Method of normal oscillations and substantiation of the choice of parameters for certain nonlinear systems with two degrees of freedom

2023;
: pp. 927–934

Revised: August 13, 2023
Accepted: August 15, 2023

Mathematical Modeling and Computing, Vol. 10, No. 3, pp. 927–934 (2023)

1
Hetman Petro Sahaidachnyi National Ground Forces Academy
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University
4
Lviv Polytechnic National University

On the example of the plane model of wheeled vehicle oscillations with adaptive power characteristic of the suspension system, the methodology for selecting its main parameters that would maximize the movement smoothness is developed.  To solve this problem, the mathematical model of relative oscillations of the sprung part is constructed, provided that they are carried out in the vertical plane.  The latter represents the system of two nonlinear differential equations describing the relative displacement of the center of mass of the sprung part and the angle of rotation of the latter around the transverse axis passing through the center of mass of the specified part.  To construct the approximate analytical solution of this equations system, and thus to describe the main parameters that determine the relative position of the sprung part under reasonable assumptions, the method of normal oscillations of nonlinear systems with concentrated masses is used.  This made it possible to obtain the system of ordinary differential equations of the first order that describe the amplitude–frequency characteristics of the sprung part vibrations.  Due to the analysis of the latter it is determined that at a certain ratio between the parameters describing the power characteristics of the suspension system, it can perform isochronous vertical and longitudinal–angular oscillations, and thus it is possible to achieve maximum comfort in transporting passengers or dangerous cargo over rough terrain.  The main obtained results can be used to create the software product for adaptive suspension, and their validity is confirmed by: a) in passing to the limit, obtaining results known from literary sources; b) generalization, based on the use of periodic Ateb-functions, of widely tested analytical methods for constructing solutions of differential equations with strong nonlinearity.

