NUMERICAL ANALYSIS ON NONLINEAR FREE VIBRATION OF CARBON NANOTUBE REINFORCED COMPOSITE BEAMS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350056 ◽  
Author(s):  
F. LIN ◽  
Y. XIANG
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Volume Fraction ◽  
Ritz Method ◽  
Axial Direction ◽  
Functionally Graded ◽  
Nonlinear Free Vibration ◽  
Third Order ◽  
First Order ◽  
Beam Theories

This paper investigates the free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). The distribution of the SWCNTs may vary through the thickness of a beam and are aligned along the beam axial direction. The virtual strain and kinetic energies of the carbon nanotube (CNT) composite beam are obtained using the classical variational method of Hamilton's principle, and the geometric nonlinearity of von Kármán sense is also included. The eigenvalue equation for free vibration of the beam is derived by the p-Ritz method. Vibration frequency parameters for the uniformly distributed (UD) and functionally graded (FG) CNT beams based on the first-order and third-order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam length to depth ratio and end support conditions on the nonlinear free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first-order and the third-order beam theories are also performed and the differences in vibration frequencies and the nonlinear to linear frequency ratios between these two theories are highlighted.

Bending and free vibration analyses of antisymmetrically laminated carbon nanotube-reinforced functionally graded plates

2016 ◽  
Vol 51 (22) ◽  
pp. 3111-3125 ◽  
Author(s):  
Bin Huang ◽  
Yan Guo ◽  
Ji Wang ◽  
Jianke Du ◽  
Zhenghua Qian ◽  
...  
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Volume Fraction ◽  
Shear Deformation Theory ◽  
Composite Plates ◽  
Functionally Graded ◽  
Width Ratio ◽  
Variable Theory ◽  
First Order ◽  
First Order Theory

In this paper, a simple four-variable first-order shear deformation theory is further applied to solve the bending and free vibration problems of antisymmetrically laminated functionally graded carbon nanotube (FG-CNT)-reinforced composite plates. The adopted four-variable theory contains only four unknowns in its displacement field which is less than the Reddy’s first-order theory. The equations of motion are derived from the Hamilton’s principle with the help of specific boundary conditions. Laminated FG-CNT-reinforced plates with different distribution types of carbon nanotube through the thickness are considered. The material properties of individual layer are estimated by using the extended rule of mixture. Analytical solutions of various simply supported antisymmetric cross-ply and angle-ply laminates are given for case study. The effects of carbon nanotube volume fraction, length to width ratio and thickness to width ratio on the non-dimensional fundamental frequency and the central deflection are investigated for antisymmetrically laminated FG-CNT-reinforced plates.

scholarly journals Free vibration of functionally graded carbon-nanotube-reinforced composite plates with cutout

2016 ◽  
Vol 7 ◽  
pp. 511-523 ◽  
Author(s):  
Mostafa Mirzaei ◽  
Yaser Kiani
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Free Vibration ◽  
Solution Method ◽  
Volume Fraction ◽  
Plate Theory ◽  
Ritz Method ◽  
Composite Plates ◽  
Functionally Graded ◽  
Plane Boundary

During the past five years, it has been shown that carbon nanotubes act as an exceptional reinforcement for composites. For this reason, a large number of investigations have been devoted to analysis of fundamental, structural behavior of solid structures made of carbon-nanotube-reinforced composites (CNTRC). The present research, as an extension of the available works on the vibration analysis of CNTRC structures, examines the free vibration characteristics of plates containing a cutout that are reinforced with uniform or nonuniform distribution of carbon nanotubes. The first-order shear deformation plate theory is used to estimate the kinematics of the plate. The solution method is based on the Ritz method with Chebyshev basis polynomials. Such a solution method is suitable for arbitrary in-plane and out-of-plane boundary conditions of the plate. It is shown that through a functionally graded distribution of carbon nanotubes across the thickness of the plate, the fundamental frequency of a rectangular plate with or without a cutout may be enhanced. Furthermore, the frequencies are highly dependent on the volume fraction of carbon nanotubes and may be increased upon using more carbon nanotubes as reinforcement.

