10.5772/intechopen.95536*

**Figure 4.**

*Research on the mode conversion piezoelectric stick–slip actuator. (a) The detailed structure of the actuator and (b) the stator. (c) Operation principle of the actuator. (d)–(f) The experimental results of the actuator [13].*

mode conversion flexure hinge can be produced easily due to four circular notch joins. A symmetrical waveform voltage is chosen to excite the stack. As shown in **Figure 4(c)**, the operation principle of the prototype can be described in the following three steps:

Step 1: no voltage is applied to the stack, and its original length remains. The contact point *P* is between the driving foot and slider where there is an initial preload force *F*0.

Step 2: the point *P* will generate a displacement of positive directions on both *ox* and *oy* axis with the gradual increase of voltage. The pressure force *F*y between the mode conversion flexure hinge and slider improves due to the displacement of the positive direction on the *oy* axis. The slider produces a displacement of the positive direction on the *ox* axis. Within a certain range, as the pressure force *F*y increases, the output force of the actuator will be improved. Thereby, the static friction force *f*<sup>s</sup> makes the slider generate a displacement *d*1.

Step 3: the stack restores to its original condition with the decrease of voltage. At this moment, kinetic friction force *f*d as a resistance force produces between the mode conversion flexure hinge and the slider. The decrease in pressure force *F*y makes the force *f*d decrease. Then, a slight back displacement *d*2 of the slider is produced, which is opposite to the direction of *d*1.

Finally, a displacement of the slider *d* (*d* = *d*1-*d*2) is produced. By repeating these steps, the prototype can achieve a large motion in the positive direction of the *ox* axis.

It is an important parameter for the angle *θ* of chutes to influent its output performance. Different angles of chutes (30°, 45° and 60°) are applied to the proposed mode conversion flexure hinge, which is simulated by the finite element method to obtain an optimal angle parameter. And the displacement *U*x of the point *P* in the positive direction of the *ox* axis and the maximum equivalent stress with different angles are listed in **Table 1**. The largest deformation of 2.85 *μ*m is obtained for the *ox* axis displacement *U*x of the point *P* under the angle of 45°, and the maximum equivalent stress is 105.86 MPa.

With the increase of frequency, the velocity of the prototype first increases and then decreases, whose peak can be reached at 415 Hz, as illustrated in **Figure 4(d)**. For the designed actuator, the maximum velocity of 18.84 mm/s, 17.82 mm/s and 14.04 mm/s are obtained respectively under the locking force of 3 N, 4 N and 5 N.

Moreover, the vibrations and the fitting results in *xoy* plane of the driving foot are shown in **Figure 4(e)** and **(f )**. An ellipse motion track obtained by the driving foot of the actuator is beneficial to the motion of the slider. It can be seen from experiment results that the maximum velocity and efficiency is 18.84 mm/s and 2.07% when the voltage of the symmetrical waveform and the frequency are 100 Vp-p and of 415 Hz, respectively. Besides, the maximum load and the displacement resolution can reach 360 g and 50 nm in terms of the actuator.

#### **2.4 Bidirectional piezoelectric stick-slip actuator**

The asymmetric flexure hinges are generally structurally asymmetric, making it difficult to achieve good consistency in bidirectional output performance. In this work, a coupled asymmetrical flexure hinge mechanism is developed to realize the bidirectional motion of the piezoelectric stick–slip actuator [14], as shown in **Figure 5**.

The bidirectional stick–slip actuator uses coupled asymmetrical flexure hinge mechanisms and symmetrical indenter to generate the controllable tangential displacement. As shown in **Figure 5(a)**, the static simulation of the FEM is used to analyze the deformation of flexure hinges with different sizes. Since a larger oblique displacement will provide better output performance, *A*1 = 1 mm and *K* = *A*2/*A*3 = 0.2 with the larger oblique displacement are selected for the final Structure design.

In this section, an MDR-based model is proposed to display the contact and frictional force between the slider and indenter for stick–slip actuators.

The framework of the MDR consists of two preliminary steps to be performed. As shown in **Figure 5(c)**, a one-dimensional profile *g*(*x*) and the elastic half-space by an elastic foundation which are in possession of normal stiffness *k*y and tangential stiffness *k*x is adopted to replace the three-dimensional profile *y* = *f* (*x*, *z*).


**Table 1.** *The simulation results of the proposed actuator.* *The Asymmetric Flexure Hinge Structures and the Hybrid Excitation Methods for Piezoelectric… DOI: http://dx.doi.org/10.5772/intechopen.95536*

**Figure 5.**

*Research on the bidirectional piezoelectric stick–slip actuator. (a) The static simulation deformation of the flexure hinges. (b) The prototype of the actuator. (c)–(d) MDR model. (e)–(g) The experimental results of the actuator [14].*

The stiffness of every individual spring can be expressed as follows:

$$
\boldsymbol{k}\_y = \boldsymbol{E}^\* \boldsymbol{\Delta x}, \boldsymbol{k}\_x = \mathbf{G}^\* \boldsymbol{\Delta x} \tag{1}
$$

