SpringerLink
Log in

Frequency Analysis of Variable Thickness Kirchhoff Plates by Isogeometric Approach

  • Original Contribution
  • Published:
Journal of The Institution of Engineers (India): Series C Aims and scope Submit manuscript

Abstract

Vibration analysis of thin plates was of great interest for ages. With the application of different computation techniques, vibrational analysis of thin plates was done. In this paper, an isogeometric approach has been taken for vibrational calculation of thin plates with variation in thickness. The change in thickness is varied as linear and parabolic profile with different tapering factors (0.3, 0.6, 0.9). Non-Uniform Rational B Splines basis functions are used to approximate the design as well as to approximate the unknown solution. The governing equation of motion of plates has been derived on the platform of classical plate theory with the application of Galerkin method. Stiffness and mass matrices are developed for vibrational analysis and applied to plates of different thicknesses. Results obtained were in the region of great accuracy with literature and error was within the range of 1%. Isogeometric approach is found to be very effective, accurate, precise, and computationally efficient.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Figure 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. A.W. Liessa, The free vibration of rectangular plates. J. Sound Vib. 31(3), 257–293 (1973)

    Article  Google Scholar 

  2. D.J. Dawe, O.L. Roufaeil, Rayleigh-Ritz vibration analysis of plates. J. Sound Vib. 69(3), 345–359 (1980)

    Article  MATH  Google Scholar 

  3. M. Tanaka, K. Yamagiwa, K. Miyazaki, T. Ueda, Free vibration analysis of elastic plate structures by boundary element method. Eng. Ananl. 5(4), 182–188 (1988)

    Article  Google Scholar 

  4. Y.K. Cheung, L.G. Tham, W.Y. Li, Free vibration and static analysis of general plate by spline finite strip. Comput. Mech. 3, 187–197 (1988)

    Article  MATH  Google Scholar 

  5. K.Y. Lam, K.C. Hung, S.T. Chow, Vibration analysis of plates with cutouts by the modified Rayleigh-Ritz method. Appl. Acoust. 28, 49–60 (1989)

    Article  Google Scholar 

  6. K.M. Liew, Vibration of thick skew plates based on mindlin shear deformation plate theory. J. Sound Vib. 168(1), 39–69 (1993)

    Article  MATH  Google Scholar 

  7. C.S. Huang, O.G. Mcgee, A.W. Liessa, Accurate vibration analysis of simply supported rhombic plates by considering stress singularities. J. Vib. Acoust. 117, 245–251 (1995)

    Article  Google Scholar 

  8. S. Kitipornchai, W. Karunasena, Free vibration of shear-deformable General triangular plates. J. Sound Vib. 199(4), 595–613 (1997)

    Article  Google Scholar 

  9. G.R. Liu, X.L. Chen, A mesh-free method for static and free vibration analyses of thin plates of complicated shape. J. Sound Vib. 241(5), 839–855 (2001)

    Article  Google Scholar 

  10. F.C. Appl, N.R. Byers, Fundamental frequency of simply supported rectangular plates with linearly varying thickness. J. Appl. Mech. 32(163), 168 (1965)

    MathSciNet  Google Scholar 

  11. D.J. Dawe, Vibration of rectangular plates of variable thickness. J. Mech. Eng. Sci. 8(1), 42–51 (1966)

    Article  Google Scholar 

  12. T. Mikami, J. Yoshimura, Application of the collocation method to vibration analysis of rectangular mindlin plates. Comput. Struct. 18(3), 422–431 (1984)

    MATH  Google Scholar 

  13. P. Malekzadeh, G. Karamib, Large amplitude flexural vibration analysis of tapered plates with edges elastically restrained against rotation using DQM. Eng. Struct. 30, 2850–2858 (2008)

    Article  Google Scholar 

  14. P. Malekzadeh, Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM. Thin-Walled Struct. 46, 11–26 (2008)

    Article  Google Scholar 

  15. P. Edwin Sudhagar, A. Ananda Babu, R. Vasudevan, P. Jeyaraj, Vibration analysis of a tapered laminated thick composite plate with ply drop-offs. Arch Appl Mech 85, 969–990 (2015)

    Article  Google Scholar 

  16. T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. V. Agrawal, S.S. Gautam, IGA: a simplified introduction and implementation details for finite element users. J. Inst. Eng. India Ser. C 100, 561–585 (2019)

    Article  Google Scholar 

  18. V. Agrawal, S.S. Gautam, Varying-order NURBS discretization: An accurate and efficient method for isogeometric analysis of large deformation contact problems. Comput. Methods Appl. Mech. Engrg. 367, 113125 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. D. S. Bombarde, S. S. Gautam and A. Nandy, A novel hybrid isogeometric element based on two-field Hellinger-Reissner principle to alleviate different types of locking, https://arxiv.org/abs/2103.09543.

