Article 7 # 2'2024

© Yaroslav Romanchuk, PhD in Physics and
Mathematics Sciences, Senior Research Fellow,
Associate professor,
ORCID: 0000-0003-3993-0128,
e-mail: romanchuky@ukr.net
(Hetman Petro Sahaidachny
National Military Academy);
© Mariia Sokil, PhD in Technical Sciences,
Associate professor, associate professor,
ORCID: 0000-0003-3352-2131,
e-mail: mariia.b.sokil@lpnu.ua
(Lviv Polytechnic National University)
RESEARCH METHODOLOGY AND CHARACTERISTICS OF NORMAL
NONLINEAR RESONANT OSCILLATIONS OF THE SYSTEM
” SPRUNG AND UNSPRUNG PARTS OF WHEELED VEHICLES”
DOI: 10.33868/0365-8392-2024-2-279-61-69

Abstract. The authors have discussed the method applicable for the system of sprung-unsprung parts of the vehicle, carrying out vertical oscillations under nonlinear elastic force interactions, which allow evaluating the influence of the elastic characteristics of tires, the suspension system, and road irregularities on the determining parameters of the oscilla-tions of the sprung-unsprung parts. What’s obtained: a) the analytical dependence of natural oscillation frequencies on the system amplitudes and parameters; b) the relationship between the levels of amplitude of the sprung and unsprung parts oscillations, at which an internal resonance occurs; c) the condition for occurrence of the resonance in case of the motor vehicle movement along the road, which is described by the ordered system of inequalities; d) the ratios that set the main parameters of vibrations of the sprung and unsprung parts in case of the resonant and non-resonant vibrations. It is shown that a larger value of the oscillation amplitude of the sprung part corresponds to a larger value of the natural frequency in a suspension system with a progressive law of changes in the elasticity of shock absorbers, and the opposite is true when a regressive law; speaking of shock absorbers and elastic tires with their progressive or regressive power characteristics, a larger value of the oscillation amplitude of the unsprung part corresponds to a larger value of the oscil-lation amplitude of the sprung part, when an internal resonance is possible; for regressive power characteristics of tires and progressive power characteristics of shock absorbers – vice versa; the resonance value of the oscillation amplitude of the unsprung part is higher for the lower speeds of movement with regressive power characteristics of the tires and, on the contrary, – for the progressive power characteristics of tires; the amplitude of the transition through the resonance of the unsprung part takes a larger value for elastic tires with a smaller value of their static deformation and a slower transition to resonance.
Keywords: sprung and unsprung parts of the wheeled vehicle, natural frequencies of oscillations, resonant oscillations, internal resonance of the unsprung and sprung parts of the wheeled vehicle.

