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Free Vibration of Rotating Twisted Composite Stiffened Plate

  • Mrutyunjay RoutEmail author
  • Amit Karmakar
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)

Abstract

The paper presents a first-order shear deformation-based finite element model to investigate the free vibration response of the rotating twisted composite stiffened plate. An eight-noded isoparametric plate element having five degrees of freedom per node is combined with an isoparametric three-noded beam element of four degrees of freedom per node for modelling the plate and the stiffener element, respectively. The present formulation employs a constraint method to calculate the stiffness and mass matrices of the stiffener element attached to the plate element, wherein the degrees of freedom of the nodes of the stiffener element are transferred to the respective nodes of the plate element considering eccentricity to maintain the compatibility between plate and stiffener. The advantage of such method is that total number of degrees of freedom due to addition of the stiffener does not increase, thereby reducing the computational time. The governing equilibrium equation is derived from Lagrangian equation of motion. The Coriolis effect is not considered as the stiffened plate is allowed to rotate at moderate rotational speed only. Parametric studies have been conducted to investigate the effect of angle of fibre orientation, pretwist angle, stiffener depth-to-plate thickness ratio and rotational speeds on the fundamental frequency and mode shapes of the stiffened plate.

Keywords

Composite Stiffener Finite element Vibration Twisted plate 

Nomenclature

L

Length of the plate (m)

b

Width of the plate (m)

h

Thickness of the plate (m)

bst

Width of stiffener (m)

dst

Depth of stiffener (m)

ϕ

Pretwist angle of the plate (deg.)

ωn

Fundamental natural frequency of the stiffened plate without rotation (rad/s)

\( \Omega^{\prime } \)

Actual rotational speed (rad/s)

Ω

Non-dimensional rotational speed \( \left( {\Omega =\Omega ^{{\prime }} /\omega_{\text{n}} } \right) \) (dimensionless)

