scholarly journals Transverse Vibration of Axially Moving Functionally Graded Materials Based on Timoshenko Beam Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Suihan Sui ◽  
Ling Chen ◽  
Cheng Li ◽  
Xinpei Liu
Keyword(s):  
Power Law ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Functionally Graded ◽  
Beam Model ◽  
Vibration Modes ◽  
Graded Materials ◽  
Power Law Exponent ◽  
Axially Moving

The transverse free vibration of an axially moving beam made of functionally graded materials (FGM) is investigated using a Timoshenko beam theory. Natural frequencies, vibration modes, and critical speeds of such axially moving systems are determined and discussed in detail. The material properties are assumed to vary continuously through the thickness of the beam according to a power law distribution. Hamilton’s principle is employed to derive the governing equation and a complex mode approach is utilized to obtain the transverse dynamical behaviors including the vibration modes and natural frequencies. Effects of the axially moving speed and the power-law exponent on the dynamic responses are examined. Some numerical examples are presented to reveal the differences of natural frequencies for Timoshenko beam model and Euler beam model. Moreover, the critical speed is determined numerically to indicate its variation with respect to the power-law exponent, axial initial stress, and length to thickness ratio.

Vibration of Double-Walled Carbon Nanotube-Based Mass Sensor via Nonlocal Timoshenko Beam Theory

2011 ◽  
Vol 2 (3) ◽  
Author(s):  
Zhi-Bin Shen ◽  
Bin Deng ◽  
Xian-Fang Li ◽  
Guo-Jin Tang
Keyword(s):  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Tube Length ◽  
Beam Model ◽  
Mass Sensor ◽  
Nonlocal Timoshenko Beam Theory ◽  
Double Walled Carbon Nanotubes ◽  
Tip Mass

The potential of double-walled carbon nanotubes (DWCNTs) as a micromass sensor is explored. A nonlocal Timoshenko beam carrying a micromass at the free end of the inner tube is used to analyze the vibration of DWCNT-based mass sensor. The length of the outer tube is not equal to that of the inner tube, and the interaction between two tubes is governed by van der Waals force (vdW). Using the transfer function method, the natural frequencies of a nonlocal cantilever with a tip mass are computed. The effects of the attached mass and the outer-to-inner tube length ratio on the natural frequencies are discussed. When the nonlocal parameter is neglected, the frequencies reduce to the classical results, in agreement with those using the finite element method. The obtained results show that increasing the attached micromass decreases the natural frequency but increases frequency shift. The mass sensitivity improves for short DWCNTs used in mass sensor. The nonlocal Timoshenko beam model is more adequate than the nonlocal Euler-Bernoulli beam model for short DWCNT sensors. Obtained results are helpful to the design of DWCNT-based resonator as micromass sensor.

Natural Frequency Improvement of a Suspended FGM Bridge

2011 ◽  
Author(s):  
M. Hadipour ◽  
M. T. Ahmadian ◽  
S. G. Lashkari ◽  
A. Barari
Keyword(s):  
Functionally Graded Materials ◽  
Power Law ◽  
Natural Frequencies ◽  
Classical Equation ◽  
Suspension Bridge ◽  
Vertical Vibration ◽  
Functionally Graded ◽  
Graded Materials ◽  
Bridge Structures ◽  
Environmental Vibration

In this paper application of Functionally Graded Materials (FGMs) in suspension bridge structures for the purpose of vertical vibration improvement is investigated. Functionally graded materials are inhomogeneous composites, which are usually made from a mixture of metal and ceramic. Initially the classical equation of motion is modified based on the FGM model and natural frequencies of structure are extracted by Galerkin method. The material properties of structure vary continuously in the longitude direction according to power law form. In this regard, combination of materials is structured in such a way that the desired frequencies of the bridge are achieved by designing the proper power law of the FGM. Natural frequencies are evaluated to avoid any resonant due to environmental vibration. A bridge structure based on FGM made of steel and aluminum oxide is designed and improved for proper frequency. For the simple case of homogenous structures results are compared with those reported in the literature and very good agreements are obtained.

scholarly journals A study on dynamic response of functionally graded sandwich beams under different dynamic loadings

