Skip to main content
SpringerLink
Log in

Frequency equations for the in-plane vibration of orthotropic circular annular plate

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Orthotropic circular annular plates have a lot of applications in engineering such as space structures and rotary machines. In this paper, frequency equations for the in-plane vibration of the orthotropic circular annular plate for general boundary conditions were derived. To obtain the frequency equation, first the equation of motion for the circular annular plate in the cylindrical coordinate is derived by using the stress-strain- displacement expressions. Helmholtz decomposition is used to uncouple the equations of motion. The wave equation is obtained by assumption a harmonic solution for the uncoupled equations. Using the separation of the variables leads to the general wave equation solution and the in-plane displacements in the r and θ directions. Finally, boundary conditions are exerted and the natural frequency is derived for general boundary conditions. The obtained results are validated by comparing with the previously reported and those from finite element analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

R :

Outer radius

R 0 :

Inner radius

ξ :

Dimensionless coordinates r/R

η :

Radius ratio R 0/R

ρ :

Density

E r , E θ :

Youngs moduli related to the r and θ directions, respectively

ν r , ν θ :

Poisson ratios related to r and θ directions, respectively

G :

Modulus of elasticity in shear

b :

Stiffness ratio E θ /E r

g :

Dimensionless shear modulus defined in text

t :

Time

σ r , σ θ :

Normal stress in r and θ direction respectively

τ r θ :

shear stress

ε r , ε θ :

Normal strain in r and θ direction respectively

γ r θ :

Shear strain

u, v :

Displacements in r and θ directions, respectively

\({\varphi, {\bf H}}\) :

Scalar and vector potentials

Ω:

Non-dimensional natural frequencies defined in text

\({\nabla}\) :

Gradient operator

References

  1. Irie T., Yamada G., Muramoto Y.: Natural frequencies of in-plane vibration of annular plates. J. Sound Vib. 97, 171–175 (1984)

    Article  Google Scholar 

  2. Ambati G., Bell J.F.W., Sharp J.C.K.: In-plane vibrations of annular rings. J. Sound Vib. 47, 415–432 (1976)

    Article  MATH  Google Scholar 

  3. Farag N.H., Pan J.: Modal characteristics of in-plane vibration of circular plates clamped at the outer edge. J. Acous. Soc. Am. 113, 1935–1946 (2003)

    Article  Google Scholar 

  4. Holland R.: Numerical studies of elastic-disk contour modes lacking axial symmetry. J. Acous. Soc. Am. 40, 1051–1057 (1966)

    Article  Google Scholar 

  5. Kane T.R., Mindlin R.D.: High-frequency extensional vibrations of plates. J. Appl. Mech. 23, 277–283 (1956)

    MATH  MathSciNet  Google Scholar 

  6. Chen S.S.H., Liu T.M.: Extensional vibration of thin plates of various shapes. J. Acous. Soc. Am. 58(4), 828–831 (1975)

    Article  Google Scholar 

  7. Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity. 4th edn. Dover, New York (1944)

    MATH  Google Scholar 

  8. Bashmal S., Bhat R., Rakheja S.: In-plane free vibration of circular annular disks. J. Sound Vib. 322, 216–226 (2009)

    Article  Google Scholar 

  9. Chen J.S., Jhu J.L.: On the in-plane vibration and stability of a spinning annular disk. J. Sound Vib. 195(4), 585–593 (1996)

    Article  Google Scholar 

  10. Park C.: Frequency equation for the in-plane vibration of a clamped circular plate. J. Sound Vib. 313, 325–333 (2008)

    Article  Google Scholar 

  11. Michael W., Rubin D., Krempl E.: Introduction to Continuum Mechanics. 3rd edn. Pergamo, The united states of America (1993)

    Google Scholar 

  12. Koo K.N.: Vibration analysis and critical speeds of polar orthotropic annular disks in rotation. Compos. Struct. 76, 67–72 (2006)

    Article  Google Scholar 

  13. Doyle J.F.: Wave Propagation in Structures. 2nd edn. Springer, New York (1997)

    Book  MATH  Google Scholar 

  14. Achenbach J.D.: Wave Propagation in Elastic Solid. North-Holland Publishing, Amsterdam (1973)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Isfahan University of Technology, 84156-83111, Isfahan, Iran

    M. R. Karamooz Ravari & M. R. Forouzan

Authors
  1. M. R. Karamooz Ravari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. R. Forouzan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. R. Karamooz Ravari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karamooz Ravari, M.R., Forouzan, M.R. Frequency equations for the in-plane vibration of orthotropic circular annular plate. Arch Appl Mech 81, 1307–1322 (2011). https://doi.org/10.1007/s00419-010-0488-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-010-0488-6

Keywords

Search

Navigation