Mathematical modeling of a vehicle based on methods of structural theory of vibration protection systems

Introduction. The purpose of the research is to evaluate the capabilities of the structural theory of vibration protection systems as applied to mathematical modeling the spring suspension of a vehicle. This is an urgent problem since there is obvious lack in theoretical knowledge in this area.

Materials and methods. Within the framework of the structural theory, the analogy method has been used to compare the calculation schemes of technical objects in the form of mechanical oscillatory systems and automatic control systems. In this case, automatic control systems are equivalent in dynamic terms to the original calculation schemes. This approach allows to use automatic control theory methods in the analysis of technical objects.

Results. An analysis of scientific literature in the field of vehicle suspension design theory has been carried out. The possibility of using the methodology of structural mathematical modeling in the development of approaches to rational vehicle suspension design has been assessed. An approach to selecting vehicle suspension parameters has been proposed. In a linear formulation, a mathematical model of the system has been constructed in the form of operator equations of motion, elastic and damping elements being taken into account. Based on the equations of motion, transfer functions of the system in two coordinates have been obtained.

Discussions and Conclusion. Analytical relationships are given taking into account the linkage coefficient of technical objects motion coordinates. The transfer functions of the relationship between vehicle elements motion coordinates were obtained taking into account the effect of damping elements and conditional random signals of road surface irregularities.

The analysis suggests that the solution to the problem of mathematical modeling a rational vehicle suspension can be implemented through using methods of the structural theory of vibration protection systems even in a linear formulation. The results obtained have made it possible to improve the dynamic characteristics of the car suspension as an automatic control system in the first approximation. Further research will be aimed at evaluating the capabilities of active and semi-active vibration protection systems.

1. Kotiev G.O., Dyakov A.S., Dubrovsky A.F. [et al.] To analyze the effectiveness of the pneumatic suspension of a bus based on the Ural four–wheel drive car . Truck. 2023; No. 10. pp. 3-8. Https://doi.org/10.36652/1684-1298-2023-10-3-8

2. Evseev K.B., Kositsyn B.B., Kotiev G.O., Stadukhin A.A. On the issue of caterpillar trains controllability evaluation at the design stage using a complex of natural-mathematical modeling. Trudy NAMI. 2022;(1): 35-51. (In Russ.) https://doi.org/10.51187/0135-3152-2022-1-35-51

3. Zheglov L., Fominykh A. Vehicle vibration safety estimation area. IOP Conference Series: Materials Science and Engineering : Design Technologies for Wheeled and Tracked Vehicles, MMBC 2019, Moscow, 01–02 октября 2019 года. Vol. 820. Moscow: Institute of Physics Publishing, 2020; P. 012028. Https://doi.org/10.1088/1757-899X/820/1/012028

4. Lahtyukhov M., Zheglov L. Automatization of calculation of dynamic characteristics of the car suspension system. IOP Conference Series: Materials Science and Engineering : Design Technologies for Wheeled and Tracked Vehicles, MMBC 2019, Moscow, 01–02 октября 2019 года. Vol. 820. Moscow: Institute of Physics Publishing, 2020; P. 012031. Https://doi.org/10.1088/1757-899X/820/1/012031

5. Makhutov N.A. Safety and risks: system research and development. Novosibirsk: Nauka Publ., 2017; 724 p.

6. Deubel C., Ernst S., Prokop G., Objective evaluationmethods of vehicle ride comfort–a literature review. J. Sound Vib. 2023; 548: 117515. https://doi.org/10.1016/j.jsv.2022.117515

7. Zhang B., Wang H., Li Z. et al. Stiffness design and mechanical performance analysis of transverse leaf spring suspension. J Mech Sci Technol. 2023; 37:1339–1348. https://doi.org/10.1007/s12206-023-0220-47. Clarence W. de Silva. Vibration. Fundamentals and Practice. Boca Raton, London, New York, Washington, D.C.: CRC Press, 2000; 957 p.

8. Clarence W. de Silva. Vibration. Fundamentals and Practice. Boca Raton, London, New York, Washington, D.C.: CRC Press, 2000; 957 p.

9. Harris S.M., Crede C.E. Shock and Vibration Handbook. New York: McGraw. Hill Book Co, 2002; 1457 p.

10. Eliseev S.V., Eliseev A.V. Theory of Oscillations. Structural Mathematical Modeling in Problems of Dynamics of Technical Objects. Series: Studies in Systems, Decision and Control. 2020; Vol.252, Springer International Publishing, Cham. 521 p.

11. Gorobtsov A.S., Kartsov S.K., Polyakov Y.A. Estimation of the vibration loading vehicle with pneumohydraulic suspensions // IOP Conference Series: Materials Science and Engineering, Tomsk, 27–29 октября 2016 года. Vol. 177. Tomsk: Institute of Physics Publishing. 2017; P. 012086. Https://doi.org/10.1088/1757-899X/177/1/012086. EDN YVLWDD.

12. Polyakov Y.A. The choice of rational stiffness joints parameters of the cabin suspension levers in the vehicle // IOP Conference Series: Materials Science and Engineering, Novosibirsk, 12–14 December 2018. Vol. 560. Novosibirsk: Institute of Physics Publishing. 2019; P. 012151. Https://doi.org/10.1088/1757-899X/560/1/012151. – EDN TOWPUZ.

13. Zhang B., Su X. Dynamic modeling and elastic characteristic analysis of the transverse leaf spring suspension. J Mech Sci Technol. 2024; 38: 1051–1058. https://doi.org/10.1007/s12206-024-0203-0

14. Li P., Xu X., Liang C. et al. Modeling and Control of the Linear Motor Active Suspension with Quasi-zero Stiffness Air Spring System Using Polynomial Chaos Expansion. Chin. J. Mech. Eng. 2025. 38: 105. https://doi.org/10.1186/s10033-025-01273-z

15. Li Q, Zhao W.H. Uncertainty analysis of motion accuracy on single-axis feed drive systems. Advances in Mechanical Engineering. 2024; 16(1): 1-15. https://doi.org/10.1177/16878132231222790

16. Xu X, Liu H, Jiang X, et al. Uncertainty analysis and optimization of quasi-zero stiffness air suspension based on polynomial chaos method. Chinese Journal of Mechanical Engineering. 2022; 35(1): 1–19. https://doi.org/10.1186/s10033-022-00758-5

17. Ma Z., Xu X., Xie J. et al. Negative Stiffness Control of Quasi-Zero Stiffness Air Suspension via Data-Driven Approach with Adaptive Fuzzy Neural Network Method. Int. J. Fuzzy Syst.2022; 24: 3715–3730. https://doi.org/10.1007/s40815-022-01357-1

18. Jiang X.W, Xu X, Liang C, et al. Robust controller design of a semi-active quasi-zero stiffness air suspension based on PCE. Journal of Vibration and Control. 2024; 30(3-4): 906-925. https://doi.org/10.1177/107754632311537

19. Chen L, Xu X, Liang C, et al. Semi-active control of a new quasi-zero stiffness air suspension for commercial vehicles based on h2H∞ state feedback. Journal of Vibration and Control 2022; 10775463211073193. https://doi.org/10.1177/10775463211073193

20. Kulkarni C.D., Sail P.P., Kalaskar A.V. et al. Real-time road testing and analysis of adjustable passive suspension system with variable spring stiffness. JMST Adv.2024; 6: 233–246. https://doi.org/10.1007/s42791-024-00082-0