References

  1. 1.
    Leissa, A.W., Ewing, M.S.: Comparison of beam and shell theories for the vibrations of thin turbomachinery blades. J. Eng. Power 105(2), 383–392 (1983)CrossRefGoogle Scholar
  2. 2.
    Kielb, R.E., Leissa, A.W., Macbain, J.C.: Vibrations of twisted cantilever plates—a comparison of theoretical results. Int. J. Numer. Meth. Eng. 21(8), 1365–1380 (1985)CrossRefGoogle Scholar
  3. 3.
    Qatu, M.S., Leissa, A.W.: Vibration studies for laminated composite twisted cantilever plates. Int. J. Mech. Sci. 33(11), 927–940 (1991)CrossRefGoogle Scholar
  4. 4.
    Liew, K.M., Lim, C.W., Ong, L.S.: Vibration of pretwisted cantilever shallow conical shells. Int. J. Solids Struct. 31(18), 2463–2476 (1994)CrossRefGoogle Scholar
  5. 5.
    Kuang, J.H., Hsu, M.H.: The effect of fiber angle on the natural frequencies of orthotropic composite pre-twisted blades. Compos. Struct. 58(4), 457–468 (2002)CrossRefGoogle Scholar
  6. 6.
    Lee, J.J., Yeom, C.H., Lee, I.: Vibration analysis of twisted cantilevered conical composite shells. J. Sound Vib. 255(5), 965–982 (2002)CrossRefGoogle Scholar
  7. 7.
    Sreenivasamurthy, S., Ramamurti, V.: Coriolis effect on the vibration of flat rotating low aspect ratio cantilever plates. J. Strain Anal. Eng. Des. 16(2), 97–106 (1981)CrossRefGoogle Scholar
  8. 8.
    Ramamurti, V., Kielb, R.: Natural frequencies of twisted rotating plates. J. Sound Vib. 97(3), 429–449 (1984)CrossRefGoogle Scholar
  9. 9.
    Karmakar, A., Sinha, P.K.: Finite element free vibration analysis of rotating laminated composite pretwisted cantilever plates. J. Reinf. Plast. Compos. 16(16), 1461–1491 (1997)CrossRefGoogle Scholar
  10. 10.
    Rao, S.S., Gupta, R.S.: Finite element vibration analysis of rotating Timoshenko beams. J. Sound Vib. 242(1), 103–124 (2001)CrossRefGoogle Scholar
  11. 11.
    Hu, X.X., Sakiyama, T., Matsuda, H., Morita, C.: Fundamental vibration of rotating cantilever blades with pre-twist. J. Sound Vib. 271(1–2), 47–66 (2004)CrossRefGoogle Scholar
  12. 12.
    Kee, Y.J., Kim, J.H.: Vibration characteristics of initially twisted rotating shell type composite blades. Compos. Struct. 64(2), 151–159 (2004)CrossRefGoogle Scholar
  13. 13.
    Joseph, S.V., Mohanty, S.C.: Free vibration of a rotating sandwich plate with viscoelastic core and functionally graded material constraining layer. Int. J. Struct. Stab. Dyn. 17(10), 1750114 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chandrashekhara, K., Kolli, M.: Free vibration of eccentrically stiffened laminated plates. J. Reinf. Plast. Compos. 16(10), 884–902 (1997)CrossRefGoogle Scholar
  15. 15.
    Prusty, B.G., Satsangi, S.K.: Finite element transient dynamic analysis of laminated stiffened shells. J. Sound Vib. 248(2), 215–233 (2001)CrossRefGoogle Scholar
  16. 16.
    Nayak, A.N., Bandyopadhyay, J.N.: Free vibration analysis of laminated stiffened shells. J. Eng. Mech. 131(1), 100–105 (2005)CrossRefGoogle Scholar
  17. 17.
    Sahoo, S., Chakravorty, D.: Stiffened composite hypar shell roofs under free vibration: Behaviour and optimization aids. J. Sound Vib. 295(1), 362–377 (2006)CrossRefGoogle Scholar
  18. 18.
    Sadek, E.A., Tawfik, S.A.: A finite element model for the analysis of stiffened laminated plates. Comput. Struct. 75(4), 369–383 (2000)CrossRefGoogle Scholar
  19. 19.
    Qing, G., Qiu, J., Liu, Y.: Free vibration analysis of stiffened laminated plates. Int. J. Solids Struct. 43(6), 1357–1371 (2006)CrossRefGoogle Scholar
  20. 20.
    Yuan, W.X., Dawe, D.J.: Free vibration and stability analysis of stiffened sandwich plates. Compos. Struct. 63(1), 123–137 (2004)CrossRefGoogle Scholar
  21. 21.
    Guo, M., Harik, I.E., Ren, W.X.: Free vibration analysis of stiffened laminated plates using layered finite element method. Struct. Eng. Mech. 14(3), 245–262 (2002)CrossRefGoogle Scholar
  22. 22.
    Li, D., Qing, G., Liu, Y.: A layerwise/solid-element method for the composite stiffened laminated cylindrical shell structures. Compos. Struct. 98, 215–227 (2013)CrossRefGoogle Scholar
  23. 23.
    Bhar, A., Phoenix, S.S., Satsangi, S.K.: Finite element analysis of laminated composite stiffened plates using FSDT and HSDT: A comparative perspective. Compos. Struct. 92(2), 312–321 (2010)CrossRefGoogle Scholar
  24. 24.
    Zhao, W., Kapania, R.K.: Vibrational analysis of unitized curvilinearly stiffened composite panels subjected to in-plane loads. In: 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 1500 (2016)Google Scholar
  25. 25.
    Castro, S.G., Donadon, M.V.: Assembly of semi-analytical models to address linear buckling and vibration of stiffened composite panels with debonding defect. Compos. Struct. 160, 232–247 (2017)CrossRefGoogle Scholar
  26. 26.
    Rout, M., Bandyopadhyay, T., Karmakar, A.: Free vibration analysis of pretwisted delaminated composite stiffened shallow shells: a finite element approach. J. Reinf. Plast. Compos. 36(8), 619–636 (2017)CrossRefGoogle Scholar
  27. 27.
    Rout, M., Hota, S.S., Karmakar, A.: Free vibration characteristics of delaminated composite pretwisted stiffened cylindrical shell. In: Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, doi: 0954406216686389 (2017)Google Scholar
  28. 28.
    Damnjanović, E., Marjanović, M., Nefovska-Danilović, M.: Free vibration analysis of stiffened and cracked laminated composite plate assemblies using shear-deformable dynamic stiffness elements. Compos. Struct. 180, 723–740 (2017)CrossRefGoogle Scholar
  29. 29.
    Cook, R.D.: Concepts and Applications of Finite Element Analysis. Wiley, Hoboken (2007)Google Scholar
  30. 30.
    Bathe, K.J.: Finite Element Procedures in Engineering Analysis. PHI, New Delhi (1990)Google Scholar
  31. 31.
    Das, H.S., Chakravorty, D.: Bending analysis of stiffened composite conoidal shell roofs through finite element application. J. Compos. Mater. 45, 525–542 (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia

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