*4.1. Analysis of Average Cutting Force*

The cutting force of the main turning direction is extracted from 50 sampling points when the tool is turned for 5 mm for analysis. The average cutting force of the LUAT and the CT at different *v<sup>w</sup>* is shown in Figure 8. ௪ݒ

**Figure 8.** Average cutting force of LUAT and conventional turning (CT).

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It can be observed from figure that when *v<sup>w</sup>* is small, the average cutting force of LUAT is significantly smaller than that of CT, and the smaller *v<sup>w</sup>* is, the more obvious the reduction of the average cutting force of longitudinal vibration is. When *v<sup>w</sup>* is close to *v<sup>t</sup>* , the effect of longitudinal ultrasonic vibration is gradually reduced. When *v<sup>w</sup>* is greater than *v<sup>t</sup>* , the turning force of LUAT has no obvious reduction compared with CT.

The reason for this phenomenon can be the contact-separation phenomenon between the tool tip and the workpiece. In the process of LUAT, the displacement where the tool nose cuts into the workpiece is extracted, and the displacement curve when the speed is 200 mm/s and 800 mm/s is shown in Figure 9. Where (a) is the displacement curve at 200 mm/s, when *v<sup>w</sup>* is less than *v<sup>t</sup>* , the direction of resultant velocity changes periodically under longitudinal vibration, and the tip of the tool contacts the workpiece discontinuously.

(a) 200 mm/s (b) 800 mm/s

**Figure 9.** Displacement trajectory of the tool tip. (**a**) 200 mm/s; (**b**) 800 mm/s.

<sup>ଵ</sup>ݐ /ܶଵݐ=ߙ <sup>௧</sup>ݒ <sup>௪</sup>ݒ ܶ When *v<sup>w</sup>* < *v<sup>t</sup>* , the duty cycle α = *t*1/*T*, where *t*<sup>1</sup> represents the time when the tool movement has the same direction with the turning direction in a cycle and *T* is the cycle. From Equation (5), we can get,

$$t\_1 = \frac{1}{\pi f} \cos^{-1} \left( -\frac{v\_w}{v\_t} \right) \tag{8}$$

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Then,

$$\alpha = \frac{1}{\pi} \cos^{-1} \left( -\frac{v\_w}{v\_t} \right) \tag{9}$$

= ߙ 1 ߨ cosିଵ(െ ௪ݒ ௧ݒ ) α ݒ௪⁄ݒ<sup>௧</sup> ௪ݒ It can be seen from Equation (9) that α decreases with the decrease of *vw*/*v<sup>t</sup>* , and the decrease of instantaneous speed *v<sup>w</sup>* of workpiece will increase the time when the resultant velocity is opposite to the main cutting speed in a cycle, and the time ratio of tool tip backward increases in displacement. The cutting force decreases significantly when the tool tip recedes, and then the average cutting force decreases.
