**4. Discussion**

### *4.1. Estimation of the Obtained E*ff*ects*

The provided study allows researching the possible steps of post-processing for the parts produced from structural anticorrosion chrome-nickel steels—20kH13 (AISI 420) and 12kH18N9T (AISI 321). The samples' detected fine structure was obtained due to high cooling rates [18,77–79]—the liquid quenching mechanism is implemented at growing solids by the laser powder bed fusion method.

The presence of inhomogeneities in the technological liquid medium during ultrasonic cavitation abrasive finishing led to a decrease in the liquid's cavitation strength and an increase in a cavitation centers number [80]. The abrasive particles acquired acceleration because of the transfer of impulse energy from shock waves of acoustic micro- and macroflows [81]. It was expressed quantitatively in a decrease in the roughness parameters ( *Ra*, *Rz*, *Rtm*) by 50%–60%, a two-fold reduction in the average pitch of irregularities, and an increase in the profile reference length by 10%. Practice showed that extended processing time, which exceeded 120 s, did not increase the obtained e ffect.

Besides, abrasive particles have damping functions and protected the surface from cavitation destruction. The surface had no traces of cavitation erosion; there were no grinding marks, which indicated a uniform e ffect over the entire processing area and intensive removal of the material layer. The layer change occured due to the destruction of the protrusions of the grain surface. The most significant e ffect was achieved by ultrasonic cavitation abrasive finishing (Table 5, Figure 8).

The analysis of the results (Table 7) shows that the hardness of 20kH13 (AISI 420) samples produced by the laser powder bed fusion method was higher than that of the cast and heat-treated 20kH13 (AISI 420) samples. The low tempering of the laser powder bed-fused samples led to a slight decrease in hardness. Given that low tempering reduces quenching stresses, this kind of heat treatment can be recommended for parts that work for wear, particularly for those chosen in this study part as a locking washer. The hardness after high tempering was significantly lower than after a low one, which was explained by the decomposition of martensite and its transition to sorbitol tempering.

Investigation of the e ffect of heat treatment on the wear resistance of 20kH13 (AISI 420) steel specimens produced by the laser powder bed fusion method showed that the wear resistance without subsequent heat treatment was higher than that of samples obtained by the traditional casting with special heat treatment (Figures 10 and 11). The abrasion resistance correlates with the measured hardness. The wear resistance decreases when the tempering temperature increases for all the samples under study. The values of the wear resistance of specimens obtained by the laser powder bed fusion method and by the laser powder bed fusion method with low tempering di ffer slightly. Simultaneously, they are slightly higher than the wear resistance of cast specimens after traditional quenching and low tempering.

### *4.2. Quantituve Evaluation of Cavitation-Abrasive Finishing Factors*

Let us quantitatively evaluate the achieved results. The sti ffness of the system for two types of steel samples produced by additive manufacturing will be:

$$K\_{420} = 4.18 \cdot 10^9 \left[\frac{\text{N}}{\text{m}}\right] \tag{15}$$

$$K\_{321} = 3.86 \cdot 10^9 \left[\frac{\text{N}}{\text{m}}\right]. \tag{16}$$

Then the period of the natural oscillations will be

$$T\_{420} = \sqrt{4\pi^2 \frac{m\_{420}}{k\_{420}}} = \sqrt{4\pi^2 \frac{0.0616 \text{[kg]}}{4.18 \cdot 10^9 \text{[m]}}} = 2.412 \cdot 10^{-5} \text{[s]},\tag{17}$$

$$T\_{321} = \sqrt{4\pi^2 \frac{0.0632 \text{[kg]}}{3.86 \cdot 10^9 \text{[}\frac{\text{N}}{\text{m}}\text{]}}} = 2.542 \cdot 10^{-5} \text{[s]},\tag{18}$$

when the period of forced oscillations is:

$$
\pi = \frac{1}{f\_{fc}} = 4.762 \cdot 10^{-5} \text{[s]}.\tag{19}
$$

The amplitude of the vibration can expressed as follows [82,83]:

$$
\overline{A\_{\mathfrak{m}}} = \overline{A\_0} \cdot \overline{\mathfrak{e}^{\mathfrak{k}\_{\mathfrak{r}}}},
\tag{20}
$$

where β is the damping coefficient expressed as a complex number in the form *a* + *bi*:

$$
\overline{\beta} = \overline{q} - \overline{\mu},
\tag{21}
$$

where *q* is the index of excited oscillations, and μ is the coefficient of medium resistance, let us allow that:

$$
\overline{\mu} \to 0 \tag{22}
$$

and *q* can be expressed as follows:

$$
\overline{q} = \sqrt{\frac{\sqrt{h} - k}{2 \cdot m\_n}},
\tag{23}
$$

where:

$$h = \frac{H\_m}{H\_0} \tag{24}$$

and

$$k = \frac{K\_m}{K\_0} = 1\tag{25}$$

where the values indexed *m* are for the actual conditions and the values indexed 0 for conditions at the initial stage of processing. If the amplitude of the vibration *Sm* is 20 μm transit to the sample in the ideal conditions, then:

