Aim. Development of a mathematical model of dynamic processes in speed change devices with multistage gear differential transmissions with closed-loop hydraulic systems on an example of a concrete design. Method. A device with a multistage differential has been considered, in which the gear wheel - the ring gear of the first stage is connected to the sun gear wheel of the second stage, the gear wheel - the ring gear of the second stage is connected to the sun gear wheel of the third stage, and so on, depending on the number of steps. Speed control is performed at the expense of carriers of each stage by means of the closed hydraulic systems established on them. On the basis of the Lagrange equation of the second kind the equations of dynamics of such devices depending on conditions of their work have been derived and solved. Results. The mathematical dynamic model have been obtained of gear differential motion with the possibility of controlling the movement of reserved hydraulic systems in order to provide the necessary law of load change on the driven link - ring gear, and the results can be the basis for quantitative analysis of power dependencies of mechanical drive with hydraulic control. Scientific novelty. For the first time a dynamic model of a speed change device in mechanical drives of machines with a multistage gear differential has been built, which allows to determine the speed of the drive link and select the required closed-loop hydraulic system to control the speed of its driven link. Practical significance. The results obtained can be the basis for a quantitative analysis of the power dependences of a mechanical drive with hydraulic control through the carriers, when the torque of the resistance changes periodically over a long period of time; or the magnitude of the impact torque of the resistance after a sharp increase remains unchanged for a long time; or the magnitude of the impact torque of the resistance after a sharp increase is maintained for a short time; or the actuator stops immediately due to significant overload.
1. V. Malashchenko, О. Strilets and V. Strilets, "Fundamentals of Creation of New Devices for Speed Change Management", Ukrainian Journal of Mechanical Engineering and Materials Science, vol. 1, no. 2, рp. 11-20, 2015.
2. A.V. Vavilov, et al., "Sovershenstvovaniye transmissiy dorozhnykh mashin dlya povysheniya ikh konkurentosposobnosti i obespecheniya importozameshcheniya" ["Improvement of road cars transmissions to increase their competitiveness and ensure import substitution"], Avtomobilnyye dorogi i mosty [Roads and Bridges], no. 2(18), pp. 102-108, 2016. [In Russian].
3. Malashchenko V., et al., "Investigation of the energy effectiveness of multistage differential gears when the speed is changed by the carrier". Diagnostyka. vol. 20. no. 6. pp. 57-64, 2019
4. C.-J Bahk and R.G. Parker, "Analytical investigation of tooth profile modification effects on planetary gear dynamics", Mechanism and Machine Theory, no. 70. pp. 298-319, 2013.
5. H. Qilin, et al., "Nonlinear Dynamic Analysis and Optimization of Closed-Form Planetary Gear System", Mathematical Problems in Engineering, vol. 2013, 12 p. doi: 10.1155/2013/149046.
6. D.R. Salgado and J. M. Castillo, "Analysis of the transmission ratio and efficiency ranges of the four-, five-, and six-link planetary gear trains", Mechanism and Machine Theory, vol. 73, pp. 218-243, 2014. doi: 10.1016.j.mechmachtheory.2013.11.001.
7. G. Peruń, "Verification Of Gear Dynamic Model In Different Operating Conditions", Scientific Journal of Silesian University of Technology. Series Transport, no. 84, pp. 99-104, 2014.
8. Y. Fuchun, F. Jianxiong and Zh. Hongcai, "Power flow and efficiency analysis of multi-flow planetary gear trains", Mechanism and Machine Theory, vol. 92, pp. 86-99, 2015. doi: 10.1016/j.mechmachtheory.2015.05.003
9. P.V. Pawar1 and P.R. Kulkarni, "Design of two stage planetary gear train for high reduction ratio", International Journal of Research in Engineering and Technology, vol. 4, iss. 6, pp. 150-157, 2015. doi: 10.15623/ijret.2015.0406025
10. Ch. Chao and Ch. Jiabin "Efficiency analysis of two degrees of freedom epicyclic gear transmission and experimental", Mechanism and Machine Theory, vol. 87, pp. 115-130, 2014. doi: 10.1016/j.mechmachtheory.2014.12.017
11. X. Tianli, et al., "Synthesis of seven-speed planetary gear trains for heavy-duty commercial vehicle", Mechanism and Machine Theory, vol. 90, pp. 230-239, 2014. doi: 10.1016/j.mechmachtheory. 2014.12.012.
