Vibrations in engineering and technology

Dmytriv Vasyl - Doctor of Engineering Sciences, Professor, Head of Department of Desing Machine and Automotive Engineering, Lviv Polytechnic National University, 79013, 12 Stepan Bandera Str., Lviv, Ukraine, е-mail: Vasyl.T.Dmytriv@lpnu.ua,  https://orcid.org/0000-0001-9361-6418

Berehuliak Stepan - recipient of the PhD degree of Department of Desing Machine and Automotive Engineering, Lviv Polytechnic National University, 79013, 12 Stepan Bandera Str., Lviv, Ukraine, е-mail: Stepan.T.Berehuliak@lpnu.ua,  https://orcid.org/0009-0005-6597-9752

The paper considers a method for reducing the number of factors for conducting a planned factorial experiment based on the use of dimensionality theory. The factor space is converted into dimensionless criteria.

The purpose of the study is to develop criterion dependencies for a computational experiment based on the results of modeling the amplitude of oscillations of a sprung mass of an elastic-damper system under a non-harmonic disturbing force pulse of a given amplitude and duration.

An example of transforming factors into dimensionless criteria is given. The transformation of the multifactor space of the planned experiment is shown on the example of an elastic-damper system. The process was subjected to the method of proportionality and a combination of similarity numbers through the equation of relations:  , where m1 - the sprung mass, kg; m2 - the unsprung mass, kg; F(t) - the perturbation force of the elastic-damper system, N; Kpr - the elasticity coefficient, N/m; Kdamp - the damping coefficient, N·s/m; trl - the relaxation time (damping of the oscillation pulse), s; g - the acceleration of free fall, m/s2. To reduce the number of similarity criteria as dimensionless factors, a simplification was adopted in the form of a ratio of criteria that are the same in nature. In the equation of relations, the first component is the ratio of masses, which characterizes the force load on the system, the second component is the ratio between the energy of oscillation and the energy of oscillation damping, which characterizes the time constant for restoring the equilibrium of the masses of the system, the third component has the physical content of the impulse of the disturbing force. The impulse of the disturbing force acting on the unsprung mass was taken to be F(t) = 500 N. The sprung and unsprung masses, the coefficients of elasticity and damping were taken within the limits corresponding to real elastic-damping systems: m1 = 400…1000 kg; m2 = 40…50 kg; Kpr = 20…40 KN/m; Kdamp = 2…2.5 KN·s/m; the duration of the oscillation damping was taken to be trl = 1 s.

Analysis of the deviation between the results of the simulation of the amplitude of oscillations and the results of the calculation by the regression model based on the computational experiment does not exceed 5%. 

1. Yildirim V. Exact Determination of the Global Tip Deflection of both Close-Coiled and Open-Coiled Cylindrical Helical Compression Springs having Arbitrary Doubly-Symmetric Cross-Sections. Interna-tional Journal of Mechanical Sciences. 2016;115–116: 280–298. https://doi.org/10.1016/j.ijmecsci.2016.06.022 

2. Paredes M. Enhanced Formulae for Determining Both Free Length and Rate of Cylindrical Compression Springs. Journal of Mechanical Design. 2016;138(2): 021404. https://doi.org/10.1115/1.4032094 

3. Liu H., Kim D. Effects of end Coils on the Natural Frequency of Automotive Engine Valve Springs. International Journal of Automo-tive Technology. 2009;10(4): 413–420. https://doi.org/10.1007/s12239-009-0047-8 

4. Liu M, Gu F, Zhang Y (2017) Ride comfort optimization of in-wheel-motor electric vehicles with in-wheel vibration absorbers. Energies 10(10): 1647. https://doi.org/10.3390/en10101647

5. Qin Y, He C, Ding P, et al. (2018) Suspension hybrid control for in-wheel motor driven electric vehicle with dynamic vibration absorbing structures. IFAC-PapersOnLine 51(31): 973–978.  https://doi.org/10.1016/j.ifacol.2018.10.054

6. Xu, Z.D.; Xu, C.; Hu, J. Equivalent fractional Kelvin model and experimental study on viscoelastic damper. J. Vib. Control. 2015, 21, 2536–2552. https://doi.org/10.1177/1077546313513604

7. Xu, Z.D.; Liao, Y.X.; Ge, T.; Xu, C. Experimental and theoretical study on viscoelastic dampers with different matrix rubbers. J. Eng. Mech.ASCE 2016, 142, 04016051. https://ascelibrary.org/doi/10.1061/(ASCE)EM.1943-7889.0001101.

