×

zbMATH — the first resource for mathematics

Axial audio-frequency stiffness of a bush mounting – the waveguide solution. (English) Zbl 1197.78044
Summary: 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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Morman, K.N.; Pan, T.Y., Application of finite-element analysis in the design of automotive elastomeric components, Rubber chem. technol., 61, 503-533, (1988)
[2] Chen, C.K.; Wu, C.T., On computational issues in large deformation analysis of rubber bushings, Mech. struct. Mach., 33, 327-349, (1997)
[3] Busfield, J.J.C.; Davies, C.K.L., Stiffness of simple bonded elastomer bushes part 1—initial behaviour, Plast. rubber compos., 30, 243-257, (2001)
[4] Kadlowec, J.; Wineman, A.; Hulbert, G., Elastomer bushing response: experiments and finite element modeling, Acta mech., 163, 25-38, (2003) · Zbl 1064.74502
[5] Adkins, J.E.; Gent, A.N., Load – deflection relations of rubber bush mountings, Br. J. appl. phys., 5, 354-358, (1954)
[6] Petek, N.K.; Kicher, T.P., An empirical model for the design of rubber shear bushings, Rubber chem. technol., 60, 298-309, (1987)
[7] Hill, J.M., Radial deflections of rubber bush mountings of finite lengths, Int. J. eng. sci., 13, 407-422, (1975) · Zbl 0307.73033
[8] Hill, J.M., Radial deflections of thin pre-compressed cylindrical rubber bush mountings, Int. J. solids struct., 13, 93-104, (1977) · Zbl 0344.73068
[9] Hill, J.M., Load – deflection relations of long cylindrical rubber bush mountings constructed from rectangular blocks, J. appl. polym. sci., 21, 1459-1467, (1977)
[10] Hill, J.M., Effect of precompression on load – deflection relations of long rubber bush mountings, J. appl. polym. sci., 19, 747-755, (1975)
[11] Hill, J.M., Load – deflection relations of bonded pre-compressed spherical rubber bush mountings, Q. J. mech. appl. math., 28, 261-270, (1975) · Zbl 0325.73078
[12] Wineman, A.; van Dyke, T.; Shi, S., A nonlinear viscoelastic model for one dimensional response of elastomeric bushings, Int. J. eng. sci., 40, 1295-1305, (1998) · Zbl 0961.74505
[13] Pipkin, A.C.; Rogers, T.G., A non-linear integral representation for viscoelastic behaviour, J. mech. phys. solids, 16, 59-72, (1968) · Zbl 0158.43601
[14] Kadlowec, J.; Wineman, A.; Hulbert, G., Coupled response model for elastomeric bushings, Rubber chem. technol., 74, 338-352, (2001)
[15] Horton, J.M.; Gover, M.J.C.; Tupholme, G.E., Stiffness of rubber bush mountings subjected to radial loading, Rubber chem. technol., 73, 253-264, (2000)
[16] Horton, J.M.; Gover, M.J.C.; Tupholme, G.E., Stiffness of rubber bush mountings subjected to tilting deflection, Rubber chem. technol., 73, 619-633, (2000)
[17] Gent, A.N., Engineering with rubber, (1992), Carl Hansen Verlag Munich
[18] Lindley, P.B., Engineering design with natural rubber, (1992), MRPRA
[19] Göbel, E.F., Rubber springs design, (1974), Newnes-Butterworths London
[20] Payne, A.R.; Scott, J.R., Engineering design with rubber, (1960), Interscience Publishers New York
[21] Freakley, P.K.; Payne, A.R., Theory and practice of engineering with rubber, (1978), Applied Science Publishers London
[22] Kari, L., Dynamic stiffness matrix of a long rubber bush mounting, Rubber chem. technol., 75, 747-770, (2002)
[23] Payne, A.R.; Whittaker, R.E., Low strain dynamic properties of filled rubbers, Rubber chem. technol., 44, 440-478, (1971)
[24] Fung, Y.C., Foundations of solid mechanics, (1965), Prentice Hall New Jersey
[25] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, ninth printing, (1972), Dover Publications New York
[26] Koeller, R.C., Applications of fractional calculus to the theory of viscoelasticity, J. appl. mech., 51, 299-307, (1984) · Zbl 0544.73052
[27] Kari, L.; Eriksson, P.; Stenberg, B., Dynamic stiffness of natural rubber cylinders in the audible frequency range using wave guides, Kautsch. gummi kunstst., 54, 106-113, (2001)
[28] Rayleigh, L., On the free vibrations of an infinite plate of homogeneous isotropic matter, Proc. London math. soc., 20, 225-234, (1889) · JFM 21.1040.01
[29] Lamb, H., On the flexure of an elastic plate, Proc. London math. soc., 21, 70-90, (1889) · JFM 22.1007.02
[30] Lamb, H., On waves in an elastic plate, Proc. roy. soc. A, 93, 114-128, (1917) · JFM 46.1232.01
[31] Goodman, L.E., Circular-crested vibrations of an elastic solid bounded by two parallel planes, Proc. 1st natl. congr. appl. mech., 65-73, (1952)
[32] 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)
[33] 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.