  20. Y. Bazilevs, T.J.R. Hughes, NURBS-based isogeometric analysis for the computation of flows about rotating components. Comput. Struct. 43, 143–150 (2008)

    MATH  Google Scholar 

  21. Y. Bazilevs, V.M. Calo, Y. Zhang, T.J.R. Hughes, Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput. Struct. 38, 310–322 (2006)

    MATH  Google Scholar 

  22. Y. Bazilevs, V.M. Calo, T.J.R. Hughes, Y. Zhang, Isogeometric fluid-structure interaction: theory, algorithms and computations. Comput. Struct. 43, 3–37 (2008)

    MathSciNet  MATH  Google Scholar 

  23. W.A. Wall, M.A. Frenzel, C. Cyron, Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput. Methods Appl. Mech. Eng. 197, 2976–2988 (2008)

    Article  MATH  Google Scholar 

  24. X. Qian, Isogeometric structural shape optimization. Comput. Methods Appl. Mech. Eng. 199, 2059–2071 (2010)

    Article  MATH  Google Scholar 

  25. S. Shojaee, N. Valizadeh, M. Arjomand, Isogeometric structural shape optimization using particle swarm algorithm. Int. J. Optim. CivilEng. 4, 633–645 (2011)

    Google Scholar 

  26. D.J. Benson, Y. Bazilevs, M.C. Hsu, T.J.R. Hughes, Isogeometric shell analysis the Reissner-Mindlin shell. Comput. Methods Appl. Mech. Eng. 199, 276–289 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. D.J. Benson, Y. Bazilevs, M.C. Hsu, T.J.R. Hughes, A large deformation rotation-free isogeometric shell. Comput. Methods Appl. Mech. Eng. 200, 1367–1378 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Kiendl, K.U. Bletzinger, J. Linhard, R. Wuchner, Isogeometric shell analysis with Kirchhoff-Love elements. Comput. Methods Appl. Mech. Eng. 198, 3902–3914 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. T.K. Uhm, S.K. Youn, T-spline finite element method for the analysis of shell structures. Int. J. Numer. Methods Eng. 80, 507–536 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. C.V. Verhoosel, M.A. Scott, T.J.R. Hughes, R. deBorst, An isogeometric analysis approach to gradient damage models. Int. J. Numer. Methods Eng. 86, 115–134 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. J.A. Cottrell, A. Reali, Y. Bazilevs, T.J.R. Hughes, Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Eng. 195, 5257–5296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Shojaee, E. Izadpanah, N. Valizadeh, J. Kiendl, Free vibration analysis of thin plates by using a NURBS-based isogeometric approach. Finite Elem. Anal. Des. 61, 23–34 (2012)

    Article  MathSciNet  Google Scholar 

  33. L. Piegl, W. Tiller, The NURBS book (monographs in visual communication), 2nd edn. (Springer-Verlag, NewYork, 1997)

    MATH  Google Scholar 

  34. D.F. Rogers, An introduction to NURBS with historical perspective (Academic Press, San Diego, CA, 2001)

    Google Scholar 

  35. G.E. Farin, NURBS curves and surfaces: from projective geometry to practical use (A.K. Peters Ltd., Natick, MA, 1995)

    MATH  Google Scholar 

  36. H.Y. Roh, M. Cho, The application of geometrically exact shell elements to B-spline surfaces. Comput. Methods Appl. Mech. Eng. 193, 2261–2299 (2004)

    Article  MATH  Google Scholar 

  37. V.P. Nguyen, R.N. Simpson, S.P.A. Bordas, T. Rabczuk, Isogeometric analysis: an overview and computer implementation aspects. Math. Comput. Simul. 117, 89–116 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. G.R. Liu, Mesh free methods: moving beyond the finite element method (CRC Press, Boca Raton, 2003)

    MATH  Google Scholar 

  39. T. Belytschko, Y.Y. Lu, L. Gu, Element free Galerkin method. Int. J Numer. Methods Eng. 37, 229–256 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Zhu, S.N. Atluri, A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Comput. Struct. 21, 211–222 (1998)

    MathSciNet  MATH  Google Scholar 

  41. S. Fernandez-Mendez, A. Huerta, Imposing essential boundary conditions in mesh-free methods. Comput. Methods Appl. Mech. Eng. 193, 1257–1275 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Y. Liu, Y.C. Hon, K.M. Liew, A mesh-free Hermite type radial point interpolation method for Kirchhoff platep roblems. Int. J. Numer. Methods Eng. 66, 1153–1178 (2006)

    Article  MATH  Google Scholar 

  43. J. Kiendl, Y. Bazilevs, M.C. Hsu, R. Wuchner, K.U. Bletzinger, The bending strip method for isogeometric analysis of Kirchhoff Love shell structures comprised of multiple patches. Comput. Methods Appl. Mech. Eng. 199, 2403–2416 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. S.C. Yen and H.Y. Cheng, Finite eiement anaiysis of tapered laminated composite piates, 8th International Conference on composite material, Vol. 1, Article 16, (1991)

  45. P. Jeyaraj, Vibro-acoustic behavior of an isotropic plate with arbitrarily varying thickness. Eur. J. Mech. A/Solids 29, 1088–1094 (2010)

    Article  MATH  Google Scholar 

Download references

Funding

The authors confirm that this work has not been attached to any funding support.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology, Jamshedpur, India

    Gourav Prasad Sinha & Bipin Kumar

Authors
  1. Gourav Prasad Sinha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bipin Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gourav Prasad Sinha.

Ethics declarations

Conflict of interest

The authors confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sinha, G.P., Kumar, B. Frequency Analysis of Variable Thickness Kirchhoff Plates by Isogeometric Approach. J. Inst. Eng. India Ser. C 104, 271–280 (2023). https://doi.org/10.1007/s40032-023-00910-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40032-023-00910-7

Keywords

Search

Navigation