1. Mandryka V. R., Shlykova V. G.  Controllability and stability of B-class car with the ESP system.  Bulletin of NTU "KhPI".  31 (1004), 69–65 (2013).
2. Pavlenko V. M., Krivoruchko O. O.  The current state of development of active suspensions of passenger cars.  Bulletin of NTU "KhPI", Automotive Engineering.  9 (1052), 54–60 (2014).
3. Xie Z., Wong K., Jing Z., Xu T., Wong K. I., Wong H. C.  A noise-insensitive semi-active air suspension for heavy-duty vehicles with an integrated fuzzy-wheelbase preview control. Mathematical Problems in Engineering.  2013, 121953 (2013).
4. Tsampardoukas G., Stammers C. W., Guglielmino E.  Hybrid balance control of a magnetorheological truck suspension.  Journal of Sound and Vibration.  317 (3–5), 514–536 (2008).
5. Kaidalov R. O., Bashtovoi V. M., Larin O. O., Vodka O. O.  Mathematical modeling of oscillations of specialized vehicle with  two-level nonlinear springing system when crossing a single road bump.  Systems of armament and military equipment.  3 (47), 14–20 (2016).
6. Nguyen V., Zhang J., et al.  Performance analysis of air suspension of heavy truck with semi-active fuzzy control.  Journal of Southeast University.  33 (2), 159–165 (2017).
7. Soliman A., Moustafa S., Shogae A.  Parameters Affecting Vehicle Ride Comfort using Half Vehicle Model.  SAE International (2008).
8. Park S., Popov A. A., Cole D. J.  Influence of soil deformation in off-road heavy vehicle suspension vibration.  Journal of Terramechanics.  41 (1), 41–68 (2004).
9. Lyashuk O. L., Sokil M. B., Marunych O. P.  Research of longitudinal-angular oscillations of wheeled vehicles.  Materials of the XIX Scientific Conference of TNTU named after I. Puluj. Ternopil. 62–63 (2016).
10. Nguyen V., Zhang J., et al.  Effect of the off-road terrains on the ride comfort of construction vehicles.  Journal of Southeast University.  35 (2), 191–197 (2019).
11. Sokil B. I., Nanivskyi R. A., Hrubel M. G.  Own vertical oscillations of the car body taking into account the nonlinear characteristics of the elastic suspension.  Scientific and production journal "Automobile transport".   5 (235), 15–18 (2013).
12. Melnychuk S. V., Podchashivskyi Y. O., Vitiuk I. V., Bovsunovskyi I. A.  Determination of the parameters of smoothness of the suspension model based on a four-link lever mechanism.  Bulletin of the ZhSTU.  4 (55), 25–27 (2010).
13. Pavlenko V. P.  The state of development of methods for diagnosing car suspension.  Bulletin of NTU "KhPI" series Automobile and tractor construction.  64 (970), 63–69 (2012).
14. Lobas L. G., Verbitskyi V. G.  Qualitative and analytical methods in the dynamics of wheeled vehicles. Kyiv, Naukova Dumka (1990).
15. Grifin M.  Handbook of Human Vibration. Academic Press, London, UK (1990).
16. Artiushchenko A. D., Suiarkov O. G.  Study of the influence of the characteristics of the suspension of a small class car on the smoothness of the ride and its modernization.  Bulletin of NTU "KhPI".  32 (1004), 21–27 (2013).
17. Podrigalo M. A., Volkov V. P., Boboshko A. A., Pavlenko V. A., Baitsur M. V., Nazarov A. I., Alekseiev V. O. Stability of wheeled vehicles against skidding during braking and ways to improve it. Kharkiv, KhNADU (2006).
18. Podryhalo M. A., Korbko M. I., Klets D. M.  Assessment of the dynamic stability of the car.  "Automobile and tractor construction". Bulletin of NTU "KhPI".  58, 134–137 (2008).
19. Hrubel M. G., Nanivsky R. A., Sokil M. B.  Oscillations of the spring-loaded part of the BCM and their influence on the stability of movement along the curved section of the track.  Scientific Bulletin of the National Laboratory of Technical Sciences of Ukraine: a collection of scientific and technical works.  24 (1), 155–162 (2014).
20. NanivskyI R. A.  Influence on the introduction of oscillations of the spring-loaded part of the BCM during its movement along a curved section of the track.  Scientific Bulletin of the National Technical University of Ukraine: Collection of scientific and technical works.  24 (3), 366–372 (2014).
21. Dushchenko V. V.  Systems of military tracked and wheeled vehicles: calculation and synthesis.  NTU "KhPI".  Kharkiv (2018).
22. Pisariev V. P., Horbunov A. P.  Opportunities for the layout of new elastic suspension elements with progressive characteristics within the existing design solution of the APCS-60.  Mechanics and Engineering.  2, 51–56 (2009).
23. Sokil B. I., Senyk A. P., Sokil M. B., Andrukhiv A. I.  Methods for studying the influence of oscillations of the spring-loaded part of wheeled vehicles on the stability of movement.  Modern technologies in mechanical engineering and transport. Scientific journal.  1 (18), 167–176 (2022).
24. Sokil B. I., Pukach P. Ya., Senyk A. P., Sokil M. B., Andrukhiv A. I., Vovk M. I.  Asymptotic method and wave theory of motion in studying the effect of periodic impulse forces on systems characterized by longitudinal motion velocity.  Mathematical Modeling and Computing.  9 (4), 909–920 (2022).
25. Huzyk N., Pukach P., Sokil B., Sokil M., Vovk M.  On the external and internal resonance phenomena of the elastic bodies with the complex oscillations.  Mathematical Modeling and Computing.  9 {1}, 152–158 (2022).
26. Uspenskyi B. V., Avramov K. V., Nikonov O. Y.  Nonlinear normal forms of forced vibrations of piecewise linear systems at superharmonic resonances.  Probl. mashinobuduvannya.  20 (4), 24–30 (2017).
27. Uspensky B. V., Avramov K. V.  On the nonlinear normal modes of free vibration of piecewise linear systems.  Journal of Sound and Vibration.  333 (14), 3252–3265 (2014).
28. Jiang D., Pierre C., Shaw S. W.  Large amplitude non-linear normal modes of piecewise linear systems.  Journal of Sound and Vibration.  272 (3–5), 869–891 (2004).
29. Senyk P. M.  Inversion of the incomplete beta function.  Ukrainian Mathematical Journal.  21, 271–278 (1969).
30. Olshanskiу V. P., Olshanskiу S. V.  Pro rukh ostsylyatora zi stepenevoyu kharakterystykoyu pruzhnosti.  Vibrations in engineering and technology.  3 (86), 34–40 (2017).