Free Vibration Characteristics of Materially Monoclinic Circular Cylinders

  • M. Darvizeh
  • C. B. Sharma


The free vibration characteristics (natural frequencies, mode shapes, modal forces and moments) for materially monoclinic cylindrical shells are analysed here using an exact approach. Axial dependence of modal forms is taken in the form of simple Fourier series instead of an exponential dependence used previously. Transverse shear deformation and rotary inertia terms are included in the analysis for a more reliable prediction of response characteristics of such high modulus composite shells. Analytical frequencies obtained from the present study are shown to be in good agreement with some previously published experimental and theoretical results.


Cylindrical Shell Mode Shape Modal Force Circular Cylindrical Shell Transverse Shear Deformation 
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Shell radius


Matrix elements


Shell thickness


Shell length


Axial wave number

Mx, Mθ, Mxθ

Moment resultants

\( \tilde{M}_{x}^{0} \)

—03C0M x (0, θ)/(l cos pθ)

\( \tilde{M}_{x}^{l} \)

πM x (l, θ)/(l cos pθ)

Nx, Nθ, Nxθ

Stress resultants

\( \tilde{N}_{x}^{0} \)

N x (0, θ)/(cos pθ)

\( \tilde{N}_{x}^{l} \)

N x (l, θ)/(cos pθ)


Circumferential wave number


Transverse shear force


Time variable

u, v, w

Inplane and radial displacement components

x, θ

Axial and circumferential co-ordinates

x , βθ

Axial and circumferential rotations respectively


Circular natural frequency


Frequency parameter ( = ρa 2ω2)

Subscripts and Superscripts


Take the values 1–10

u, v, w, βx , βθ

Denote a variable corresponding to longitudinal modal forms

0, l

Indicate the values at x = 0, l


Denotes transpose of a matrix

x, θ

Indicate respective directions


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  1. 1.
    Bert, C. W., Advances in dynamics of composite structures. In Composite Structures 4, ed. I. H. Marshall. Elsevier Applied Science, London and New York, 1987, pp. 1–17.Google Scholar
  2. 2.
    Vanderpool, M. E. and Bert, C. W., Vibration of a materially monoclinic, thick wall circular cylindrical shell. AIAA J., 19 (1981) 634–41.CrossRefGoogle Scholar
  3. 3.
    Forsberg, K., Influence of boundary conditions on the modal characteristics of thin cylindrical shells. AIAA J., 2 (1964) 2150–7.CrossRefGoogle Scholar
  4. 4.
    Darvizeh, M., Free vibration characteristics of orthotropic thin circular cylindrical shells. PhD thesis, University of Manchester, 1986.Google Scholar
  5. 5.
    Greif, R. and Mittendorf, S. C., Structural vibration and Fourier series. J. Sound Vib., 48 (1916) 113–22.CrossRefGoogle Scholar

Copyright information

© Elsevier Science Publishers Ltd 1989

Authors and Affiliations

  • M. Darvizeh
    • 1
  • C. B. Sharma
    • 2
  1. 1.Department of Mechanical EngineeringGilan UniversityRashtIran
  2. 2.Department of MathematicsUniversity of Manchester (UMIST), Institute of Science and Technology (UMIST)ManchesterUK

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