2018 ◽  
Vol 192 ◽  
pp. 02011
Author(s):  
Wachirawit Songsuwan ◽  
Monsak Pimsarn ◽  
Nuttawit Wattanasakulpong
Keyword(s):  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Functionally Graded ◽  
Transverse Shear Deformation ◽  
Sandwich Beams ◽  
Governing Equations ◽  
Dynamic Loadings ◽  
Parametric Studies

In this research, free and forced vibration of functionally graded sandwich beams is considered using Timoshenko beam theory which takes into account the significant effects of transverse shear deformation and rotary inertia. The governing equations of motion are formulated from Lagrange's equations and they are solved by using The Ritz and Newmark methods. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, etc. on natural frequencies and dynamic deflections of the beams. According to the numerical results, all parametric studies considered in this research have significant impact on free and forced behaviour of the beams; for example, the frequency is low and the dynamic deflection is large for the beams which are hinged at both ends.

Radial Basis Function-Based Stochastic Natural Frequencies Analysis of Functionally Graded Plates

2019 ◽  
Vol 17 (09) ◽  
pp. 1950061 ◽  
Author(s):  
P. K. Karsh ◽  
R. R. Kumar ◽  
S. Dey
Keyword(s):  
Radial Basis Function ◽  
Power Law ◽  
Basis Function ◽  
Natural Frequencies ◽  
Functionally Graded ◽  
Direct Monte Carlo Simulation ◽  
Distribution Law ◽  
Stochastic Variation ◽  
Graded Materials ◽  
Radial Basis

This paper deals with portraying the stochastic natural frequencies of cantilever plates made up of functionally graded materials (FGMs) by employing the radial basis function (RBF)-based finite element (FE) approach. The material modeling of FGM plates is carried out by employing three different distribution laws, namely power law, sigmoid law, and exponential law. A generalized algorithm is developed for uncertainty quantification of natural frequencies of the FGM structures due to stochastic variation in the material properties and temperature. The deterministic FE code is validated with the previous literature, whereas convergence study is carried out in between stochastic results obtained from full scale direct Monte Carlo Simulation (MCS) and MCS results obtained from RBF surrogate model of different sample sizes. The percentage of error present in the RBF model is also determined. The influence of crucial parameters such as distribution law, degree of stochasticity, power law index and temperature are determined for natural frequencies analysis of FGMs plates. The results illustrate the input parameters considered in the present study have significant effects on the first three stochastic natural frequencies of cantilever FGM plates.

scholarly journals Formulation and experimental validation of space-fractional Timoshenko beam model with functionally graded materials effects

2021 ◽  
Author(s):  
Paulina Stempin ◽  
Wojciech Sumelka
Keyword(s):  
Functionally Graded Materials ◽  
Timoshenko Beam ◽  
Functionally Graded ◽  
Beam Model ◽  
Material Parameter Identification ◽  
Graded Materials ◽  
Size Dependent ◽  
Non Local ◽  
Thick Beam ◽  
Euler Bernoulli Beam

AbstractIn this study, the static bending behaviour of a size-dependent thick beam is considered including FGM (Functionally Graded Materials) effects. The presented theory is a further development and extension of the space-fractional (non-local) Euler–Bernoulli beam model (s-FEBB) to space-fractional Timoshenko beam (s-FTB) one by proper taking into account shear deformation. Furthermore, a detailed parametric study on the influence of length scale and order of fractional continua for different boundary conditions demonstrates, how the non-locality affects the static bending response of the s-FTB model. The differences in results between s-FTB and s-FEBB models are shown as well to indicate when shear deformations need to be considered. Finally, material parameter identification and validation based on the bending of SU-8 polymer microbeams confirm the effectiveness of the presented model.

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

2019 ◽  
Vol 11 (04) ◽  
pp. 1950040 ◽  
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.

