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


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.


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)


Material properties of the material A


Material properties of the material B

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

Layer thickness increment (n)


Layer Index


Number of the layer


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


Rotation angles of the cross section about the y-axis


Rotation angles of the cross section about the z-axis


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

(x, r, θ)

Cylindrical coordinates


Young modulus of the layer (n)


Poisson coefficient of the layer (n)


Shear correction factor


The density mass of the layer (n)


Length of the shaft


Conicity angle of the shaft


Left shaft diameter


Thickness of the shaft


The nth inner radius of the shaft in FGM


The nth outer radius of the shaft in FGM


Moment of mass inertia of the rotating shaft


Diametral moment of inertia of the rotating shaft per unit length


Moment of polar inertia of the rotating shaft per unit length


Kinetic energy of the shaft


Deformation energy


Deformation tensor


Stress tensor


Frequency, eigenvalue


Rotation speed


Matrix of the shape functions


Matrix mass of the FGM conical shaft


Stiffness matrix of the FGM conical shaft


Gyroscopic matrix of the FGM conical shaft


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




Virtual work of generalized forces


Generalized forces


Shape functions


Number of the shape functions or number of hierarchical terms



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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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