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Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic, Fractional Derivative, Generalized Friction Model

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Abstract

In presenting a nonlinear dynamic model of a rubber vibrationisolator, the quasistatic and dynamic motion influences on theforce response are investigated within the time and frequencydomain. It is found that the dynamic stiffness at the frequency ofa harmonic displacement excitation, superimposed upon the longterm isolator response, is strongly dependent on staticprecompression, dynamic amplitude and frequency. The problems ofsimultaneously modelling the elastic, viscoelastic and frictionforces are removed by additively splitting them, modelling theelastic force response by a nonlinear, shape factor basedapproach, displaying results that agree with those of aneo-Hookean hyperelastic isolator at a long term precompression.The viscoelastic force is modeled by a fractional derivativeelement, while the friction force governs from a generalizedfriction element displaying a smoothed Coulomb force. A harmonicdisplacement excitation is shown to result in a force responsecontaining the excitation frequency and its every otherhigher-order harmonic, while using a linearized elastic forceresponse model, whereas all higher-order harmonics are present forthe fully nonlinear case. It is furthermore found that the dynamicstiffness magnitude increases with static precompression andfrequency, while decreasing with dynamic excitationamplitude – eventually increasing at the highest amplitudes due tononlinear elastic effects – with its loss angle displaying amaximum at an intermediate amplitude. Finally, the dynamicstiffness at a static precompression, using a linearized elasticforce response model, is shown to agree with the fully nonlinearmodel except at the highest dynamic amplitudes.

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References

  1. Harris, J. and Stevenson, A., 'On the role of nonlinearity in the dynamic behaviour of rubber components', Rubber Chemistry and Technology 59(5), 1986, 740–764.

    Google Scholar 

  2. Ravindra, B. and Mallik, A. K., 'Hard Duffing-type vibration isolator with combined Coulomb and viscous damping', International Journal of Non-Linear Mechanics 28, 1993, 427–440.

    Google Scholar 

  3. Ravindra, B. and Mallik, A. K., 'Performance of non-linear vibration isolators under harmonic excitation', Journal of Sound and Vibration 170(3), 1994, 325–337.

    Google Scholar 

  4. Ravindra, B. and Mallik, A. K., 'Chaotic response of a harmonically excited mass on an isolator with nonlinear stiffness and damping characteristics', Journal of Sound and Vibration 182(3), 1995, 345–353.

    Google Scholar 

  5. Natsiavas, S. and Tratskas, P., 'On vibration isolation of mechanical systems with non-linear foundations', Journal of Sound and Vibration 194(2), 1996, 173–185.

    Google Scholar 

  6. Gjika, K., Dufour, R., and Ferraris, G., 'Transient response of structures on viscoelastic or elastoplastic mounts: Prediction and experiments', Journal of Sound and Vibration 198(3), 1996, 361–278.

    Google Scholar 

  7. Shekhar, N. C., Hatwal, H., and Mallik, A. K., 'Response of non-linear dissipative shock isolators', Journal of Sound and Vibration 214(4), 1998, 589–603.

    Google Scholar 

  8. Mallik, A. K., Kher, V., Puri, M., and Hatwal, H., 'On the modelling of non-linear elastomeric vibration isolators', Journal of Sound and Vibration 219(2), 1999, 239–253.

    Google Scholar 

  9. Shekhar, N. C., Hatwal, H., and Mallik, A. K., 'Performance of non-linear isolators and absorbers to shock excitations', Journal of Sound and Vibration 227(2), 1999, 293–307.

    Google Scholar 

  10. Richards, C.M. and Singh, R., 'Characterization of rubber isolator nonlinearities in the context of single-and multi-degree-of-freedom experimental systems', Journal of Sound and Vibration 247(5), 2001, 807–834.

    Google Scholar 

  11. Gent, A. N., Engineering with Rubber, Carl Hansen Verlag, Munich, 1992.

    Google Scholar 

  12. Kari, L., 'The non-linear temperature dependent stiffness of precompressed rubber cylinders', Kautschuk Gummi Kunststoffe 55(3), 2002, 76–81.

    Google Scholar 

  13. Kari, L., 'An analytical temperature dependent collocation model for preloaded rubber cylinders', Journal of Strain Analysis 37(4), 2002, 289–299.

    Google Scholar 

  14. Hill, J. M., 'A review of partial solutions of finite elasticity and their applications', International Journal of Non-Linear Mechanics 36, 2001, 447–463.

