*3.2. Analysis of LUAT Process*

ݐ

തതത௨ݒ

In conventional turning (CT), the absolute speed of the tool tip in the peripheral direction of the workpiece is a fixed value. The introduction of ultrasonic vibration triggers the tool tip to produce high frequency reciprocating vibration in the peripheral direction. The average speed in a cycle *v<sup>u</sup>* can be obtained from Equation (4):

$$
\overline{\upsilon\_{\iota}} = \mathfrak{Z} \rtimes A \rtimes f \tag{4}
$$

݂ × ܣ ×2= തതത௨ݒ where, *A* and *f* represent the amplitude and the frequency of the tool tip vibration.

௨ݒ

ݐ߱ cos ܽ = ௨ݒ <sup>௨</sup>ݒ

തതത௨ݒ <sup>௪</sup>ݒ ݒ According to the modal analysis and harmonic response analysis, the suitable resonant frequency *f* is 18,260 Hz, and the displacement *A* is 12 µm. Consequently, the average speed *v<sup>u</sup>* = 438.24 mm/s. The turning motion is equivalent to the fixed workpiece and the moving tool. At this time, the resultant velocity *v* can be regarded as the superposition of the workpiece velocity *v<sup>w</sup>* and the instantaneous velocity of the tool *v<sup>u</sup>* which comes from ultrasonic vibration.

$$
v = v\_{\overline{w}} + v\_{\overline{u}} \tag{5}$$

௨ݒ

ܽ =

688.04 cos 114,731ݐ ݒ<sup>௧</sup> = max ݒ<sup>௨</sup> = 688.04 

௪ݒ

144

௧

߱×ݐ× തതത௨ݒ 1 െ cos ߱ݐ

= <sup>௨</sup>ݒ 688.04 = ܽ ݂ߨ2 = ߱

ݒ <sup>௨</sup>ݒ <sup>௪</sup>ݒ ௪ݒ

Let *v<sup>u</sup>* be cosine function and *v<sup>u</sup>* = *a* cos ω*t*, then there is:

$$
\overline{v\_{\mathsf{U}}} \times t = \int\_0^t a \cos \omega t dt \tag{6}
$$

Take *t* as half of the ultrasonic vibration period, then:

$$a = \frac{\overline{v\_{\mu}} \times t \times \omega}{1 - \cos \omega t} \tag{7}$$

Plugging ω = 2π*f*, it can be easily obtained that *a* = 688.04 mm/s. Therefor *v<sup>u</sup>* = 688.04 cos 114, 731*t*, at this time the critical turning speed *v<sup>t</sup>* = max*v<sup>u</sup>* = 688.04 mm/s.

The relationship between tool tip and workpiece is affected by the critical turning speed *v<sup>u</sup>* of tool tip and workpiece velocity *vw*. Specifically,


In the range of 100 mm/s to 1800 mm/s, different *v<sup>w</sup>* are selected to carry out the turning simulation experiment of BMG. The interval is 100 mm/s when *v<sup>w</sup>* is below 600 mm/s and then rise to 200 mm/s when *v<sup>w</sup>* is above 600 mm/s. ௪ݒ ௪ݒ ௪ݒ