Natural frequency analysis of a functionally graded rotor-bearing system with a slant crack subjected to thermal gradients

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Arnab Bose ◽  
Prabhakar Sathujoda ◽  
Giacomo Canale
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Power Law ◽  
Beam Theory ◽  
Functionally Graded ◽  
Thermal Gradients ◽  
Element Analysis ◽  
Rotor Bearing ◽  
Natural Frequency Analysis ◽  
Bearing System

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.

scholarly journals UNCOUPLED VIBRATIONS IN FUNCTIONALLY GRADED TIMOSHENKO BEAM

2016 ◽  
Vol 54 (6) ◽  
pp. 785 ◽  
Author(s):  
Nguyen Tien Khiem ◽  
Nguyen Ngoc Huyen
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Flexural Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Power Law Distribution ◽  
Material Parameters ◽  
Flexural Vibrations ◽  
Fgm Beam

Free vibration of FGM Timoshenko beam is investigated on the base of the power law distribution of FGM. Taking into account the actual position of neutral plane enables to obtain general condition for uncoupling of axial and flexural vibrations in FGM beam. This condition defines a class of functionally graded beams for which axial and flexural vibrations are completely uncoupled likely to the homogeneous beams. Natural frequencies and mode shapes of uncoupled flexural vibration of beams from the class are examined in dependence on material parameters and slendernes

Vibration Response of a Thin-Walled Cylindrical Pipe with Liquid

2012 ◽  
Vol 189 ◽  
pp. 345-349
Author(s):  
Yu Lan Wei ◽  
Bing Li ◽  
Li Gao ◽  
Ying Jun Dai
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Element Model ◽  
Beam Model ◽  
Vibration Modes ◽  
Timoshenko Beam Model ◽  
Cylindrical Pipe ◽  
Bending Vibration ◽  
Beam Bending ◽  
Thin Walled

Vibration characteristics of the thin-walled cylindrical pipe are affected by the liquid within the pipe. The natural frequencies and vibration modes of the pipe without liquid are analyzed by the theory of beam bending vibration and finite element model, which is based on the Timoshenko beam model. The first three natural frequencies and vibration modes of the pipe with or without liquid are acquired by experiments. As shown in the experiment results, the natural frequencies of the containing liquid pipe are lower than the natural frequencies of the pipe without liquid.

Vibration analysis of a thin functionally graded plate having an out of plane material inhomogeneity resting on Winkler–Pasternak foundation under different combinations of boundary conditions

2018 ◽  
Vol 233 (8) ◽  
pp. 2636-2662 ◽  
Author(s):  
Piyush Pratap Singh ◽  
Mohammad Sikandar Azam ◽  
Vinayak Ranjan
Keyword(s):  
Boundary Conditions ◽  
Power Law ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Pasternak Foundation ◽  
Functionally Graded Plate ◽  
Material Inhomogeneity ◽  
Power Law Exponent ◽  
Out Of Plane ◽  
Graded Material

In the present research article, classical plate theory has been adopted to analyze functionally graded material plate, having out of plane material inhomogeneity, resting on Winkler–Pasternak foundation under different combinations of boundary conditions. The material properties of the functionally graded material plate vary according to power law in the thickness direction. Rayleigh–Ritz method in conjugation with polynomial displacement functions has been used to develop a computationally efficient mathematical model to study free vibration characteristics of the plate. Convergence of frequency parameters (nondimensional natural frequencies) has been attained by increasing the number of polynomials of displacement function. The frequency parameters of the functionally graded material plate obtained by proposed method are compared with the open literature to validate the present model. Firstly, the present model is used to calculate first six natural frequencies of the functionally graded plate under all possible combinations of boundary conditions for the constant value of stiffness of Winkler and Pasternak foundation moduli. Further, the effects of density, aspect ratio, power law exponent, Young’s modulus on frequency parameters of the functionally graded plate resting on Winkler–Pasternak foundation under specific boundary conditions viz. CCCC (all edges clamped), SSSS (all edges simply supported), CFFF (cantilever), SCSF (simply supported-clamped-free) are studied extensively. Furthermore, effect of stiffness of elastic foundation moduli (kp and kw) on frequency parameters are analyzed. It has been observed that effects of aspect ratios, boundary conditions, Young’s modulus and density on frequency parameters are significant at lower value of the power law exponent. It has also been noted from present investigation that Pasternak foundation modulus has greater effect on frequency parameters as compared to the Winkler foundation modulus. Most of the results presented in this paper are novel and may be used for the validation purpose by researchers. Three dimensional mode shapes for the functionally graded plate resting on elastic foundation have also been presented in this article.

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