Prediction of the dynamic equivalent stiffness for a rubber bushing using the finite element method and empirical modeling

  • Hyun Seong Lee
  • Jae Kyong Shin
  • Sabeur Msolli
  • Heung Soo KimEmail author


A hybrid method using an approximation that is based on the finite element analysis and empirical modeling is proposed to analyze the dynamic characteristics of a rubber bushing. The hyperelastic–viscoplastic model and an overlay method are used to obtain the hysteresis of the rubber bushing in the finite element analysis. A spring, fractional derivatives, and frictional components are used in the empirical model to obtain the dynamic stiffness in wide ranges of the excitation frequencies and amplitudes. The parameters of the proposed empirical model are determined using the hysteresis curves that were obtained from the finite element analysis. The dynamic stiffness of the rubber bushing in the wide ranges of the frequencies and amplitudes was predicted using the proposed hybrid method and was validated using lower arm bushing experiments. The proposed hybrid method can predict the dynamic stiffness of a rubber bushing without the performance of iterative experiments and the incurrence of a high computational cost, making it applicable to analyses of full-size vehicles with numerous rubber bushings under various vibrational loading conditions.


Rubber bushing Finite element method Overlay method Empirical model Dynamic equivalent stiffness 



This work was supported by the Technology Innovation Program (10048305, Launching Plug-in Digital Analysis Framework for Modular System Design) funded by the Ministry of Trade, Industry & Energy (MI, Korea).