12. J. Drewniak, P. Garlicka, and P. Kolber "Design for the bi-planetary gear train", Scientific Journal of Silesian University of Technology. Series Transport, no. 91, pp. 5-17, 2016. doi: 10.20858/sjsutst.2016.91.1
13. J. Li, et al., "Power Analysis and Efficiency Calculation of Multistage Micro-planetary Transmission", Energy Procedia, no. 141, pp. 654-659, 2017. doi: 10.1016/j.egypro.2017.11.088
14. Wenjian Yang and Huafeng Ding, "Automatic detection of degenerate planetary gear trains with different degree of freedoms", Applied Mathematical Modelling, no. 64, pp. 320-332, 2018. doi: 10.1016/j.apm.2018.07.038.
15. E.L. Esmail, E. Pennestrì and A. H. Juber, "Power losses in two-degrees-of-freedom planetary gear trains: A critical analysis of Radzimovsky's formulas", Mechanism and Machine Theory, vol. 128, pp. 191-204, 2018. doi: 10.1016/j.mechmachtheory.2018.05.015.
16. A.M. Dankov, "Planetary Continuously Adjustable Gear Train With Force Closure Of Planet Gear And Central Gear: From Idea To Design", Science & Technique, no. 17(3), pp. 228-237, 2018. doi: 10.21122/2227-1031-2018-17-3-228-237.
17. M. Dobariya, "Design of Compound Planetary Gear Train", International Journal for Research in Applied Science and Engineering Technology, vol. 6, iss. 4, pp. 3179-3184, 2018. doi: 10.22214/ijraset.2018.452.
18. О. Strilets, et al., "Dynamichna model keruvannya shvydkosti cherez epitsykl pryvoda iz zubchastoyu dyferentsialnoyu peredacheyu" ["Dynamic model of speed control through the ring gear of the drive with gear differential transmission"], Visnyk Natsionalnoho universytetu "Lvivska politekhnika". "Dynamika, mitsnist ta proektuvannya mashyn i pryladiv" [Bulletin of the Lviv Polytechnic National University". "Dynamics, strength and design of machines and devices"], no. 911. pp. 63-67, 2019. [In Ukrainian].
19. О. Strilets, V. Malashchenko and V. Strilets, "Dynamika prystroyu dlya keruvannya zminamy shvydkosti z zubchastoyu dyferentsialnoyu peredacheyu i zamknutoyu hidrosystemoyu cherez sonyachne zubchaste koleso" ["Dynamics of the device for control of changes of speed with a gear differential transfer and the closed hydraulic system through a sun gear wheel"]. Visnyk NTU "KPI". Seriya: Mashynoznavstvo ta SAPR [Bulletin of the National Technical University "KhPI". Series: Mechanical Engineering and CAD], no. 1'2020. pp. 93-98, 2020. [In Ukrainian].
20. О. Strilets, V. Malashchenko and V. Strilets, "Dynamika prystroyu dlya keruvannya zminamy shvydkosti z zubchastoyu dyferentsialnoyu peredacheyu i zamknutoyu hidrosystemoyu cherez vodylo" ["Dynamics of the speed changes control device with a gear differential and the closed hydraulic system through the carrier"], Naukovyy visnyk KhDMU [Scientific Bulletin of KhDMU], no. 2 (7). pp. 176-182, 2020. [In Ukrainian].
21. О. Strilets, V. Malashchenko and V. Strilets, "Vyznachennya zvedenykh obertalnykh momentiv rivnyan dynamiky prystroyiv zminy shvydkosti cherez zubchasti dyferentsialy z zamknutymy hidrosystemamy" ["Determination of the consolidated torques for the equations of dynamics of speed change devices through gear differentials with closed hydraulic systems"], Herald of Khmelnytskyi National University. Technical sciences, iss. 4. pp. 18-23, 2020. [In Ukrainian].
22. О. Strilets, V. Malashchenko and V. Strilets, "Dynamic model of a closed-loop hydraulic system for speed control through gear differential" Scientific Journal of TNTU. - Tern. : TNTU- Vol 98. - no. 2, pp. 91-98, 2020.