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Dynamic Study of a Functionally Graded Material Rotating Conical Shaft Based on a New Model of Variation by Slice in the Material Properties

  • Abdelhak ElhannaniEmail author
  • Kaddour Refassi
  • Abbès Elmeiche
Technical Article---Peer-Reviewed
  • 14 Downloads

Abstract

This article is concerned with the dynamic behavior study of a rotating conical shaft bi-supported in functionally graded material (FGM), using the Timoshenko beam theory (FSDBT), with consideration of the gyroscopic effect. The material properties are varied through the thickness direction of the shaft based on a new model which defines the variation per slice in the volume fraction of the two extreme materials according to the exponential law distribution (E-FGM). This dynamic system is examined from a mathematical development and a hierarchical finite element modeling p-version. To validate our model, the numerical results obtained are compared with those available in the literature. Several examples are treated and a discussion is held to determine the influence of the internal properties (FGM), the conicity (α) and the slenderness ratio on the eigen-frequencies and consequently on the dynamic behavior of FGM rotating conical shafts.

Keywords

Dynamic behavior study Rotating conical shaft FGM Variation per slice Gyroscopic Hierarchical finite element method Eigen-frequency 

List of Symbols

\(P\left( n \right)\)

Material properties of the layer (n)

\(P_{A}\)

Material properties of the material A

\(P_{B}\)

Material properties of the material B

\(r\left( n \right)\)

Layer thickness increment (n)

\(n\)

Layer Index

Nc

Number of the layer

DR

Right diameter of the shaft

U(x, y, z)

Displacement in X direction

V(x, y, z)

Displacement in Y direction

W(x, y, z)

Displacement in Z direction

βx

Rotation angles of the cross section about the y-axis

βy

Rotation angles of the cross section about the z-axis

ϕ

Angular displacement of the cross section due to the torsion deformation of the shaft

(x, r, θ)

Cylindrical coordinates

E(n)

Young modulus of the layer (n)

ν(n)

Poisson coefficient of the layer (n)

ks

Shear correction factor

ρ(n)

The density mass of the layer (n)

L

Length of the shaft

α

Conicity angle of the shaft

DL

Left shaft diameter

h

Thickness of the shaft

Rn−1

The nth inner radius of the shaft in FGM

Rn

The nth outer radius of the shaft in FGM

Im

Moment of mass inertia of the rotating shaft

Id

Diametral moment of inertia of the rotating shaft per unit length

Ip

Moment of polar inertia of the rotating shaft per unit length

Ec

Kinetic energy of the shaft

Ed

Deformation energy

εij

Deformation tensor

σij

Stress tensor

ϖ

Frequency, eigenvalue

Ω

Rotation speed

[N]

Matrix of the shape functions

[M]

Matrix mass of the FGM conical shaft

[K]

Stiffness matrix of the FGM conical shaft

[G]

Gyroscopic matrix of the FGM conical shaft

{qi}

Generalized coordinates, with (\(i = U_{0} ,V_{0} ,W_{0} ,\beta_{x} , \beta_{y} ,\phi\))

\(\left\{ {\dot{q}_{i} } \right\}\)

Generalized speeds

\(\left\{ {\ddot{q}_{i} } \right\}\)

Generalized accelerations

t

Time

δA

Virtual work of generalized forces

Fi(t)

Generalized forces

g(x)

Shape functions

p

Number of the shape functions or number of hierarchical terms

Notes

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Copyright information

© ASM International 2019

Authors and Affiliations

  • Abdelhak Elhannani
    • 1
    Email author
  • Kaddour Refassi
    • 1
  • Abbès Elmeiche
    • 1
  1. 1.Laboratory of Solids and Structures Mechanics, Faculty of TechnologyUniversity of Sidi-Bel AbbesSidi Bel AbbèsAlgeria

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