Axial audio-frequency stiffness of a bush mounting – the waveguide solution.

*(English)*Zbl 1197.78044Summary: An axial, dynamic stiffness model of an arbitrary wide and long rubber bush mounting is developed within the audible-frequency range, where influences of audible frequencies, material properties, bush mounting length and radius, are investigated. The problems of simultaneously satisfying the locally non-mixed boundary conditions at the radial and end surfaces are solved by adopting a waveguide approach, using the dispersion relation for axially symmetric waves in thick-walled infinite plates, while satisfying the radial boundary conditions by mode matching. The rubber is assumed nearly incompressible, displaying dilatation elasticity and deviatoric viscoelasticity based on a fractional derivative, standard linear solid embodying a Mittag-Leffler relaxation kernel, the main advantage being the minimum parameter number required to successfully model wide-frequency band material properties. The stiffness is found to depend strongly on frequency, displaying acoustical resonance phenomena; such as stiffness peaks and troughs. The presented model agrees fully with a simplified, long-bush model while diverging from it for increased diameter-to-length ratios. To a great extent, the increased influences of higher order modes and dispersion explain the discrepancies reported for the approximate approach.

##### MSC:

78A55 | Technical applications of optics and electromagnetic theory |

##### Keywords:

waveguide; non-mixed boundary condition; Mittag-Leffler; fractional derivative; audible frequency; bush mounting
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\textit{M. Coja} and \textit{L. Kari}, Appl. Math. Modelling 31, No. 1, 38--53 (2007; Zbl 1197.78044)

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