Frequency Equations for Vibration Analysis of Fuel Rod

2005 ◽  
Author(s):  
Hyeong Koo Kim ◽  
Jae Ik Kim ◽  
Kyu Tae Kim ◽  
Moon Saeng Kim
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Sine Series ◽  
Various Boundary Conditions ◽  
Fuel Rod ◽  
Conventional Analysis ◽  
Fourier Sine Series ◽  
Fuel Design ◽  
Parametric Analyses

In this study, the frequency equations for calculating the natural frequencies of the beams with generally restrained boundary conditions by both translational and rotational springs are derived in matrix form using Fourier sine series. In order to show the validation of the solution, numerical results for two degenerate cases are compared with existing results for natural frequency obtained by the conventional analysis. And as a specific application, the natural frequencies of fuel rod for KSNP (Korean Standard Nuclear Plant) fuel assembly are calculated and compared with the external excitations. As a result, the frequency equation derived by present paper seems to be very useful to evaluate the natural frequencies of the double span beams with various boundary conditions. Especially, when some parametric analyses are needed to modify fuel design, the equation can be applied very easily.

scholarly journals Torsional vibration of cracked carbon nanotubes with torsional restraints using Eringen’s nonlocal differential model

2018 ◽  
Vol 38 (1) ◽  
pp. 70-87 ◽  
Author(s):  
Mustafa Ö Yayli ◽  
Suheyla Y Kandemir ◽  
Ali E Çerçevik
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Torsional Vibration ◽  
Nonlocal Boundary ◽  
Coefficient Matrix ◽  
Vibration Frequencies ◽  
Sine Series ◽  
Differential Model ◽  
Fourier Sine ◽  
Fourier Sine Series

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

Node Control of Uniform Beams Subject to Various Boundary Conditions

1992 ◽  
Vol 59 (4) ◽  
pp. 983-990 ◽  
Author(s):  
L. Weaver ◽  
L. Silverberg
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Controlled System ◽  
Free Boundary Conditions ◽  
Various Boundary Conditions ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Control Forces ◽  
Direct Feedback

This paper introduces node control, whereby discrete direct feedback control forces are placed at the nodes of the N+1th mode (the lowest N modes participate in the response). Node control is motivated by the node control theorem which states, under certain conditions, that node control preserves the natural frequencies and natural modes of vibration of the controlled system while achieving uniform damping. The node control theorem is verified for uniform beams with pinned-pinned, cantilevered, and free-free boundary conditions, and two cases of beams with springs on the boundaries. A general proof of the node control theorem remains elusive.

FREE VIBRATION ANALYSIS OF DISCRETELY STIFFENED SKEW PLATES

2001 ◽  
Vol 01 (01) ◽  
pp. 125-144 ◽  
Author(s):  
HUAN ZENG ◽  
CHARLES W. BERT
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Vibration Characteristics ◽  
Skew Angle ◽  
Various Boundary Conditions ◽  
Skew Plate ◽  
Convergence Study

Stiffened skew plates find application in various engineering fields. The free vibration characteristics of such plates have been studied by various methods. An orthogonally stiffened skew plate is a skew plate with stiffeners running orthogonal to two opposite edges. To the best knowledge of the present investigators, no previous work has been done for free vibration characteristics of skew plates of such stiffening geometry. The present work studies the free vibration of such plates. The pb-2 Rayleigh–Ritz method was employed due to its accuracy and computational efficiency. The conventional finite element method was also used as a comparative check. A convergence study was first performed for various boundary conditions. Then the vibration of orthogonally stiffened skew plates with different boundary conditions was studied. Close agreement was found between these two methods. The variations of natural frequencies with different parameters, including skew angle ϕ, edge ratio b/a, and height-thickness ratio f/h, were investigated for three types of boundary conditions.

Exact Nonlinear Dynamic Analysis of a Beam With a Nonlinear Vibration Absorber and With Various Boundary Conditions

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Mohammad Bukhari ◽  
Oumar Barry
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Vibration ◽  
Nonlinear Dynamic ◽  
Microelectromechanical Systems ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Vibration Absorber ◽  
Nonlinear Frequency ◽  
Nonlinear Vibration Absorber ◽  
Various Boundary Conditions

