*3.2.2 A flexure-based EVC vibrator actuated by two parallel and one perpendicular piezoelectric actuators*

In this work, a flexure-based EVC vibrator actuated by two parallel and one perpendicular piezoelectric actuator was proposed [15]. The 3D model of the developed

#### **Figure 11.**

*Illustration of 3D model, experimental setup and testing results. (a) 3D model of the developed EVC vibrator. (b) Testing experimental setup. (c) Results of motion stroke. (d) Results of the frequency responses. (e) Results of the resolution test. (f) the synthesized cutting tool locus and the projections [15].*

EVC vibrator is shown in **Figure 11(a)**. The EVC vibrator mainly consists of three PEAs (piezoelectric actuators) and capacity probes, two compliant mechanisms. Two compliant mechanisms were independently manufactured considering the complexity of machining. Then the compliant mechanisms were assembled by screws connection. For easy analysis, three axes were defined as follows: axis along PEA1 motion direction was defined as *x1*, axis along PEA2 motion direction was defined as *x2*,

## **Figure 12.**

*Illustration of 3D model, prototype and testing results. (a) 3D model of the developed EVC vibrator. (b) Mechanical structure of compliant mechanism. (c) Fabricated prototype. (d) Simulated tool locus. (e) Actual tool locus. (f) Tracking performance. (g) Step response [16].*

## *Design, Analysis and Testing of Piezoelectric Tool Actuator for Elliptical Vibration Cutting DOI: http://dx.doi.org/10.5772/intechopen.103837*

and axis along PEA3 motion direction was defined as *z*. Additionally, the kinematic modeling was established. The offline tests were carried out to investigate the performance based on the experimental setup shown in **Figure 11(b)**. The motion stroke performances along three axes are shown in **Figure 11(c)**. It can be seen that the maximum displacement along *z-*axis can reach up to 26 μm. The maximum displacements along *x1* and *x2* axes are not the same which are 22 μm and 24 *μm*, respectively, due to the manufacturing error. A swept excitation method with frequency from 100 Hz to 3000 Hz was performed to assess the dynamic characteristics. **Figure 11(d)** shows that the natural frequency along *z*, *x1* and *x2* axes are about 1901 Hz, 1889 Hz and 1895 Hz, respectively, which is enough for ultra-precision machining. The resolution performance results are shown in **Figure 11(e)** through stair excitation tests. The resolution of motion axes along *z*, *x1* and *x2* axes are about 33 nm, 35 nm and 36 nm, respectively. Meanwhile, the synthesized cutting tool locus and the projections are shown in **Figure 11(f )** which validate the correctness of the kinematic model and feasibility of the developed EVC vibrator.

### *3.2.3 A flexure-based EVC vibrator actuated by four parallel piezoelectric actuators*

In this work, a flexure-based EVC vibrator actuated by four parallel piezoelectric actuators was proposed [16]. The mechanical structure of the developed EVC vibrator is shown in **Figure 12(a)**. It consists of one fixture base, one compliant mechanism, four piezoelectric actuators and capacity sensors. The compliant mechanism is shown in **Figure 12(b)**, which was fabricated by wire EDM. The prototype of developed EVC vibrator was fabricated and assembled as shown in **Figure 12(c)**. In order to investigate the kinematic characteristics, the kinematic model was established and the simulation results are shown in **Figure 12(d)**, of which I represents reference ellipse, II represents ellipse obtained by changing the acting locations of the four piezoelectric actuators, III represents ellipse obtained by changing the phase shifts of actuated signals, IV represents ellipse obtained by changing the amplitudes of actuated signals. **Figure 12(e)** shows the measured tool locus which is an ellipse in 3D space. The measured tool locus are in accordance with the simulated locus shown in **Figure 12(d)**. **Figure 12(f)** shows the tracking performance in *x-*direction. A sine wave with amplitude of 6 *μ*m and frequency of 1 Hz was adopted as the command signal. The maximum following error is about 12 nm, which is less than 4% of the full span. The result of step response along *x-*direction is shown in **Figure 12(g)**. There are almost no overshoot and steady errors by utilizing a typical PID controller.
