**5. Conclusions**

In this study we showed that the experimentally observed improving of an as-cast structure at low-frequency and low-intensity vibration can be reasonably described by the thermal model (Stefan problem with mechanical vibration).

Optimal conditions for low-frequency and low-intensity vibration of the melt in a cylindrical chill mold were determined by theoretical calculation and verified by experimental testing, that is, a vibration frequency of about 60 Hz and an amplitude of about 0.5 mm. Under such conditions, there is a significant decrease in the grain size in the ingot as well as improved soundness (density) of the as-cast metal. The yield strength increased by ~80% as compared to the ingot cast without vibration.

The optimal conditions within the thermal model are explained by the reduction of the solidification time with increasing frequency on the one hand, and by the growing integral mechanical stresses during solidification on the other hand. The specific integral of mechanical stresses in the melt subjected to vibration during solidification can be considered as a measure of the effectiveness of vibration within the thermal approach for the process description.

Theoretical calculation allows one to optimize the time of the vibration effect. The vibration has no practical value if applied above the liquidus temperature. Moreover, it can act as a source of the undesirable phenomena of gas capture. The vibration should be started just at the beginning of solidification and finish with the complete metal solidification.

The mathematical model presents a cylindrical mold with horizontal vibration. The model does not take into account other geometric variations, nor the vertical component of vibration. In future, it would be useful to simulate other conditions of metal solidification with vibration.

**Author Contributions:** Conceptualization, A.V. and D.E.; Data curation, V.D. and D.E.; Formal analysis, O.K.; Funding acquisition, A.V.; Investigation, D.E.; Methodology, O.K. and P.D.; Project administration, A.V.; Resources, V.D.; Software, O.K.; Supervision, A.V.; Validation, M.K. and V.D.; Writing—original draft, O.K.; Writing—review & editing, M.K. and P.D.

**Funding:** This research was funded by Russian Science Foundation, grant number 17-13-01252.

**Conflicts of Interest:** The authors declare no conflict of interest.
