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Free Vibration Characteristics of Functionally Graded Pre-twisted Conical Shells under Rotation

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Abstract

This article deals with effect of rotation and pretwist angle on free vibration characteristics of functionally graded conical shells. The dynamic equilibrium equation is derived from Lagrange’s equation neglecting the Coriolis effect for moderate rotational speeds. The materials properties of conical shell are varied with a power-law distribution of the volume fractions of their constituents through its thickness. Convergence studies are performed in respect of mesh sizes, and comparisons of the present solutions and those reported in open literature are provided to substantiate the accuracy of the proposed method. Computer codes developed to obtain the numerical results for the combined effects of twist angle and rotational speed on the natural frequencies of functionally graded conical shells. The mode shapes for a typical laminate configuration under different conditions are also illustrated. Numerical results are obtained for the non-dimensional fundamental (NDFF) and second frequencies (NDSF).

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Abbreviations

E:

Young’s modulus

G:

Shear modulus

N:

Material property graded index (or power law exponent)

T:

Temperature in Kelvin

NDFF, NDSF:

Non-dimensional fundamental and second natural frequency, respectively

Si :

Material property

Vf :

Volume fraction

Lo :

Length

Qj :

Shape functions

Lo/s:

Aspect ratio

[J]:

Jacobian matrix

[Me], [M]:

Element mass matrix and global mass matrix

[Ce], [C]:

Element coriolis matrix and global coriolis matrix

[Ke], [K]:

Element elastic stiffness matrix and global elastic stiffness matrix

[Kσe], [Kσ]:

Element geometric stiffness matrix, global geometric stiffness matrix

[KRe], [KR]:

Element rotational stiffness matrix and global rotational stiffness matrix

{F(Ω2)}:

Global vector of nodal equivalent centrifugal forces

{Fce}, {Fe}, {F}:

Element load vector due to centrifugal force, element load vector due to externally applied load and global vector of externally applied load, respectively

h:

Thickness

x, y, z:

Local coordinate axes (plate coordinate system)

r1, r2 :

Radius of curvature (r1 < r2)

rx :

Radius of curvature in x-direction

ry :

Radius of curvature in y-direction

rxy :

Radius of twist

bo :

Reference width

Ψ:

Twist angle

Ω:

Non-dimensional speed of rotation (Ω′/ωo)

Ω′:

Actual angular speed of rotation

ς, χ:

Local natural coordinates of the element

υ:

Poisson’s ratio

ρ:

Mass density

ω:

Non-dimensional frequency parameter

λ:

Non-dimensional frequency

\(\upphi_{\text{ve}}\) :

Vertex angle

\(\upphi_{\text{o}}\) :

Base subtended angle of cone

ωo :

Fundamental natural frequency of a non-rotating shell

ωn :

Natural frequency of rotating shell

{δe},{δ}:

Element displacement vector and global displacement vector

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  1. Department of Mechanical Engineering, Jadavpur University, Kolkata, 700032, India

    Apurba Das & Amit Karmakar

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  1. Apurba Das
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Correspondence to Amit Karmakar.

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Das, A., Karmakar, A. Free Vibration Characteristics of Functionally Graded Pre-twisted Conical Shells under Rotation. J. Inst. Eng. India Ser. C 99, 681–692 (2018). https://doi.org/10.1007/s40032-017-0378-6

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