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Journal of Elasticity

, Volume 37, Issue 3, pp 179–242 | Cite as

Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

  • Millard F. Beatty
  • Rezaul A. Khan
Article

Abstract

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.

Keywords

Finite Amplitude Elastic Tube Shear Response Simple Harmonic Oscillator Duffing Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J.E. Long, Bearings in Structural Engineering. Newnes-Butterworth, London (1974).Google Scholar
  2. 2.
    A. Major, Dynamics in Civil Engineering—Analysis and Design, Vol II, Chap. V. Akademiai Kiado, Budapest (1980).Google Scholar
  3. 3.
    P.M. Sheridan, F.O. James and T.S. Miller, Design of components. In A.N. Gent (ed.), Engineering with Rubber. Hanser Verlag, New York (1992) Chapter 8.Google Scholar
  4. 4.
    M.F. Beatty, Finite amplitude vibrations of a body supported by simple shear springs. J. Appl. Mech. 51 (1984) 361–366.Google Scholar
  5. 5.
    M.F. Beatty, Finite amplitude, periodic motion of a body supported by arbitrary isotropic, elastic shear mountings. J. Elasticity 20 (1988) 203–230.Google Scholar
  6. 6.
    M.F. Beatty, Stability of a body supported by a simple vehicular shear suspension system. Int. J. Non-Linear Mech. 24 (1989) 65–77.Google Scholar
  7. 7.
    R.S. Rivlin, Large elastic deformation of isotropic materials. VI-Further results in the theory of torsion, shear and flexure. Phil. Trans. Roy. Soc., London A 242 (1949) 173–195.Google Scholar
  8. 8.
    A.E. Green and W. Zerna, Theoretical Elasticity, 2nd edn. Clarendon Press, Oxford (1968).Google Scholar
  9. 9.
    A. Mioduchowski and J.B. Haddow, Finite telescopic shear of a compressible hyperelastic tube. Int. J. Non-Linear Mech. 9 (1974) 209–220.Google Scholar
  10. 10.
    D.A. Polignone and C.O. Horgan, Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes. Quart. Appl. Math. 50 (1992) 323–341.Google Scholar
  11. 11.
    D.A. Polignone and C.O. Horgan, Pure azimuthal shear of compressible nonlinearly elastic circular tubes. Quart. Appl. Math. 52 (1994) 113–131.Google Scholar
  12. 12.
    M.M. Carroll and C.O. Horgan, Finite static solutions for a compressible elastic solid. Quart. Appl. Math. 48 (1990) 767–780.Google Scholar
  13. 13.
    A. Ertepinar, Finite deformations of compressible hyperelastic tubes subjected to circumferential shear. Int. J. Engng. Sci. 23 (1985) 1187–1195.Google Scholar
  14. 14.
    A. Ertepinar, On the finite circumferential shearing of compressible hyperelastic tubes. Int. J. Engng. Sci. 28 (1990) 889–896.Google Scholar
  15. 15.
    J.L. Nowinski and A.B. Schultz, Note on a class of finite longitudinal oscillations of thick-walled cylinders. Proceedings of the Indian Congress of Applied Mechanics, Kharagpur, India (1964) pp. 31–44.Google Scholar
  16. 16.
    J.L. Nowinski, On a dynamic problem in finite elastic shear. Int. J. Engng. Sci. 4 (1966) 501–510.Google Scholar
  17. 17.
    A.S.D. Wang, On free oscillations of elastic incompressible bodies in finite shear. Int. J. Engng. Sci. 7 (1969) 1199–1212.Google Scholar
  18. 18.
    S.S. Antman and G. Zhong-Heng, Large shearing oscillations of incompressible nonlinearly elastic solids. J. Elasticity 14 (1984) 249–262.Google Scholar
  19. 19.
    A. Ertepinar and T. Tokdemir, Simultaneous, finite, gyroscopic and radial oscillations of hyperelastic cylindrical tubes. Int. J. Engng. Sci. 22 (1984) 375–382.Google Scholar
  20. 20.
    R.L. Fosdick and G.P. MacSithigh, Shearing motion and the formation of shock waves in an elastic circular tube. J. Elasticity 8 (1978) 191–267.Google Scholar
  21. 21.
    J.K. Knowles, Large amplitude oscillations of a tube of incompressible elastic material. Quart. Appl. Math. 18 (1960) 71–77.Google Scholar
  22. 22.
    M.F. Beatty, Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues-with examples. Appl. Mech. Revs. 40 (1987) 1699–1734.Google Scholar
  23. 23.
    V.K. Agarwal, On finite anti-plane shear for a compressible elastic circular tube. J. Elasticity 9 (1979) 311–319.Google Scholar
  24. 24.
    J.E. Adkins and A.N. Gent, Load deflexion relations of rubber bush mountings. Br. J. Appl. Physics 5 (1954) 354–358.Google Scholar
  25. 25.
    A.C. Stevenson, Some boundary value problems in two-dimensional elasticity. Phil. Mag. 34 (1943) 766–793.Google Scholar
  26. 26.
    J.D. Logan, Applied Mathematics—a Contemporary Approach. John Wiley and Sons, Inc., New York (1987).Google Scholar
  27. 27.
    W. Magnus and S. winkler, Hill's Equation. Dover Publications, New York (1979).Google Scholar
  28. 28.
    A.G. Greenhill, The Application of Elliptic Functions. Dover Publications, New York (1959).Google Scholar
  29. 29.
    J.E. Adkins, Some general results in the theory of large elastic deformations. Proc. Roy. Soc. London 231 (1955) 75–99.Google Scholar
  30. 30.
    R.W. Ogden, P. Chadwick and E.W. Haddon, Combined axial and torsional shear of a tube of incompressible isotropic elastic material. Quart. J. Mech. Appl. Math. 26 (1973) 23–41.Google Scholar
  31. 31.
    R.J. Atkin and N. Fox, An Introduction to the Theory of Elasticity. Longman, New York (1980).Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Millard F. Beatty
    • 1
  • Rezaul A. Khan
    • 1
  1. 1.Department of Engineering MechanicsUniversity of Nebraska-LincolnLincolnUSA

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