8. Sato, D.; Osabel, D.M.; Kasai, K. Evaluation method for practical application of viscoelastic damper using equivalent sinusoidal waveforms of long-duration random excitations in along- and across-wind directions. Eng. Struct. 2022, 254, 113735. https://doi.org/10.1016/j.engstruct.2021.113735

9. Lu, D.; Xue, B.; Cao, Y.; Chen, B. Constitutive theory for direct coupling of molecular frictions and the viscoelasticity of soft materials. J. Appl. Mech. ASME 2022, 89, 051007. https://doi.org/10.1115/1.4053728

10. Dityatyev O. About the question of selection of the type of vibrostan for diagnostics of vehicle suspension. Suchasni tehnologii v mashinobuduvanni ta transporti.  2023, 2(21), 91-100. DOI: 10.36910/automash.v2i21.1213

11. Dityatyev O.  The influence of an untested axle wheel on the error of determining the damping of a wheel tested by the eusama method EuSaMA. Suchasni tehnologii v mashinobuduvanni ta transporti. 2024, 1(22), 139-146. DOI: 10.36910/automash.v1i22.1354

12. Sonin A.A. The Physical Basis of Dimensional Analysis. 2nd Edition, Department of Mechanical Engineering, MIT, Cambridge. 2001. http://goo.gl/2BaQM6

13. Shen W., and Lin D. K. J. Statistical theories for dimensional analysis, Statistica Sinica. 2019, 29(2), pp. 527–550. https://www.jstor.org/stable/26705477

14. Albrecht M. C., Albrecht T. A., Nachtsheim C. J. and Cook R. D. Experimental Design for Engineering Dimensional Analysis, Technometrics. 2013, 55(3), 257–270. http://www.jstor.org/stable/24587346

15. Dmytriv V.T., Dmytriv I.V., Horodetskyy I.M., Horodniak R.V., Dmytriv T.V. Method of theory of dimensions in experimental research of systems and processes. INMATEH - Agricultural Engineering. 2021, 65(3), 233 - 240. DOI: 10.35633/INMATEH-65-24

16. Genove J., Tashkov S., Traykov B., Arnaudov K., Venkov G. Anaysis and control laws of semi-active vehicle suspension systems with rending an account of the physical characteristics of magneto-rheological dampers. The International Conference on Applied Mechanics and Mechanical Engineering. 2008. 13(13). 15-31. DOI: 10.21608/amme.2008.39370

17. Hyniova K. Experiments Taken on Energy Management in Active Suspension of Vehicles. WSEAS Transactions on Systems. 2012, 11(8), 326-335. 

18. Jesus Lozoya-Santos J., Morales-Menendez R., Tudón-Martínez J. C., Sename O., Dugard L., Ramirez-Mendoza R. Control Strategies for an Automotive Suspension with a MR Damper. Preprints of the 18th IFAC World Congress, Milano (Italy) August 28 - September 2, 2011, 1820-1825. https://doi.org/10.3182/20110828-6-IT-1002.03641

19. Tao, Y., Rui, X.; Liu, F.; Shan, J.; Zhang, J. Analysis of the Vibration Characteristics of a Moving Tracked Vehicle Considering the Powertrain Magnetorheological Damping System. Appl. Sci. 2025, 15, 8997. https://doi.org/10.3390/app15168997

20. Zhang, H., Wang, E., Min, F., Subash R., Su C. Skyhook-based semi-active control of full-vehicle suspension with magneto-rheological dampers. Chin. J. Mech. Eng. 2013, 26, 498–505. https://doi.org/10.3901/CJME.2013.03.498

21. Dmytriv V., Berehuliak S. a mathematical model of an elastic-damper system based on the example of a kelvin-voigt body. Vibrations   in engineering  and technology. 2025, 2(17), 30-38. DOI: 10.37128/2306-8744-2025-2-3

22. Dmytriv V.T., Dmytriv I.V., Yatsunskyi P.P. Experimental pulse generator combined with the milking machine collector, INMATEH - Agricultural Engineering. 2019, 59(3), 219-226. DOI: 10.35633/INMATEH-59-24