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Free Vibration Frequencies of a Variable Cross-Section Timoshenko-Ehrenfest Beam using Fourier-p Element

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Abstract

An efficient method has been developed to analyze the natural frequencies of a Timoshenko-Ehrenfest beam with variable cross-sections undergoing free flexural vibration. The method uses a single element to discretize the beam and combines polynomial and trigonometric shape functions to represent the displacement fields, with the trigonometric shape functions representing the internal degrees of freedom (DOF) of the beam. By incorporating trigonometric shape functions as enriching functions alongside polynomials, the potential issues related to ill-conditioning associated with higher-order polynomials are avoided. The differential equations of motion for the free vibration are derived using Lagrange’s equation. The generalized eigenvalue problem is then formulated. Non-dimensional natural frequencies are computed considering different parameter variations such as taper ratios, shear deformation, rotary inertia parameters, as well as end conditions. Incorporating rotary inertia and shear deformation in the analysis of Timoshenko-Ehrenfest beam lowers its natural frequencies compared to simpler beam theories like Euler–Bernoulli. It is observed that as taper ratios increase, the beam’s mass decreases toward the tip, causing higher frequency parameter values for the fundamental mode. Increasing rotary inertia parameter leads to a decreasing trend in frequency parameters especially in higher modes. The combined effect of rotary inertia and tapering significantly reduces frequency parameter values for all modes, with the rotary inertia counteracting the tapering influence. The proposed Fourier-p element model accurately predicts Timoshenko-Ehrenfest beam frequencies with high precision compared to other existing models.

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Abbreviations

A :

Cross-sectional area of the beam at an arbitrary location \(x\) (m2)

A 0 :

Cross-sectional area at the root of the beam \(= b_{ \circ } h_{ \circ }\) (m2)

[B]:

Strain displacement matrix

b :

Beam breadth at an arbitrary location \(x\) (m)

b 0 :

Beam breadth at the root of the beam (m)

E :

Modulus of elasticity (N/m2)

G :

Modulus of rigidity (N/m2)

h :

Beam depth at an arbitrary location \(x\) (m)

h 0 :

Beam depth at the root of the beam (m)

I :

Second moment of area of the beam at an arbitrary location \(x\) (m4)

I 0 :

Second moment of area at the root (thick end) of the beam \(= \left( {{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0pt} {12}}} \right)b_{ \circ } h_{ \circ }^{3}\) (m4)

j :

Number of trigonometric terms used

[K]:

Stiffness matrix \(= \left[ {K_{e} } \right] + \left[ {K_{s} } \right]\) (N/m)

[K e]:

Elastic stiffness matrix (N/m)

[K s]:

Shear stiffness matrix (N/m)

\(\kappa\) :

Shear correction factor

L :

Length of the beam (m)

\(\fancyscript{L}\) :

Lagrangian

[M]:

Mass matrix \(= \left[ {M_{t} } \right] + \left[ {M_{r} } \right]\) (kg)

[M t]:

Translational mass matrix (kg)

[M r]:

Rotary inertia mass matrix (kg)

n :

Number of terms used in the Fourier shape functions

\(\left[ {N_{w} } \right]\) :

Transverse displacement shape functions matrix

\(\left[ {N_{\theta } } \right]\) :

Rotation shape functions matrix

\(r_{g}\) :

Radius of gyration of the cross section \(= \sqrt {{I \mathord{\left/ {\vphantom {I A}} \right. \kern-0pt} A}}\) (m)

\(\overline{r}_{g}\) :

Rotary inertia parameter \(= {{r_{g} } \mathord{\left/ {\vphantom {{r_{g} } L}} \right. \kern-0pt} L}\)

\(\left\{ q \right\}\) :

Vector of nodal coordinates (m)

\(\left\{ {\overline{q}} \right\}\) :

Vector of displacement amplitudes of vibration (m)

T :

Kinetic energy (J)

t :

Time (s)

U :

Strain energy (J)

\(\gamma\) :

Shear strain (rad)

\(\varepsilon\) :

Normal strain

\(u_{x}\), \(u_{z}\) :

Components of the displacement vector in the two coordinate directions \(x\) and \(z\) (m)

\(\theta\) :

Rotation angle due to bending (rad)

\(\nu\) :

Poisson’s ratio

\(\rho\) :

Mass density (Kg/m3)

\(\sigma\) :

Normal stress (N/m2)

\(\tau\) :

Shear stress (N/m2)

\(\tau_{b}\) :

Breadth taper ratio

\(\tau_{h}\) :

Depth taper ratio

\(\omega\) :

Natural frequency (rad/s)

\(\overline{\omega }\) :

Frequency parameter \(= \omega L^{2} \sqrt {{{\rho A_{ \circ } } \mathord{\left/ {\vphantom {{\rho A_{ \circ } } {EI_{ \circ } }}} \right. \kern-0pt} {EI_{ \circ } }}}\)

\(\zeta\) :

Non-dimensional length \(= {x \mathord{\left/ {\vphantom {x L}} \right. \kern-0pt} L}\)

\(\left[ {} \right]^{T}\) :

Transpose of []

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Acknowledgements

The support of King Fahd University of Petroleum & Minerals is greatly appreciated.

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Authors and Affiliations

  1. Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, KFUPM BOX 579, 31261, Dhahran, Saudi Arabia

    Abdelaziz Bazoune

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Appendix

Appendix

The non-zeros entries of the beam enriched Fourier-\(p\) element shape functions are given by

$$\begin{aligned} & N_{w1} = N_{\theta 2} = \frac{1}{24}\left( {24 - 274\xi + 1125\xi^{2} - 2125\xi^{3} + 1875\xi^{4} - 625\xi^{5} } \right) \\ & N_{w3} = N_{\theta 4} = \frac{1}{24}\left( {600\xi - 3850\xi^{2} + 8875\xi^{3} - 8750\xi^{4} + 3125\xi^{5} } \right) \\ & N_{w5} = N_{\theta 6} = \frac{1}{12}\left( { - 300\xi + 2675\xi^{2} - 7375\xi^{3} - 8125\xi^{4} - 3125\xi^{5} } \right) \\ & N_{w7} = N_{\theta 8} = \frac{1}{12}\left( {200\xi + 1950\xi^{2} + 6125\xi^{3} - 7500\xi^{4} + 3125\xi^{5} } \right) \\ & N_{wj1} = N_{\theta j2} = \sin \left( {j\pi \xi } \right) \\ & N_{w9} = N_{\theta 10} = \frac{1}{24}\left( { - 150\xi + 1525\xi^{2} - 5125\xi^{3} + 6875\xi^{4} - 3125\xi^{5} } \right) \\ & N_{w11} = N_{\theta 12} = \frac{1}{24}\left( {24\xi - 250\xi^{2} + 875\xi^{3} - 1250\xi^{4} + 625\xi^{5} } \right) \\ \end{aligned}$$

where \(\xi = {x \mathord{\left/ {\vphantom {x L}} \right. \kern-0pt} L};\;\left( {0 \le \xi \le 1} \right)\) is the non-dimensional length.

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Bazoune, A. Free Vibration Frequencies of a Variable Cross-Section Timoshenko-Ehrenfest Beam using Fourier-p Element. Arab J Sci Eng 49, 2831–2851 (2024). https://doi.org/10.1007/s13369-023-08289-4

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