  1. Bagley, R.L., TORVIK, J.: Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA J. 21(5), 741–748 (1983)zbMATHGoogle Scholar
  2. Banks, H.T., Hu, S., Kenz, Z.R.: A brief review of elasticity and viscoelasticity for solids. Adv. Appl. Math. Mech. 3(01), 1–51 (2011)MathSciNetGoogle Scholar
  3. Berg, M.: A non-linear rubber spring model for rail vehicle dynamics analysis. Veh. Syst. Dyn. 30(3–4), 197–212 (1998)Google Scholar
  4. Cao, L., Sadeghi, F., Stacke, L.-E.: An explicit finite-element model to investigate the effects of elastomeric bushing on bearing dynamics. J. Tribol. 138(3), 031104 (2016)Google Scholar
  5. Coveney, V., Johnson, D., Turner, D.: A triboelastic model for the cyclic mechanical behavior of filled vulcanizates. Rubber Chem. Technol. 68(4), 660–670 (1995)Google Scholar
  6. D5992-96, A.: Standard guide for dynamic testing of vulcanized rubber and rubber-like materials using vibratory methods. (1996)Google Scholar
  7. Dean, G., Duncan, J., Johnson, A.: Determination of non-linear dynamic properties of carbon-filled rubbers. Polym. Test. 4(2–4), 225–249 (1984)Google Scholar
  8. Dzierzek, S.: Experiment-based modeling of cylindrical rubber bushings for the simulation of wheel suspension dynamic behavior. In: SAE Technical Paper, (2000)Google Scholar
  9. Findley, W.N., Davis, F.A.: Creep and relaxation of nonlinear viscoelastic materials. Dover Publications, Mineola (1989)Google Scholar
  10. Fletcher, W., Gent, A.: Nonlinearity in the dynamic properties of vulcanized rubber compounds. Rubber Chem. Technol. 27(1), 209–222 (1954)Google Scholar
  11. García Tárrago, M.J., Kari, L., Vinolas, J., Gil-Negrete, N.: Frequency and amplitude dependence of the axial and radial stiffness of carbon-black filled rubber bushings. Polym. Test. 26(5), 629–638 (2007a)Google Scholar
  12. García Tárrago, M.J., Kari, L., Viñolas, J., Gil-Negrete, N.: Torsion stiffness of a rubber bushing: a simple engineering design formula including the amplitude dependence. J. Strain Anal. Eng. Des. 42(1), 13–21 (2007b)Google Scholar
  13. García Tárrago, M.J., Vinolas, J., Kari, L.: Axial stiffness of carbon black filled rubber bushings: frequency and amplitude dependence. KGK. Kautschuk, Gummi Kunststoffe 60(1–2), 43–48 (2007c)Google Scholar
  14. Govindjee, S., Simo, J.C.: Mullins’ effect and the strain amplitude dependence of the storage modulus. Int. J. Solids Struct. 29(14–15), 1737–1751 (1992)zbMATHGoogle Scholar
  15. Gracia, L., Liarte, E., Pelegay, J., Calvo, B.: Finite element simulation of the hysteretic behaviour of an industrial rubber. Application to design of rubber components. Finite Elem. Anal. Des. 46(4), 357–368 (2010)Google Scholar
  16. Horton, J., Gover, M., Tupholme, G.: Stiffness of rubber bush mountings subjected to radial loading. Rubber Chem. Technol. 73(2), 253–264 (2000a)Google Scholar
  17. Horton, J., Gover, M., Tupholme, G.: Stiffness of rubber bush mountings subjected to tilting deflection. Rubber Chem. Technol. 73(4), 619–633 (2000b)Google Scholar
  18. Kaliske, M., Rothert, H.: Formulation and implementation of three-dimensional viscoelasticity at small and finite strains. Comput. Mech. 19(3), 228–239 (1997)zbMATHGoogle Scholar
  19. Kaya, N., Erkek, M.Y., Güven, C.: Hyperelastic modelling and shape optimisation of vehicle rubber bushings. Int. J. Veh. Des. 71(1–4), 212–225 (2016)Google Scholar
  20. Khajehsaeid, H., Baghani, M., Naghdabadi, R.: Finite strain numerical analysis of elastomeric bushings under multi-axial loadings: a compressible visco-hyperelastic approach. Int. J. Mech. Mater. Des. 9(4), 385–399 (2013)Google Scholar
  21. Li, K., Liu, J., Han, X., Jiang, C., Qin, H.: Identification of oil-film coefficients for a rotor-journal bearing system based on equivalent load reconstruction. Tribol. Int. 104, 285–293 (2016)Google Scholar
  22. Lijun, Z., Zengliang, Y., Zhuoping, Y.: Novel empirical model of rubber bushing in automotive suspension system. In: Proceedings of ISMA, 20–22 September (0170) (2010)Google Scholar
  23. Liu, J., Meng, X., Jiang, C., Han, X., Zhang, D.: Time-domain Galerkin method for dynamic load identification. Int. J. Numer. Meth. Eng. 105(8), 620–640 (2016)MathSciNetGoogle Scholar
  24. Liu, J., Sun, X., Han, X., Jiang, C., Yu, D.: A novel computational inverse technique for load identification using the shape function method of moving least square fitting. Comput. Struct. 144, 127–137 (2014)Google Scholar
  25. Lu, Y.C.: Fractional derivative viscoelastic model for frequency-dependent complex moduli of automotive elastomers. Int. J. Mech. Mater. Des. 3(4), 329–336 (2006)Google Scholar
  26. Luo, Y., Liu, Y., Yin, H.: Numerical investigation of nonlinear properties of a rubber absorber in rail fastening systems. Int. J. Mech. Sci. 69, 107–113 (2013)Google Scholar
  27. Medalia, A.: Effect of carbon black on dynamic properties of rubber vulcanizates. Rubber Chem. Technol. 51(3), 437–523 (1978)Google Scholar
  28. Mullins, L.: Softening of rubber by deformation. Rubber Chem. Technol. 42(1), 339–362 (1969)Google Scholar
  29. Oldham, K., Spanier, J.: The fractional calculus. Academic Press, New York (1974)zbMATHGoogle Scholar
  30. Olsson, A.K.: Finite element procedures in modelling the dynamic properties of rubber. Lund University, Structural Mechanics (2007)Google Scholar
  31. Oscar, J., Centeno, G.: Finite Element Modeling of Rubber Bushing for Crash Simulation-Experimental Tests and Validation. Structural Mechanics, Lund University, Lund (2009)Google Scholar
  32. Payne, A., Whittaker, R.: Low strain dynamic properties of filled rubbers. Rubber Chem. Technol. 44(2), 440–478 (1971)Google Scholar
  33. Pipkin, A., Rogers, T.: A non-linear integral representation for viscoelastic behaviour. J. Mech. Phys. Solids 16(1), 59–72 (1968)zbMATHGoogle Scholar
  34. Puel, G., Bourgeteau, B., Aubry, D.: Parameter identification of nonlinear time-dependent rubber bushings models towards their integration in multibody simulations of a vehicle chassis. Mech. Syst. Signal Process. 36(2), 354–369 (2013)Google Scholar
  35. Sjöberg, M.M., Kari, L.: Non-linear behavior of a rubber isolator system using fractional derivatives. Veh. Syst. Dyn. 37(3), 217–236 (2002)Google Scholar
  36. Wineman, A., Van Dyke, T., Shi, S.: A nonlinear viscoelastic model for one dimensional response of elastomeric bushings. Int. J. Mech. Sci. 40(12), 1295–1305 (1998)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mechanical, Robotics and Energy EngineeringDongguk University-SeoulSeoulRepublic of Korea

Personalised recommendations