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

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## 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

*D*_{− R}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*β*_{x}Rotation angles of the cross section about the y-axis

*β*_{y}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

*D*_{− L}Left shaft diameter

*h*Thickness of the shaft

*R*_{n−1}The

*n*th inner radius of the shaft in FGM*Rn*The

*n*th outer radius of the shaft in FGM*I*_{m}Moment of mass inertia of the rotating shaft

*I*_{d}Diametral moment of inertia of the rotating shaft per unit length

*I*_{p}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

- {
*q*_{i}} 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

## References

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