Free vibration of functionally graded carbon nanotube reinforced composite cylindrical panels with general elastic supports

2018 ◽  
Vol 5 (1) ◽  
pp. 95-115
Author(s):  
Fei Xie ◽  
Jinyuan Tang ◽  
Ailun Wang ◽  
Cijun Shuai ◽  
Qingshan Wang
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Volume Fraction ◽  
Ritz Method ◽  
Functionally Graded ◽  
Elastic Supports ◽  
Various Boundary Conditions ◽  
Reinforced Composite ◽  
Cylindrical Panels ◽  
Classical Boundary

Abstract In this paper, a unified solution for vibration analysis of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical panels with general elastic supports is carried out via using the Ritz method. The excellent accuracy and reliability of the present method are compared with the results of the classical boundary cases found in the literature. New results are given for vibration characteristics of FG-CNTRC cylindrical panels with various boundary conditions. The effects of the elastic restraint parameters, thickness, subtended angle and volume fraction of carbon nanotubes on the free vibration characteristic of the cylindrical panels are also reported.

scholarly journals Independent coordinate coupling method for vibration analysis of a functionally graded carbon nanotube–reinforced plate with central hole

2019 ◽  
Vol 11 (8) ◽  
pp. 168781401987292 ◽  
Author(s):  
Yan Guo ◽  
Yanan Jiang ◽  
Bin Huang
Keyword(s):  
Carbon Nanotube ◽  
Volume Fraction ◽  
Ritz Method ◽  
Matching Condition ◽  
Functionally Graded ◽  
Central Hole ◽  
Lagrange’S Equation ◽  
Reinforced Plate ◽  
Distribution Type ◽  
Admissible Functions

In this article, the free vibration of a functionally graded carbon nanotube–reinforced plate with central hole is investigated by means of the independent coordinates-based Rayleigh–Ritz method. For the proposed method, the kinematic and potential energies are substituted into Lagrange’s equation in order to obtain the equation of motion. However, the total energies are computed by the difference of energies between the hole domain and the plate domain. By applying the displacement matching condition at the hole domain, two coordinate systems are coupled. For the Rayleigh–Ritz method, the mode shape functions of uniform beams are assumed as admissible functions. By this method, convergent results can be obtained with certain number of terms of admissible functions. The present results clearly reflect the effects of the carbon nanotube distribution type, carbon nanotube volume fraction, hole size, and boundary condition on the nondimensional natural frequencies. The provided results show that the present method is efficient in studying the vibration problems of functionally graded carbon nanotube–reinforced plate with central hole.

scholarly journals A Semianalytical Approach for Free Vibration Characteristics of Functionally Graded Spherical Shell Based on First-Order Shear Deformation Theory

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Fuzhen Pang ◽  
Cong Gao ◽  
Jie Cui ◽  
Yi Ren ◽  
Haichao Li ◽  
...  
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Numerical Verification ◽  
First Order

This paper describes a unified solution to investigate free vibration solutions of functionally graded (FG) spherical shell with general boundary restraints. The analytical model is established based on the first-order shear deformation theory, and the material varies uniformly along the thickness of FG spherical shell which is divided into several sections along the meridian direction. The displacement functions along circumferential and axial direction are, respectively, composed by Fourier series and Jacobi polynomial regardless of boundary restraints. The boundary restraints of FG spherical shell can be easily simulated according to penalty method of spring stiffness technique, and the vibration solutions are obtained by Rayleigh–Ritz method. To verify the reliability and accuracy of the present solutions, the convergence and numerical verification have been conducted about different boundary parameters, Jacobi parameter, etc. The results obtained by the present method closely agree with those obtained from the published literatures, experiments, and finite element method (FEM). The impacts of geometric dimensions and boundary conditions on the vibration characteristics of FG spherical shell structure are also presented.

Free vibration analysis of functionally graded double-tapered beam rotating in thermal environment considering geometric nonlinearity, shear deformability, and Coriolis effect

2017 ◽  
Vol 232 (12) ◽  
pp. 2244-2262 ◽  
Author(s):  
Suman Pal ◽  
Debabrata Das
Keyword(s):  
Mathematical Model ◽  
Free Vibration ◽  
Material Properties ◽  
Volume Fraction ◽  
Thermal Environment ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Coriolis Effect ◽  
Governing Equations

The present work investigates the free vibration behavior of double-tapered functionally graded beams rotating in thermal environment, using an improved mathematical model. The functional gradation for ceramic–metal compositions, following power-law, is considered to be symmetric with respect to the mid-plane, leading to metal-rich core and ceramic-rich outer surfaces of the beam. The temperature dependence of the material properties are considered using Touloukian model. The nonlinearity in strain–displacement relationships for both the axial and transverse shear strains are considered. Firstly, the governing equations for deformed beam configuration under time-independent centrifugal loading are obtained using minimum total potential energy principle, and the solution is obtained following Ritz method. Then the free vibration problem of the centrifugally deformed beam is formulated employing Lagrange’s principle and considering tangent stiffness of the deformed beam configuration. Coriolis effect is considered in the mathematical model, and the governing equations are transformed to the state-space for obtaining an eigenvalue problem. The results for the first two modes of both chord-wise and flap-wise vibrations are presented in nondimensional plane to show the effects of taperness parameter, root-offset parameter, volume fraction exponent, operating temperature, and functionally graded material composition. The results in comparative form are presented for both temperature-dependent and temperature-independent material properties.