Δ*x* represents the distance between the springs of elastic foundation, and the effective elastic modulus is defined as:

$$\frac{1}{E\_{\text{1}}} = \frac{1 - \nu\_{\text{1}}^2}{E\_{\text{1}}} + \frac{1 - \nu\_{\text{2}}^2}{E\_{\text{2}}} \tag{2}$$

$$\frac{1}{G} = \frac{1 - \nu\_1^2}{G\_1} + \frac{1 - \nu\_2^2}{G\_2} \tag{3}$$

$$G\_{\mathbf{i}\_{1,2}} = \frac{E\_{\mathbf{i}\_{1,2}}}{2\left(1 + \nu\_{\mathbf{i}\_{1,2}}\right)}\tag{4}$$

*E*<sup>1</sup> *G*1 and *ν*1 represent the Young's modulus of the indenter, its shear modulus and the Poisson's ratio, respectively. *E*2, *G*2 and *ν*2 are the corresponding material parameters of the half-space.

**Figure 5(d)** shows the schematic diagram of the experimental system. Here, the friction coefficient of the slider in contact with the base is set to *μ*1. The friction coefficient that cylindrical indenter drives the slider is set as *μ*. The indenter is made of a material with parameters of *E*1, *G*1 and *ν*1, and elastic parameters of the slider material are *E*2, *G*2 and *ν*2.

Further modeling is based on experimental data. Newton's second law then yields as follows:

$$M\ddot{X} = F\_x - F\_{base} - mg\tag{5}$$

*F*x represents the frictional force between the indenter and slider. With the movement of the slider, a sliding friction force *F*base goes up between the slider and the base. It is calculated as

$$F\_{\text{base}} = \text{sgn}\left(\dot{X}\right)\mu\_{\text{i}}F\_{\text{y}} \tag{6}$$

Since in the numerical calculation, we can only use a discrete model as follows. In the circumstances, the value of the elongation of spring adds relative displacement D*x* that can be calculated as follows:

$$
\Delta \ddot{\mathbf{x}} = \left[ \mathbf{x}(\mathbf{t}\_i + \Delta \mathbf{t}) - \mathbf{x}(\mathbf{t}\_i) \right] - \left[ X(\mathbf{t}\_i + \Delta \mathbf{t}) - X(\mathbf{t}\_i) \right] \tag{7}
$$

where *t*i and Δ*t* are the incrementally increasing time and the step of numerical time integration of Eq. (5) (time increment value).

The function *g*(*x*) is found for a cylinder of finite length that is indented into a half-space. The form of this function is defined as follows:

$$\log\left(\varkappa\right) = \beta \frac{L^2}{R} \left(\frac{aL}{\varkappa} + 1\right) \exp\left(-\frac{aL}{\varkappa}\right) \tag{8}$$

where *L* and *R* are the cylinder length and its radius, respectively. In addition, the ratio *L*/*R* determines the values of the parameters *α* and *β*.

The velocity is measured at a voltage with the saw-tooth wave of 100 Vp-p, and the duty ratio "*S*" is 90%, as shown in **Figure 5(e)**. As the frequency changes, the velocity trend shows a parabolic. And the different locking forces have different optimal driving frequencies. The *x* direction is taken as the moving direction of the slider. When the frequency is 850 Hz and locking force is 1 N, the maximum velocities in the positive direction and negative direction are 10.14 mm/s and 9.99 mm/s. When the frequency is 750 Hz and locking force is 2 N mm/s, the maximum velocities of the actuator are 8.56 and 8.72 mm/s in the positive direction and negative direction. When the frequency is 650 Hz and locking force is 3 N, the maximum velocities of the actuator are about 4.75 and 4.84 mm/s in the positive direction and negative direction.

As shown in **Figure 5(f )**, the solid lines and the dotted lines are the displacement curves and the simulation displacement curve of the slider under different locking forces. When the voltage is in a stage of rapid decrease, the dynamic friction *The Asymmetric Flexure Hinge Structures and the Hybrid Excitation Methods for Piezoelectric… DOI: http://dx.doi.org/10.5772/intechopen.95536*

will be generated between the indenter and the slider. Meanwhile, the difference in the direction of the dynamic friction and the slider results in the production of the backward motion. Therefore, every step can produce a backward motion. Although it can be seen from the dotted line that the slider moves in forwarding and backward directions, the average velocity that is larger than zero will result in a net directional motion.

Experimental results indicate that the maximum loads tested in the experiment are 0.6 N, 0.9 N and 1.5 N under locking forces of 1 N, 2 N and 3 N, respectively. Furthermore, when the locking force and the minimum starting voltage are 2 N and 31.9 Vp-p respectively, the forward and reversed displacement resolutions of the actuator are 91.1 nm and 74.4 nm. When the voltage and the frequency are 100 Vp-p and 850 Hz, the maximum output velocity and the maximum load of the actuator in the positive x-direction is 10.14 mm/s and 1.5 N. When the load is 90 g, a locking force is 5 N, and the velocity is 5.48 mm/s, the actuator can reach the maximum efficiency of 0.57%.