Abstract We study the nonlinear vibration of a beam with an attached grounded and ungrounded nonlinear vibration absorber (NVA) using the exact natural frequencies and mode shapes of the loaded beam. The nonlinearity in the beam is due to midplane stretching and that in the NVA is of cubic stiffness nonlinearity. We consider various boundary conditions and derive their closed-form characteristic equations and mode shapes. The method of multiple scales (MMS) is directly applied to the nonlinear partial differential equations of motion to obtain explicit expressions of the nonlinear frequency, modulation, and loci of the saddle-node bifurcation equations. Our analytical approach is validated using direct numerical simulation. Parametric studies demonstrate that the performance of the NVA does not only depend on its key design variables and location, but also on the boundary conditions, midplane stretching of the beam, and type of configuration (i.e., grounded NVA versus ungrounded NVA). Our analysis also indicates that the use of common approach such as employing approximate modes in estimating the nonlinear response of a loaded beam produces significant error (i.e., up to 1200% in some case). These observations suggest that the exact modes shape and natural frequencies are required for a precise investigation of the nonlinear dynamic of loaded beams. These findings could contribute to the design improvement of NVAs, microelectromechanical systems (MEMS), energy harvesters, and metastructures.

scholarly journals On the Flexural-Torsional Vibration and Stability of Beams Subjected to Axial Load and End Moment

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
M. Tahmaseb Towliat Kashani ◽  
Supun Jayasinghe ◽  
Seyed M. Hashemi
Keyword(s):  
Boundary Conditions ◽  
Axial Force ◽  
Axial Load ◽  
Natural Frequencies ◽  
Compressive Force ◽  
Elastic Beam ◽  
Axial Compressive Force ◽  
Various Boundary Conditions ◽  
Linear Eigenvalue Problem ◽  
The Moment

The free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MATLAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.

Vibration and Frequency Analysis of Non-Uniform Timoshenko Beams Subjected to Axial Forces

2003 ◽  
Author(s):  
A. R. Ohadi ◽  
H. Mehdigholi ◽  
E. Esmailzadeh
Keyword(s):  
Stability Analysis ◽  
Boundary Conditions ◽  
Axial Force ◽  
Frequency Equation ◽  
Axial Loads ◽  
Various Boundary Conditions ◽  
Axial Forces ◽  
First Case ◽  
The Stability ◽  
Elastically Supported

Dynamic and stability analysis of non-uniform Timoshenko beam under axial loads is carried out. In the first case of study, the axial force is assumed to be perpendicular to the shear force, while for the second case the axial force is tangent to the axis of the beam column. For each case, a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending was obtained. The parameters of the frequency equation were determined for various boundary conditions. Several illustrative examples of uniform and non-uniform beams with different boundary conditions such as clamped supported, elastically supported, and free end mass have been presented. The stability analysis, for the variation of the natural frequencies of the uniform and non-uniform beams with the axial force, has also been investigated.

On Free Vibration of Functionally Graded Mindlin Plate and Effect of In-Plane Displacements

2013 ◽  
Vol 29 (2) ◽  
pp. 373-384 ◽  
Author(s):  
A. Hasani Baferani ◽  
A.R. Saidi ◽  
H. Ehteshami
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Various Boundary Conditions ◽  
Aspect Ratios

AbstractIn this paper, free vibration analysis of functionally graded rectangular plate is investigated based on the first order shear deformation theory and the effect of in-plane displacements on the natural frequencies of such plate is studied. The governing equations of motion are obtained, which are five coupled partial differential equations, without any simplification. Some mathematical manipulation leads us to decouple the equations. The decoupled equations are solved by the Levy's method for various boundary conditions. As the results show, in some boundary conditions the in-plane displacements cause a drastic change of frequencies. In other words, neglecting the in-plane displacement, which is assumed in some papers, is not proper for these boundary conditions. However, in the other boundary conditions, the natural frequencies are not significantly affected by the in-plane displacements. The results for various boundary conditions are discussed in detail and some interpretations for these differences are provided. Besides to the comparisons, the accurate natural frequencies of the plate for six different boundary conditions with several aspect ratios, thickness-length ratios and power law indices are presented. The natural frequencies of Mindlin functionally graded rectangular plates with considering the in-plane displacements are reported for the first time and can be used as benchmark.

scholarly journals On the Flexural-Torsional Vibration and Stability of Beams Subjected to Axial Load and End Moment

2021 ◽  
Author(s):  
M. Tahmaseb Towliat Kashani ◽  
Supun Jayasinghe Jayashinghe ◽  
Seyed M. Hashemi
Keyword(s):  
Boundary Conditions ◽  
Axial Force ◽  
Axial Load ◽  
Natural Frequencies ◽  
Compressive Force ◽  
Elastic Beam ◽  
Axial Compressive Force ◽  
Various Boundary Conditions ◽  
Linear Eigenvalue Problem ◽  
The Moment

The free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MAT LAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.

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