    Google Scholar 

  15. Fenander, Å., 'Frequency dependent stiffness and damping of railpads', Proceedings of the Institution of Mechanical Engineers, Part F, Journal of Rail and Rapid Transit 211, 1997, 51–62.

    Google Scholar 

  16. Bagley, R. L. and Torvik, P. J., 'Fractional Calculus – A different approach to the analysis of viscoelastically damped structures', AIAA Journal 21, 1983, 741–748.

    Google Scholar 

  17. Rossikhin, Y. A. and Shitikova, M. V., 'Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids', Applied Mechanical Review 50(1), 1997, 15–67.

    Google Scholar 

  18. Shimizu, N. and Zhang, W., 'Fractional calculus approach to dynamic problems of viscoelastic materials', Japan Society of Mechanical Engineers C 42(4), 1999, 825–837.

    Google Scholar 

  19. Enelund, M. and Olsson, P., 'Damping described by fading memory – analysis and application to fractional derivative models', International Journal of Solids and Structures 36, 1999, 939–970.

    Google Scholar 

  20. Schmidt, A. and Gaul, L., 'Finite element formulation of viscoelastic constitutive equations using fractional time derivatives', Nonlinear Dynamics 29, 2002, 37–55.

    Google Scholar 

  21. Padovan, J. and Sawicki, J. T., 'Nonlinear vibrations of fractionally damped systems', Nonlinear Dynamics 16, 1998, 321–336.

    Google Scholar 

  22. Sjöberg, M. and Kari, L., 'Non-linear behavior of a rubber isolator system using fractional derivatives', Vehicle System Dynamics 37(3), 2002, 217–236.

    Google Scholar 

  23. Brackbill, C. R., Lesieutre, G. A., Smith, E. C., and Ruhl, L. E., 'Characterization and modeling of the low strain amplitude and frequency dependent behavior of elastomeric damper materials', Journal of the American Helicopter Society 45(1), 2000, 34–42.

    Google Scholar 

  24. Brackbill, C. R., Smith, E. C., and Lesieutre, G. A., 'Application of a refined time domain elastomeric damper model to helicopter rotor aeroelastic response and stability', Journal of the American Helicopter Society 47(3), 2002, 186–197.

    Google Scholar 

  25. Lesieutre, G. A. and Govindswamy, K., 'Finite element modeling of frequency-dependent and temperaturedependent dynamic behavior of viscoelastic materials in simple shear', International Journal of Solids and Structures 33(3), 1996, 419–432.

    Google Scholar 

  26. Bruni, S. and Collina, A., 'Modelling the viscoelastic behaviour of elastomeric components: An application to the simultion of train–track interaction', Vehicle System Dynamics 34, 2000, 283–301.

    Google Scholar 

  27. Berg, M., 'A non linear rubber spring model for rail vehicle dynamics analysis', Vehicle System Dynamics 30, 1998, 97–212.

    Google Scholar 

  28. Bouc, R., 'Modèle mathématique d'hystérésis', Acustica 24, 1971, 16–25.

    Google Scholar 

  29. Wen, Y. K., 'Method of random vibration of hysteretic systems', ASCE Journal of the Engineering Mechanics Division 102, 1976, 249–263.

    Google Scholar 

  30. Ni, Y. Q., Ko, J.M., and Wong, C.W., 'Identification of non-linear hysteretic isolators from periodic vibration tests', Journal of Sound and Vibration 217(4), 1998, 737–756.

    Google Scholar 

  31. Al Majid, A. and Dufour, R., 'Formulation of a hysteretic restoring force model. Application to vibration isolaton', Nonlinear Dynamics 27, 2002, 69–85.

    Google Scholar 

  32. Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.

    Google Scholar 

  33. Sjöberg, M., 'Rubber isolators – measurements and modelling using fractional derivatives and friction', SAE Paper No. 2000-01-3518, 2000.

  34. Fried, I., Numerical Solutions of Differential Equations, Academic Press, London, 1979.

    Google Scholar 

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Authors and Affiliations

  1. Scania CV AB, Group of Vehicle Dynamics, S-151 87, Södertälje, Sweden

    Mattias Sjöberg

  2. The Marcus Wallenberg Laboratory for Sound and Vibration Research, Department of Aeronautical and Vehicle Engineering, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Leif Kari

Authors
  1. Mattias Sjöberg
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  2. Leif Kari
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Sjöberg, M., Kari, L. Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic, Fractional Derivative, Generalized Friction Model. Nonlinear Dynamics 33, 323–336 (2003). https://doi.org/10.1023/A:1026037703124

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  • DOI: https://doi.org/10.1023/A:1026037703124

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