On the use of Chebyshev polynomials in the Rayleigh-Ritz method for vibration and buckling analyses of circular cylindrical three-dimensional graphene foam shells

The three-dimensional free vibration and time response of rotating functionally graded (FG) cantilevered beams are studied. Material properties of functionally graded beams are assumed to change gradually through both the width and the thickness in power-law form. The second-kind Lagrange’s equations are used in conjunction with the Ritz method to derive the comprehensive coupling dynamic equations for the axial, chordwise, and flapwise motions. The trial functions of deformations are taken as the products of the Chebyshev polynomials and the corresponding boundary functions. Nonlinear coupling deformations are considered to capture the dynamic stiffening effect due to the rotating motion. The influences of the material gradient index and rotational speed on modal characteristics are investigated by the state space method. The eigenvalue loci veering phenomena with modal conversions are exhibited. The time responses indicate that the deformations of rotating functionally graded beams are greatly affected by the material gradient index. It is shown that for large deformation problems, using Chebyshev polynomials is more efficient in computing precision and robustness than using other polynomials.

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Three-Dimensional Dynamics Analysis of Rotating Functionally Gradient Beams Based on Timoshenko Beam Theory

International Journal of Applied Mechanics ◽

10.1142/s1758825119500406 ◽

2019 ◽

Vol 11 (04) ◽

pp. 1950040 ◽

Cited By ~ 3

Author(s):

Ding Zhou ◽

Jianshi Fang ◽

Hongwei Wang ◽

Xiaopeng Zhang

Keyword(s):

Timoshenko Beam ◽

Chebyshev Polynomials ◽

Slenderness Ratio ◽

Ritz Method ◽

Three Dimensional ◽

Beam Theory ◽

Functionally Graded ◽

Timoshenko Beam Theory ◽

Strengthening Effect ◽

Graded Materials

Through the Timoshenko beam theory (TBT), the 3D dynamics of a rotary functional gradient (FG) cantilever beam are investigated. Material capabilities alter continuously throughout the thickness obeying the power law. It is assumed that the Poisson’s ratio does not change. Based on the von Kármán nonlinearity, the governing equation is determined through the Hamilton principle, which includes the Coriolis effects. The couplings among the axial, flapwise and chordwise deformations caused by the usage of the functionally graded materials (FGMs) are revealed. Chebyshev polynomials are utilized to construct trial functions of deformations in the Rayleigh–Ritz method. The centrifugal strengthening effect caused by the rotational motion is described through the nonlinear axial shortening deformations derived from transverse deformations. The influences of the dimensionless angular velocity, FG index and slenderness ratio on vibration characteristics are studied. It is proved that the FG index significantly affects the dynamic response of deformation. For high-frequency external excitation cases, selection of Chebyshev polynomials as trial functions is more stable and effective than other polynomials.

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A three-dimensional nerve guide conduit based on graphene foam/polycaprolactone

Materials Science and Engineering C ◽

10.1016/j.msec.2021.112110 ◽

2021 ◽

pp. 112110

Author(s):

Neda Bahremandi Tolou ◽

Hamidreza Salimijazi ◽

Mahshid Kharaziha ◽

Giuliana Faggio ◽

Rosa Chierchia ◽

...

Keyword(s):

Three Dimensional ◽

Graphene Foam

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α-MnO2 nanotube@δ-MnO2 nanoflake hierarchical structure on three-dimensional graphene foam as a lightweight and free-standing supercapacitor electrode

Journal of Alloys and Compounds ◽

10.1016/j.jallcom.2021.158934 ◽

2021 ◽

Vol 865 ◽

pp. 158934

Author(s):

Mi He ◽

Linli Cao ◽

Weilong Li ◽

Xinwei Chang ◽

Zhaoyu Ren

Keyword(s):

Hierarchical Structure ◽

Three Dimensional ◽

Supercapacitor Electrode ◽

Free Standing ◽

Graphene Foam

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Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

Journal of Applied Mechanics ◽

10.1115/1.1989355 ◽

2005 ◽

Vol 72 (5) ◽

pp. 797-800 ◽

Cited By ~ 14

Author(s):

Jae-Hoon Kang ◽

Arthur W. Leissa

Keyword(s):

Ritz Method ◽

Three Dimensional ◽

Axial Direction ◽

Free Vibrations ◽

Mode Shapes ◽

Vibration Frequencies ◽

Shells Of Revolution ◽

Dimensional Theory ◽

Conical Shells ◽

Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

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Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

Journal of Applied Mechanics ◽

10.1115/1.4007475 ◽

2013 ◽

Vol 80 (2) ◽

Cited By ~ 1

Author(s):

D. Zhou ◽

S. H. Lo

Keyword(s):

Cross Section ◽

Chebyshev Polynomial ◽

Numerical Solutions ◽

Twist Angle ◽

Ritz Method ◽

Three Dimensional ◽

Cylindrical Domain ◽

Linear Elasticity Theory ◽

Boundary Functions ◽

Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

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Electrophoresis-Microwave Synthesis of S, N-Doped Graphene Foam for High-Performance Supercapacitors

Journal of Materials Chemistry A ◽

10.1039/d1ta02877d ◽

2021 ◽

Author(s):

Wenqiang Wang ◽

Wenyi Zhang ◽

Gengchao Wang ◽

Chunzhong Li

Keyword(s):

Electrode Material ◽

Microwave Synthesis ◽

High Performance ◽

Three Dimensional ◽

Graphene Foam ◽

Composition And Structure ◽

Doped Graphene ◽

The Ideal

The three-dimensional self-standing architecture of heteroatom-doped graphene is the ideal electrode material for supercapacitors. However, the facile control of its composition and structure is still a challenge. Herein we have...

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