Abstract
The free vibration of axially functionally graded (FG) non-uniform beams with different boundary conditions is studied using Differential Transformation (DT) based Dynamic Stiffness approach. This method is capable of modeling any beam (Timoshenko or Euler, centrifugally stiffened or not) whose cross sectional area, moment of Inertia and material properties vary along the beam. The effectiveness of the method is confirmed by comparing the present results with existing closed form solutions and numerical results. In FG beams, flexural rigidity and mass density may take majority of functions including polynomials, trigonometric and exponential functions (converted to polynomial expressions). DT based Dynamic stiffness approach is proved to be a versatile and simple approach compared to many other methods already proposed.
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Abbreviations
- A 0 :
-
Area of the section at the root
- A(x):
-
Cross sectional area at any section
- b(x):
-
Breadth of the cross section at any section
- d(x)=E(x)I(x):
-
Flexural rigidity at any section
- E(x):
-
Modulus of elasticity of the axially graded material at any section
- e :
-
Taper ratio
- G(x):
-
Modulus of rigidity at any section
- h(x):
-
Depth of the cross section at any section
- I(x):
-
Moment of Inertia at any section
- I 0 :
-
Moment of Inertia of the section at the root
- K(x)=κA(x)G(x):
-
Shear rigidity at any section
- K :
-
Structural stiffness matrix
- KG :
-
Geometric stiffness matrix
- L :
-
Length of the beam
- mr=1:
-
Depth taper only
- mr=2:
-
Both width and depth taper
- M T :
-
Tip mass at the free end
- M(x):
-
Moment at any section
- M :
-
Mass matrix
- nt :
-
Number of terms
- nr :
-
Material non-homogeneity factor
- P(x):
-
Centrifugal force or compressive load
- p(x):
-
Lateral load
- \(p = \frac{\mathit{PL}^{2}}{\mathit{EI}_{0}}\) :
-
Buckling load parameter
- R :
-
Hub radius
- s :
-
Summation index
- T :
-
Typical material property
- T a ,T z :
-
Typical material property for Alumina and Zirconia respectively
- V(x):
-
Shear force
- w :
-
Lateral deflection of the centre line of the beam
- x, y and z :
-
Cartesian coordinate axes
- y(x):
-
Function
- β :
-
Tip mass parameter
- δ=R/L :
-
Non-dimensional parameter for hub radius \(\eta = \sqrt{\varOmega^{2} + \omega^{2}}\)
- κ :
-
Shear correction factor
- λ :
-
Rotating speed parameter where \(\lambda = \varOmega\sqrt{\frac{\rho_{0}A_{0}L^{4}}{E_{0}I_{0}}}\)
- μ :
-
Natural frequency parameter where \(\mu = \omega\sqrt{\frac{\rho_{0}A_{0}L^{4}}{E_{0}I_{0}}}\)
- Ω :
-
Angular rotation speed in radians/sec
- φ :
-
Shape function for θ
- ρ(x):
-
Mass density of axially graded material at any section
- ψ :
-
Shape function for bending rotation
- θ :
-
Bending rotation
- ω :
-
Natural frequency
- \(\xi = \frac{x}{L} \) :
-
Non-dimensional variable
- ς :
-
Damping factor
- \(\nabla = \frac{\mathrm{d}}{\mathrm{d}x}\) :
-
Operator
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Acknowledgements
The author thanks the management and Principal Dr. R. Rudramoorthy of PSG College of Technology for providing necessary facilities to complete the research work reported in this paper. The author also thanks the anonymous reviewers for their advice and suggestions towards enhancing the quality of the manuscript.
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Rajasekaran, S. Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach. Meccanica 48, 1053–1070 (2013). https://doi.org/10.1007/s11012-012-9651-1
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DOI: https://doi.org/10.1007/s11012-012-9651-1