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Dynamic Equations for an Isotropic Spherical Shell Using Power Series Method and Surface Differential Operators

  • Reza OkhovatEmail author
  • Anders Boström
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

Dynamic equations for an isotropic spherical shell are derived by using a series expansion technique. The displacement field is split into a scalar (radial) part and a vector (tangential) part. Surface differential operators are introduced to decrease the length of the shell equations. The starting point is a power series expansion of the displacement components in the thickness coordinate relative to the mid-surface of the shell. By using the expansions of the displacement components, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions that can be used to eliminate all but the four of lowest order and to express higher order expansion functions in terms of these of lowest orders. Applying the boundary conditions on the surfaces of the spherical shell and eliminating all but the four lowest order expansion functions give the shell equations as a power series in the shell thickness. After lengthy manipulations, the final four shell equations are obtained in a more compact form which can be represented explicitly in terms of the surface differential operators. The method is believed to be asymptotically correct to any order. The eigenfrequencies are compared to exact three-dimensional theory and membrane theory.

Keywords

Spherical shell Shell equations Surface differential operators Dynamic Eigenfrequency Power series 

Notes

Acknowledgements

The present project is funded by the Swedish Research Council and this is gratefully acknowledged.

References

  1. 1.
    Tuner CE (1965) Plate and shell theory. Longmans, LondonGoogle Scholar
  2. 2.
    Heyman J (1977) Equilibrium of shell structures. Oxford University Press, OxfordzbMATHGoogle Scholar
  3. 3.
    Niordson FI (1985) Shell theory. Elsevier Science, New YorkzbMATHGoogle Scholar
  4. 4.
    Shah AH, Ramkrishnan CV, Datta SK (1969) Three-dimensional and shell-theory analysis of elastic waves in a hollow sphere. part I: analytical foundation. J Appl Mech 36:431–439CrossRefzbMATHGoogle Scholar
  5. 5.
    Shah AH, Ramkrishnan CV, Datta SK (1969) Three-dimensional and shell-theory analysis of elastic waves in a hollow sphere. part II: numerical results. J Appl Mech 36:440–444CrossRefGoogle Scholar
  6. 6.
    Niordson FI (2001) An asymptotic theory for spherical shells. Int J Solids Struct 38:8375–8388CrossRefzbMATHGoogle Scholar
  7. 7.
    Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New YorkzbMATHGoogle Scholar
  8. 8.
    Soedel W (1993) Vibrations of shells and plates. Marcel Dekker, New YorkzbMATHGoogle Scholar
  9. 9.
    Leissa AW (1993) Vibration of shells. Acoustical Society of America, New YorkGoogle Scholar
  10. 10.
    Boström A (2000) On wave equations for elastic rods. Z Angew Math Mech 80:245–251CrossRefzbMATHGoogle Scholar
  11. 11.
    Boström A, Johansson J, Olsson P (2001) On the rational derivation of a hierarchy of dynamic equations for a homogeneous, isotropic, elastic plate. Int J Solids Struct 38:2487–2501CrossRefzbMATHGoogle Scholar
  12. 12.
    Mauritsson K, Folkow PD, Boström A (2011) Dynamic equations for a fully anisotropic elastic plate. J Sound Vib 330:2640–2654CrossRefGoogle Scholar
  13. 13.
    Mauritsson K, Boström A, Folkow PD (2008) Modelling of thin piezoelectric layers on plates. Wave Motion 45:616–628CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Martin PA (2005) On flexural waves in cylindrically anisotropic elastic rods. Int J Solids Struct 42:2161–2179CrossRefGoogle Scholar
  15. 15.
    Martin PA (2004) Waves in woods: axisymmetric waves in slender solids of revolution. Wave Motion 40:387–398CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Johanson G, Niklasson AJ (2003) Approximation dynamic boundary conditions for a thin piezoelectric layer. Int J Solids Struct 40:3477–3492CrossRefGoogle Scholar
  17. 17.
    Folkow PD, Johanson M (2009) Dynamic equations for fluid-loaded porous plates using approximate boundary conditions. J Acoust Soc Am 125:2954–2966CrossRefGoogle Scholar
  18. 18.
    Okhovat R, Boström A (2013) Dynamic equations for an anisotropic cylindrical shell using power series method. In: Topics in modal analysis, vol 7. The society for experimental mechanics. Springer, New YorkGoogle Scholar
  19. 19.
    Hägglund AM, Folkow PD (2008) Dynamic cylindrical shell equations by power series expansions. Int J Solids Struct 45:4509–4522CrossRefzbMATHGoogle Scholar
  20. 20.
    Okhovat R, Boström A (2012) Dynamic equations for an isotropic spherical shell using power series method. In: Eleventh international conference on computational structural technology. Civil-Comp PressGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2015

Authors and Affiliations

  1. 1.Department of Applied MechanicsChalmers University of TechnologyGothenburgSweden

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