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

The stiffness of the guide mechanism in the *z*2 axial direction can be considered as a statically indeterminate beam. Then the analytical stiffness modeling was carried out. Meanwhile, a constant force of 1 N was imposed on the input end. Simulation result is shown in **Figure 2(e)**. Additionally, the dynamics analysis was also conducted based on the Lagrange equation and FEA method. It can be seen that if the fixed condition was improved, the second natural frequency will become the first natural frequency, which is helpful to achieve a large working bandwidth. The results of stiffness and natural frequency along the working direction both from analytical modeling and FEA were obtained. The comparison results of FEA and analytical modeling are shown in **Table 1**.

According to the analysis above, a prototype was fabricated. Offline tests were carried out to evaluate the performances. The experimental setup is shown in **Figure 3(a)**. As shown in **Figure 3(b)** are the results of the step responses along *z*1 axis and *z*2 axis. The rising time are 1.9 ms and 1.5 ms, and the setting time are respectively 3.43 ms and 3.64 ms. There are no steady errors. The amplitudefrequency responses are shown in **Figure 3(c)**. It can be seen that the first natural frequency is about 1200 Hz. The motion stroke and resolution tests are respectively shown in **Figure 3(d)** and **(e)**, the maximum motion stroke of *z*1 axis is about 15 μm, the maximum motion stroke of *z*2 axis is about 19 μm. The resolution of *z*1 and *z*2 axes are approximately 15 nm and 30 nm. **Figure 3(f )** shows the real experimental tooltip displacement in different phase shifts, which has a good agreement with the simulation results shown in **Figure 3(g)**. Additionally, the developed EVC vibrator has a good tracking performance and very low crosstalk between the motion axes. However, this vibrator can only be used in orthogonal EVC. The mechanical structure needs further optimization to obtain better performance.

## *3.1.1.2 Improved type*

An improved EVC vibrator was developed aiming to solve the problems discussed above. The mechanical structure of compliant mechanism and the prototype are shown in **Figure 4(a)** and **(b)**. This vibrator not only can be used in orthogonal EVC but also can be used in oblique EVC through angle adjustment of the torque gauge [9]. In order to obtain the best performance, three structure parameters were needed to be further optimized considering the compact structure, high motion stroke and working bandwidth.

The optimization problem was stated first. Objective: ∆*L*(*t*1*, t*2*, r*) ≥ 30 *μ*m, *f*(*t*1*, t*2*, r*) > 3000 Hz. Constraints: ∆*Lc*/∆*L* < 5%, σ max <σ /*<sup>s</sup> n* . Within ranges: *t*1∈[0.5, 1.5] mm, *t*2∈[0.8, 2.4] mm, *r*∈[0.5, 1.5] mm. In this work, the cutting force was taken into consideration during the structure optimization. A response surface methodology was adopted to establish the relationship between the input variables and output parameters. An NSGA-II algorithm was used to perform the optimization process. The main process of the optimization can be concluded as follows:


#### **Table 1.**

*Comparisons results of analytical modeling and FEA method.*

#### **Figure 3.**

*Experimental setup and testing results. (a) Experimental setup. (b) Step responses. (c) Amplitude-frequency responses. (d) Motion stroke. (e) Resolution tests. (f) Experimental results of output tool tip locus with phase shifts. (g) Simulation results of output tool tip locus with phase shifts [8].*

Step 1: Mechanical design is finished first, static and modal analyses are conducted to obtain the response value for the initial design parameters.

Step 2: A response surfaces methodology is adopted to create a predictive model for the design points and the response values. Then, the predicted error should be checked, the other design of experiments methodology or increased experimental design points should be considered when the error is larger than the requirements.

Step 3: MOGA is adopted to deal with the optimization processes via selection, crossover, and mutation. The optimization is converged when the maximum allowable Pareto percentage is realized.

According to the optimization results, a prototype was fabricated with the 7075 T6 aluminum. Offline tests were carried out to evaluate the performance of the optimization vibrator. The results of the step responses along *z1* axis and *z2* axis are shown in **Figure 5(a)**. The rising times are 4.4 ms and 4.2 ms. The setting times are respectively 6.7 ms and 9.06 ms. There are no steady errors and overshoots. The amplitude-frequency responses are shown in **Figure 5(b)**. It can be seen that the first natural frequency is about 1800 Hz. The motion stroke and resolution tests are respectively shown in **Figure 5(c)** and **(d)** by using stair excitation signal to each axis, the maximum motion stroke of *z1* axis is about 37 μm, the maximum motion stroke of *z2* axis is about 31 μm. The resolution of the *z1* axis and *z2* axis are *Design, Analysis and Testing of Piezoelectric Tool Actuator for Elliptical Vibration Cutting DOI: http://dx.doi.org/10.5772/intechopen.103837*

**Figure 4.** *Illustration of the improved EVC device. (a) The compliant mechanism. (b) The prototype [9].*

approximately 9 nm and 10 nm. **Figure 5(e)** shows the motion tracking performance for *z1* axis and parasitic motion for *z2* axis. It can be seen that the maximum tracking error along *z1* axis is 0.7 μm, which is 2.9% of the maximum input displacement. As shown in **Figure 5(f )**, the maximum parasitic motion of *z2* axis is 0.05 μm, which is 0.21% of the maximum input displacement of *z1* axis. For *z2* axis, the maximum tracking error is 0.72 μm, which is 3% of the maximum input displacement. The maximum parasitic motion of *z1* axis is within 0.035 μm, which is about 0.15% of the maximum input displacement. **Figure 5(g)** shows the input signal, the tool vibration locus in 3D space and the projection in *xy* plane. Compared with the former one, this vibrator has higher modulation ability and commonality for lathes with different configurations.