Free Vibration and Buckling Analysis of Sandwich Beams with Functionally Graded Carbon Nanotube-Reinforced Composite Face Sheets

2015 ◽  
Vol 15 (07) ◽  
pp. 1540011 ◽  
Author(s):  
Helong Wu ◽  
Sritawat Kitipornchai ◽  
Jie Yang
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Volume Fraction ◽  
Slenderness Ratio ◽  
Micromechanical Model ◽  
Sandwich Beam ◽  
Sheet Thickness ◽  
Functionally Graded ◽  
Sandwich Beams ◽  
Reinforced Composite

This paper investigates the free vibration and elastic buckling of sandwich beams with a stiff core and functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets within the framework of Timoshenko beam theory. The material properties of FG-CNTRCs are assumed to vary in the thickness direction, and are estimated through a micromechanical model. The governing equations and boundary conditions are derived by using Hamilton's principle and discretized by employing the differential quadrature (DQ) method to obtain the natural frequency and critical buckling load of the sandwich beam. A detailed parametric study is conducted to study the effects of carbon nanotube volume fraction, core-to-face sheet thickness ratio, slenderness ratio, and end supports on the free vibration characteristics and buckling behavior of sandwich beams with FG-CNTRC face sheets. The vibration behavior of the sandwich beam under an initial axial force is also discussed. Numerical results for sandwich beams with uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) face sheets are also provided for comparison.

scholarly journals Nonlinear free vibration analysis of elastically supported carbon nanotube-reinforced composite beam with the thermal environment in non-deterministic framework

2017 ◽  
Vol 4 (1) ◽  
pp. 85-103 ◽  
Author(s):  
Virendra Kumar Chaudhari ◽  
Niranjan L. Shegokar ◽  
Achchhe Lal
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Thermal Loading ◽  
Volume Fraction ◽  
Perturbation Technique ◽  
Order Perturbation ◽  
Nonlinear Free Vibration ◽  
Reinforced Composite ◽  
Elastically Supported ◽  
Foundation Parameters

AbstractThis paper deals with the investigation of nonlinear free vibration behavior of elastically supported carbon nanotube reinforced composite (CNTRC) beam subjected to thermal loading with random system properties. Material properties of each constituent’s material, volume fraction exponent and foundation parameters are considered as uncorrelated Gaussian random input variables. The beam is supported by a Pasternak foundation with Winkler cubic nonlinearity. The higher order shear deformation theory (HSDT) with von-Karman non-linearity is used to formulate the governing equation using Hamilton principle. Convergence and validation study is carried out through the comparison with the available results in the literature for authenticity and accuracy of the present approach used in the analysis. First order perturbation technique (FOPT),Second order perturbation technique (SOPT) and Monte Carlo simulation (MCS) methods are employed to investigate the effect of geometric configuration, volume fraction exponent, foundation parameters, distribution of reinforcement and thermal loading on nonlinear vibration characteristics CNTRC beam.The present work signifies the accurate analysis of vibrational behaviour influences by different random variables. Results are presented in terms of mean, variance (COV) and probability density function (PDF) for various aforementioned parameters.

Nonlinear free vibration analysis of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes

2019 ◽  
Vol 25 (14) ◽  
pp. 2063-2078 ◽  
Author(s):  
Mahsa Heidari ◽  
Hadi Arvin
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Free Vibration ◽  
Direct Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Timoshenko Beams ◽  
Nonlinear Free Vibration ◽  
Reinforced Composite ◽  
Linear And Nonlinear

In this paper, the linear and nonlinear free vibrations of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes are presented. The formulation is based on the assumptions of Timoshenko beam theory in addition to consideration of the nonlinear von Karman strain–displacement relationship. The effective material properties of carbon nanotube reinforced composites are determined employing the Mori–Tanaka micromechanics model and the extended mixture rule. For the carbon nanotube reinforced composite beams, uniform distribution and four types of functionally graded distribution patterns of single-walled carbon nanotube reinforcements are considered. A differential transform method is applied on the nondimensionalized equations of motion to release the flapping modeshapes and the associated natural frequencies. The direct method of multiple scales is implemented to derive the effective nonlinearity and the corresponding nonlinear natural frequency. The accuracy of the present outcomes is validated by the comparison with the results given in the literature. The numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fractions, distribution types of the carbon nanotubes, and rotation speed on linear and nonlinear free vibration characteristics of carbon nanotube reinforced composite beam. The results demonstrate the important role of carbon nanotube distribution profile on linear and nonlinear free vibration features.

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