References
1. Manziak M. O., Krainyk L. V. & Hrubel M. H. (2021). Tendentsii rozvytku konstruktsii pidvisok viiskovykh avtomobiliv. [Trends in the development of military vehicle suspension structures]. Weapon systems and military equipment, 65, 1, 28–33 [in Ukrainian].
2. Voitenko V. A. (2012) Matematychne modeliuvannia pruzhnoi pidvisky kolisnoho transportnoho zasobu. [Mathematical modeling of the elastic suspension of a wheeled vehicle]. Automation of technological and business processes, 12, 11, 29–34 [in Ukrainian].
3. Pavlenko V. M. (2014) Suchasnyi stan rozvytku aktyvnykh pidvisok dlia lehkovykh avtomobiliv / Pavlenko V. M., Kryvoruchko O. O. [The current state of development of active suspensions for passenger cars]. Bulletin of NTU “KhPI”. Series: Automotive industry, 9, 54–60 [in Ukrainian].
4. Dushchenko V. V. (2018). Systemy pidresoriuvannia viis-kovykh husenychnykhi kolisnykh mashyn: rozrakhunok i syntez. [Suspension systems of military tracked and wheeled vehicles: calculation and synthesis]. Kharkiv, NTU “KhPI”, 336. [in Ukrainian]
5. Kozhushko A. P. (2018). Kolyvannia mekhanichnykh system v avtomobile- ta traktorobuduvanni: navch. posibnyk. [Oscillations of mechanical systems in automobile and tractor con-struction: training. manual]. Kharkiv, NTU “KhPI”. Individual entrepreneur Panov A. М. 316. [in Ukrainian].
6. Poliak I. Ie., Borysov O.V. & Matsaienko A.M. (2023). Modeliuvannia pidresorenoi chastyny mobilnoho transportnoho zasobu. [Modeling of the sprung part of a mobile vehicle]. Systems and technologies of communication, informatization and cyber security, 3, 66-73. [in Ukrainian].
7. Sokil B., Lyashuk O., Sokil M., Popovich P., Vovk Y. & Perenchuk O. (2018). Dynamic Effect of Cushion Part of Wheeled Vehicles on their Steerability. International Journal of Automotive and Mechanical Engineering, 15, 1, 4880-4892 March. [in English].
8. Sokil, B., Lyashuk, O., Sokil, M., Vovk, Y., Dzyura, V., Aulin, V. & Khoroshun, R. (2021). Interpreting the main power characteristics choice of the wheel vehicles guided cushioning system. Communications. Scientific Letters of the University of Zilina, 23, 2, B139-B149. [in English].
9. Hrubel M. H., , Nanivskyi R. A., Sokil M. B. (2015). Rezonansni kolyvannia pidresorenoi chastyny kolisnykh transportnykh zasobiv pid chas rukhu vzdovzh vporiadkovanoi systemy nerivnostei. [Resonant oscillations of the sprung part of wheeled vehicles during movement along an ordered system of bumps]. Bulletin of the Vinnytsia Polytechnic Institute, 1, 155–161. [in Ukrainian]
10. Lyashuk O., Klendiy V., Pyndus T., Sokil M. & Marunych O. (2016). Longitudinalangular oscillation of wheeled vehicles with non-linear power characteristics of absorber system. Scientific Journal of the Ternopil National Technical University, 2, 83, 82–89. [in English].
11. Litvinenko I., Marushchak P. & Lupenko S. (2014). Pro-cessing and modeling of ordered relief on the surface of heat-resistant steels after laser irradiation as a cyclic ran-dom process. Automatic Control and Computer Sciences, 48, 1–9. [in English].
12. Hrubel M., Nanivskyi R., Sokil M. (2018). Oscillation of a sprung part of vehicles with non-conservative specifications of shock absorbers. Coll. of scientific works of the Kharkiv National University of the Air Force, 2, 56, 132–136. [in Eng-lish].
13. Andrukhiv A. I., Sokil B. & Sokil M. (2018). Bazovi zasady shchodo obgruntuvannia vyboru sylovykh parametriv adaptyvnoi pidvisky kolisnykh transportnykh zasobiv spetsialnoho pryznachennia. [Basic principles regarding the justification of the choice of power parameters of the adaptive suspension of wheeled vehicles of special purpose]. Military-technical collection of the National Academy of Ground Forces. Lviv, 19, 38–51. [in Ukrainian].
14. Nanivskyi R. A. (2022). Modeliuvannia dynamiky kolisnykh transportnykh zasobiv pid chas rukhu shliakhom iz nerivnostiamy [Modeling the dynamics of wheeled vehicles during movement on a road with irregularities]. Visnyk mashynobuduvannia ta transport, 2, 16, 72–80. [in Ukrainian].
15. Mitropolskiy Yu. A. (2012). Izbrannyie trudyi v 2-h tomah. [Selected works in 2 volumes]. Kyiv, Naukova dumka, 504. [in Russian].
16. Senyk P. (1969). Obernennia nepovnoi Beta-funktsii. [The inverse of the incomplete Beta function]. Ukrainian Mathematical Journal. Vol. 21. No. 3. Pp. 325–333. [in Ukrainian].
17. Nazarkevych M. (2012). Doslidzhennia zalezhnostei Beta-ta Atebfunktsii. [Investigation of Beta- and Ateb-function dependencies]. Bulletin of the National University “Lviv Polytechnic”. Computer Science and Information Technology, 732, 207–216. [in Ukrainian].
18. Sokil B. I., Senyk A. P., Sokil M. B. & Andrukhiv A. I. (2022) Metodyka doslidzhennia vplyvu kolyvan pidresorenoi chastyny kolisnykh transportnykh zasobiv na stiikist rukhu. [The method of researching the influence of vibrations of the sprung part of wheeled vehicles on the stability of move-ment]. Modern technologies in mechanical engineering and transport, 1, 18, 167‒176. [in Ukrainian].
19. Baranov A. V., Romanchuk Ya. P., Sokil B. I & Sokil M. B. (2023). Dynamika ta stiikist rukhu KTZ iz nekonservatyvnoiu kharakterystykoiu systemy pidresoriuvannia. [Dynamics and stability of the movement of the vehicle with a non-conservative characteristic of the suspension system]. Science and technology today. Series: Technique, 6, 20, 76‒88. [in Ukrainian].