*2.2. Modal and Harmonic Analysis of the LUAT System*

Modal analysis is carried out to investigate the natural frequency and vibration shape of the turning system. The front face of the flange is set to be fixed to ground. After setting the contact region and meshing, 25 order resonant frequencies were calculated as shown un Figure 3. At the 12th, the 18th and the 19th modal the horn vibrates longitudinally, their resonant frequencies are 9493 Hz, 17,666 Hz and 18,315 Hz. Figure 4 describes these three modal shapes. As presented in figure, the 19th vibration mode is in positive agreement with requirements because it outputs the largest relative deformation, and its maximum vibration occurs at the front end of the horn, whereas the back end and the transducer only vibrates slightly. More detailed analysis will be examined in harmonic response analysis.

**Figure 3.** Natural frequency and vibration type of 25 orders of modals.

**Figure 4.** Modal shape of the first three longitudinal vibrations. (**a**) 9493 Hz; (**b**) 17,666 Hz; (**c**) 18,315 Hz.

In the modal analysis, the vibration mode is relative value, which cannot reflect the true amplitude of the ultrasonic turning device at the tool. It is crucial to further analyze the harmonic response of the device on the basis of the 19th mode to obtain the amplitude at the tool tip. The harmonic response analysis is then conducted by workbench. Remote force is applied along the horn direction on its back end. The sweep range is set to between 17,000 Hz and 19,000 Hz (based on the 19th mode) with 100 solution intervals. Figure 5 shows the output displacement curve of the setting frequencies regarding the turning tool installed on the top of the horn. It could be observed that near to the frequency of 18,260 Hz, the structure is in the state of resonance, where the output displacement reaches the peak as 12 µm.

**Figure 5.** Displacement curve of the tool between 17,000 Hz and 19,000 Hz.

μ The designed LUAT system can output a considerable vibration with an amplitude up to 12 µm and a frequency of 18,260 Hz in the 19th mode, which is suitable for the turning process.