$$
\sqrt{h} = \sqrt{\frac{H\_{\text{m}}}{H\_0}} = 0.9995\tag{26}
$$

and

$$
\overline{\beta\_{420}} = \overline{q\_{420}} = |0.064| \,\text{s}.\tag{27}
$$

$$
\overline{\beta\_{321}} = \overline{q\_{321}} = \|0.063\vert.\tag{28}
$$

The amplitude of the forced oscillations will be:

$$
\overline{A\_{420}} = \overline{A\_0} \cdot e^{(0.064|\cdot 4.76 \cdot 10^{-5})} = \overline{A\_0} \cdot e^{3.05 \cdot 10^{-6}} = 1.000003 \cdot \overline{A\_0} = 2.0 \cdot 10^{-6} [\text{m}],\tag{29}
$$

$$
\overline{A\_{321}} = \overline{A\_0} \cdot e^{([0.063] \cdot 4.76 \cdot 10^{-5})} = \overline{A\_0} \cdot e^{3.00 \cdot 10^{-6}} = 1.000003 \cdot \overline{A\_0} = 2.0 \cdot 10^{-6} [\text{m}].\tag{30}
$$

The applied force will be:

$$F\_{A420} = K\_{420} \cdot \delta = 4.18 \cdot 10^9 \text{[N]} \cdot \text{\(\cdot\text{\(}10^{-12}\text{[m]} = 25.08 \cdot 10^{-3}\text{[N]}\text{)}\text{.} \tag{31}$$

$$F\_{A321} = K\_{321} \cdot \delta = 3.86 \cdot 10^9 \text{[N]} \cdot \text{\(\cdot\text{l}\)} \cdot \text{\(10^{-12}\text{[m]} = 23.16 \cdot 10^{-3}\text{[N]}\text{.} \tag{32}$$

and acoustic pressure on 0.0004 m<sup>2</sup> of the sample:

$$P\_{A420} = \frac{F\_{A420}}{S\_A} = \frac{25.08 \cdot 10^{-3} \text{[N]}}{0.0004 \text{[m}^2\text{]}} = 62.7 \text{[Pa]},\tag{33}$$

$$P\_{A321} = \frac{F\_{A321}}{S\_A} = \frac{23.16 \cdot 10^{-3} \text{[N]}}{0.0004 \text{[m}^2\text{]}} = 57.9 \text{[Pa]}.\tag{34}$$

Thus, it should be noted that the period of the natural oscillations that are determined by the nature of the system was less than the chosen period (frequency) of forced oscillations. Self-oscillations resulting from the action of the internal energy of the system with a fixed frequency *fslf*, close to the natural frequency *fnat*, and a fixed amplitude; the reason, in this case, is associated with the low rigidity of the system and fluctuations in the acting force *FA*; vibration frequency increases with increasing system rigidity (stiffness *K*) and decreases with decreasing workpiece thickness. It is recommended that the cavitation finishing be performed outside the free (natural) oscillation area to exclude the resonance phenomenon [84]:

$$0.7 < \frac{f\_{frc}}{f\_{slf}} < 1.3\tag{35}$$

or

$$0.7 < \frac{T}{\pi} < 1.3\tag{36}$$

then the process can be characterized as effective and stable. In the given case:

$$\frac{T\_{420}}{\tau} = \frac{2.412 \cdot 10^{-5}}{4.762 \cdot 10^{-5}} = 0.51,\tag{37}$$

$$\frac{T\_{321}}{\tau} = \frac{2.542 \cdot 10^{-5}}{4.762 \cdot 10^{-5}} = 0.53. \tag{38}$$

As it can be seen, the chosen frequency should be enlarged for further experiments to achieve a more effective and stable processing mode in cavitation finishing; however, the introduction of ceramic abrasive in the working zone changed the conditions in the working tank and allowed reduced frequency in combination with the effectiveness of abrasive granule deformation. For improving parameters of processing up to the ratio of stable processing without adding additives that influence medium resistance, the frequency of the forced oscillations should be in the interval of 29–54 kHz for 20kH13 (AISI 420) steel and 28–52 kHz for 12kH18N9T (AISI 321) steel. If the amplitude of the forced vibrations in the working zone is small and measures in micrometers, then the difference δ between *A*0 and *Am* is extremely small and is measured in picometers, which 0.01 of an angstrom (Å). With the known acoustic pressure, it is possible to evaluate the acoustic pressure's amplitude in each moment of the cycle. At the same time, it should be noted that frequency higher than 30 kHz and up to 1 MHz could be harmful to the biological process in the human body since arising cavitation with bubble formation with a diameter less than 1 μm (ultrasound surgery) [85], when ultrasound in the range 2–29 MHz is used in echography; the works should be conducted according to the sanitary norms and rules of production.
