Abstract
The dynamic stiffness of a chemically and physically ageing rubber vibration isolator in the audible frequency range is modelled as a function of ageing temperature, ageing time, actual temperature, time, frequency and isolator dimension. In particular, the dynamic stiffness for an axially symmetric, homogeneously aged rubber vibration isolator is derived by waveguides where the eigenmodes given by the dispersion relation for an infinite cylinder satisfying traction free radial surface boundary condition are matched to satisfy the displacement boundary conditions at the lateral surface ends of the finite rubber cylinder. The constitutive equations are derived in a companion paper (Part 1). The dynamic stiffness is calculated over the whole audible frequency range 20–20,000 Hz at several physical ageing times for a temperature history starting at thermodynamic equilibrium at \(+25\,^{\circ }\hbox {C}\) and exposed by a sudden temperature step down to \(-60\,^{\circ }\hbox {C}\) and at several chemical ageing times at temperature \(+25\,^{\circ }\hbox {C}\) with simultaneous molecular network scission and reformation. The dynamic stiffness results are displaying a strong frequency dependence at a short physical ageing time, showing stiffness magnitude peaks and troughs, and a strong physical ageing time dependence, showing a large stiffness magnitude increase with the increased physical ageing time, while the peaks and troughs are smoothed out. Likewise, stiffness magnitude peaks and troughs are frequency-shifted with increased chemical ageing time. The developed model is possible to apply for dynamic stiffness prediction of rubber vibration isolator over a broad audible frequency range under realistic environmental condition of chemical ageing, mainly attributed to oxygen exposure from outside and of physical ageing, primarily perceived at low-temperature steps.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983)
Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, 133–155 (1986)
Brandrup, J., Immergut, E.H., Grulke, E.A.: Polymer Handbook, 4th edn. Wiley, New York (1999)
Cangialosi, D., Boucher, V.M., Alegría, A., Colmenero, J.: Physical aging in polymers and polymer nanocomposites: recent results and open questions. Soft Matter 9, 8619–8630 (2013)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)
Chree, C.: The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and applications. Trans. Camb. Philos. Soc. 14, 250–309 (1889)
Coja, M., Kari, L.: Axial audio-frequency stiffness of a bush mounting—the waveguide solution. Appl. Math. Modell. 31, 38–53 (2007)
Fredette, L., Singh, R.: Estimation of the transient response of a tuned, fractionally damped elastomeric isolator. J. Sound Vib. 382, 1–12 (2016)
Gaul, L.: Dynamical transfer behaviour of elastomer isolators; boundary element calculation and measurement. Mech. Syst. Signal Process. 5, 13–24 (1991)
Graff, K.F.: Wave Motion in Elastic Solids (reprint). Dover Publications, New York (1991)
Harrison, M., Sykes, A.O., Martin, M.: Wave effects in isolation mounts. J. Acoust. Soc. Am. 24, 62–71 (1952)
Greiner, R., Schwarzl, F.R.: Thermal contraction and volume relaxation of amorphous polymers. Rheol. Acta 23, 378–395 (1984)
Johlitz, M.: On the representation of ageing phenomena. J. Adhes. 88, 620–648 (2012)
Johlitz, M., Diercks, M., Lion, A.: Thermo-oxidative ageing of elastomers: a modelling approach based on finite strain theory. Int. J. Plast. 63, 131–151 (2014)
Kari, L.: On the waveguide modelling of dynamic stiffness of cylindrical vibration isolators. Part I: The model, solution and experimental comparison. J Sound. Vib. 244, 211–233 (2001)
Kari, L.: On the waveguide modelling of dynamic stiffness of cylindrical vibration isolators. Part II: The dispersion relation solution, convergence analysis and comparison with simple models. J Sound. Vib. 244, 235–257 (2001)
Kari, L.: On the dynamic stiffness of preloaded vibration isolators in the audible frequency range: Modeling and experiments. J. Acoust. Soc. Am. 113, 1909–1921 (2003)
Kari, L.: Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range. Part 1: Constitutive equations. Continuum Mech. Thermodyn. (2017). doi:10.1007/s00161-017-0569-7
Kari, L., Eriksson, P., Stenberg, B.: Dynamic stiffness of natural rubber cylinders in the audible frequency range using wave guides. Kaut. Gummi Kunstst. 54, 106–111 (2001)
Kim, S., Singh, R.: Multi-dimensional characterization of vibration isolators over a wide range of frequencies. J. Sound. Vib. 245, 877–913 (2001)
Kim, S., Singh, R.: Vibration transmission through an isolator modelled by continuous system theory. J. Sound. Vib. 248, 925–953 (2001)
Lee, J., Thompson, D.J.: Dynamic stiffness formulation, free vibration and wave motion of helical springs. J. Sound. Vib. 239, 297–320 (2001)
Miklowitz, J.: The theory of elastic waves and waveguides. In: North-Holland Series in Applied Mathematics and Mechanics, vol. 22, pp. 1–618 (1978)
Odegard, G.M., Bandyopadhyay, A.: Physical aging of epoxy polymers and their composites. J. Polym. Sci. Part B Polym. Phys. 49, 1695–1716 (2011)
Onoe, M.: Modified quotients of cylinder functions. Math. Comput. 10, 27–28 (1956)
Pochhammer, L.: Über die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiszylinder. J. Reine Angew. Math. 81, 324–336 (1876)
Ungar, E.E., Dietrich, C.W.: High-frequency vibration isolation. J. Sound. Vib. 4, 224–241 (1966)
Vakakis, A.F., Paipetis, S.A.: Transient response of unidirectional vibration isolators with many degrees of freedom. J. Sound. Vib. 99, 557–562 (1985)
Wollscheid, D., Lion, A.: Predeformation- and frequency-dependent material behaviour of filler-reinforced rubber: Experiments, constitutive modelling and parameter identification. Int. J. Solids Struct. 50, 1217–1225 (2013)
Wollscheid, D., Lion, A.: The benxefit of fractional derivatives in modelling the dynamics of filler-reinforced rubber under large strains: a comparison with the Maxwell-element approach. Comput. Mech. 53, 1015–1031 (2014)
Zemanek, J. Jr.: An experimental and theoretical investigation of elastic wave propagation in a cylinder. J. Acoust. Soc. Am. 51, 265–283 (1972)
Zhang, J., Richards, C.M.: Dynamic analysis and parameter identification of a single mass elastomeric isolation system using a Maxwell-Voigt model. J. Vib. Acoust. 128, 713–721 (2006)
Zhu, S.-J., Weng, X.-T., Chen, G.: Modelling of the stiffness of elastic body. J. Sound. Vib. 262, 1–9 (2003)
Östberg, M., Kari, L.: Transverse, tilting and cross-coupling stiffness of cylindrical rubber isolators in the audible frequency range–the wave-guide solution. J. Sound. Vib. 330, 3222–3244 (2011)
Östberg, M., Coja, M., Kari, L.: Dynamic stiffness of hollowed cylindrical rubber vibration isolators–The wave-guide solution. Int. J. Solids Struct. 50, 1791–1811 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kari, L. Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range: Part 2—waveguide solution. Continuum Mech. Thermodyn. 29, 1047–1059 (2017). https://doi.org/10.1007/s00161-017-0573-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-017-0573-y