A refined beam theory for bending and vibration of functionally graded tube-beams

Abstract The present work aims to analyze the natural and whirl frequencies of a slant-cracked functionally graded rotor-bearing system using finite element analysis for the flexural vibrations. The functionally graded shaft is modelled using two nodded beam elements formulated using the Timoshenko beam theory. The flexibility matrix of a slant-cracked functionally graded shaft element has been derived using fracture mechanics concepts, which is further used to develop the stiffness matrix of a cracked element. Material properties are temperature and position-dependent and graded in a radial direction following power-law gradation. A Python code has been developed to carry out the complete finite element analysis to determine the Eigenvalues and Eigenvectors of a slant-cracked rotor subjected to different thermal gradients. The analysis investigates and further reveals significant effect of the power-law index and thermal gradients on the local flexibility coefficients of slant-cracked element and whirl natural frequencies of the cracked functionally graded rotor system.

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A Refined Theory for Bending Vibratory Analysis of Thick Functionally Graded Beams

Mathematics ◽

10.3390/math9121422 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1422

Author(s):

Youssef Boutahar ◽

Nadhir Lebaal ◽

David Bassir

Keyword(s):

Beam Theory ◽

Functionally Graded ◽

Effective Characteristics ◽

Hyperbolic Function ◽

Transverse Displacement ◽

Refined Theory ◽

Distribution Law ◽

Fg Beam ◽

The Impact ◽

Beam Theories

A refined beam theory that takes the thickness-stretching into account is presented in this study for the bending vibratory behavior analysis of thick functionally graded (FG) beams. In this theory, the number of unknowns is reduced to four instead of five in the other approaches. Transverse displacement is expressed through a hyperbolic function and subdivided into bending, shear, and thickness-stretching components. The number of unknowns is reduced, which involves a decrease in the number of the governing equation. The boundary conditions at the top and bottom FG beam faces are satisfied without any shear correction factor. According to a distribution law, effective characteristics of FG beam material change continuously in the thickness direction depending on the constituent’s volume proportion. Equations of motion are obtained from Hamilton’s principle and are solved by assuming the Navier’s solution type, for the case of a supported FG beam that is transversely loaded. The numerical results obtained are exposed and analyzed in detail to verify the validity of the current theory and prove the influence of the material composition, geometry, and shear deformation on the vibratory responses of FG beams, showing the impact of normal deformation on these responses which is neglected in most of the beam theories. The obtained results are compared with those predicted by other beam theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of FG beams.

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Wave propagation analysis of size-dependent rotating inhomogeneous nanobeams based on nonlocal elasticity theory

Journal of Vibration and Control ◽

10.1177/1077546317711537 ◽

2017 ◽

Vol 24 (17) ◽

pp. 3809-3818 ◽

Cited By ~ 15

Author(s):

Farzad Ebrahimi ◽

Mohammad Reza Barati ◽

Parisa Haghi

Keyword(s):

Elasticity Theory ◽

Nonlocal Elasticity ◽

Nonlocal Parameter ◽

Beam Theory ◽

Nonlocal Elasticity Theory ◽

Wave Dispersion ◽

Functionally Graded ◽

Wave Number ◽

Graded Material ◽

Fg Nanobeam

The present research deals with the wave dispersion behavior of a rotating functionally graded material (FGMs) nanobeam applying nonlocal elasticity theory of Eringen. Material properties of rotating FG nanobeam are spatially graded according to a power-law model. The governing equations as functions of axial force due to centrifugal stiffening and displacements are obtained by employing Hamilton’s principle based on the Euler–Bernoulli beam theory. By using an analytical model, the dispersion relations of the FG nanobeam are derived by solving an eigenvalue problem. Numerical results clearly show that various parameters, such as angular velocity, gradient index, wave number and nonlocal parameter, are significantly effective to characteristics of wave propagations of rotating FG nanobeams. The results can be useful for next generation study and design of nanomachines, such as nanoturbines, nanoscale molecular bearings and nanogears, etc.

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Multi-moving Loads Induced Vibration of FG Sandwich Beams Resting on Pasternak Elastic Foundation

Walailak Journal of Science and Technology (WJST) ◽

10.48048/wjst.2021.9260 ◽

2021 ◽

Vol 18 (9) ◽

Author(s):

Wachirawit SONGSUWAN ◽

Monsak PIMSARN ◽

Nuttawit WATTANASAKULPONG

Keyword(s):

Dynamic Response ◽

Elastic Foundation ◽

Excitation Frequency ◽

Beam Theory ◽

Equation Of Motion ◽

Functionally Graded ◽

Sandwich Beams ◽

Moving Loads ◽

Layer Thickness Ratio ◽

Pasternak Elastic Foundation

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

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Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2015.m899 ◽

2016 ◽

Vol 9 (1) ◽

pp. 144-156 ◽

Cited By ~ 11

Author(s):

Majid Ghadiri ◽

Mohsen Safi

Keyword(s):

Differential Equation ◽

Perturbation Method ◽

Homotopy Perturbation Method ◽

Nonlinear Partial Differential Equation ◽

Beam Theory ◽

Nonlocal Elasticity Theory ◽

Functionally Graded ◽

Simply Supported ◽

Homotopy Perturbation ◽

Functionally Graded Nanobeam

AbstractIn this paper, He's homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen's nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin's method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

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Analytical and numerical investigation of the free vibration of functionally graded materials sandwich beams

Archives of Materials Science and Engineering ◽

10.5604/01.3001.0015.4314 ◽

2021 ◽

Vol 2 (110) ◽

pp. 72-85

Author(s):

S.H. Bakhy ◽

M. Al-Waily ◽

M.A. Al-Shammari

Keyword(s):

Analytical Solution ◽

Free Vibration ◽

Functionally Graded Materials ◽

Sandwich Beam ◽

Beam Theory ◽

Functionally Graded ◽

Gradient Index ◽

Sandwich Beams ◽

Graded Materials ◽

The Impact

Purpose: In this study, the free vibration analysis of functionally graded materials (FGMs) sandwich beams having different core metals and thicknesses is considered. The variation of material through the thickness of functionally graded beams follows the power-law distribution. The displacement field is based on the classical beam theory. The wide applications of functionally graded materials (FGMs) sandwich structures in automotive, marine construction, transportation, and aerospace industries have attracted much attention, because of its excellent bending rigidity, low specific weight, and distinguished vibration characteristics. Design/methodology/approach: A mathematical formulation for a sandwich beam comprised of FG core with two layers of ceramic and metal, while the face sheets are made of homogenous material has been derived based on the Euler–Bernoulli beam theory. Findings: The main objective of this work is to obtain the natural frequencies of the FG sandwich beam considering different parameters. Research limitations/implications: The important parameters are the gradient index, slenderness ratio, core metal type, and end support conditions. The finite element analysis (FEA), combined with commercial Ansys software 2021 R1, is used to verify the accuracy of the obtained analytical solution results. Practical implications: It was found that the natural frequency parameters, the mode shapes, and the dynamic response are considerably affected by the index of volume fraction, the ratio as well as face FGM core constituents. Finally, the beam thickness was dividing into frequent numbers of layers to examine the impact of many layers' effect on the obtained results. Originality/value: It is concluded, that the increase in the number of layers prompts an increment within the frequency parameter results' accuracy for the selected models. Numerical results are compared to those obtained from the analytical solution. It is found that the dimensionless fundamental frequency decreases as the material gradient index increases, and there is a good agreement between two solutions with a maximum error percentage of no more than 5%.

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