scholarly journals Linear vibrations of triple-walled carbon nanotubes

2017 ◽  
Vol 23 (11) ◽  
pp. 1456-1481 ◽  
Author(s):  
Matteo Strozzi ◽  
Francesco Pellicano
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Displacement Fields ◽  
Linear Vibrations ◽  
Longitudinal Variable ◽  
Walled Carbon Nanotubes ◽  
Linear Field

In this paper, the linear vibrations of triple-walled carbon nanotubes (TWNTs) are investigated. A multiple elastic thin shell model is applied. The TWNT dynamics is studied in the framework of the Sanders–Koiter shell theory. The van der Waals interaction between any two layers of the TWNT is modelled by a radius-dependent function. The shell deformation is described in terms of longitudinal, tangential and radial displacements. Simply supported, clamped and free boundary conditions are applied. The three displacement fields are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the tangential variable. The Rayleigh–Ritz method is applied to obtain approximate natural frequencies and mode shapes. The present model is validated in the linear field by means of comparisons with data from the literature. This study is focused on determining the effect of geometry and boundary conditions on the natural frequencies of TWNTs.

Hydroelastic Vibration of Rectangular Plates

1996 ◽  
Vol 63 (1) ◽  
pp. 110-115 ◽  
Author(s):  
Moon K. Kwak
Keyword(s):  
Boundary Conditions ◽  
Approximate Formula ◽  
Natural Frequencies ◽  
Mass Matrix ◽  
Ritz Method ◽  
Plate Boundary ◽  
Rigid Wall ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Virtual Mass

This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

scholarly journals The Effect of Volume Fraction of Single-Walled Carbon Nanotubes on Natural Frequencies of Polymer Composite Cone-Shaped Shell Made from Poly(Methyl Methacrylate)

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
A. H. Meysami ◽  
H. Razavi ◽  
I. Fakhari Golpayegani ◽  
V. Bagheri
Keyword(s):  
Carbon Nanotubes ◽  
Methyl Methacrylate ◽  
Polymer Composite ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Ritz Method ◽  
Single Walled Carbon Nanotubes ◽  
Poly Methyl Methacrylate ◽  
Single Walled Carbon ◽  
Walled Carbon Nanotubes

In this paper, the effect of volume fraction of single-walled carbon nanotubes on natural frequencies of polymer composite cone-shaped shells made from Poly(Methyl Methacrylate) (PMMA) is studied. In order to determine the characterization of materials reinforced with nanoparticles, the molecular dynamics and mixture rule has been used. The motion equations of composite shell based on the classical thin shells theory using Hamilton’s principle are obtained. Then, using the Ritz method, approximate analytical solution of the natural frequency is presented. Results indicate that the nanotubes have a noticeable effect on the natural frequencies.

scholarly journals Vibration analysis of carbon nanotubes-based zeptogram masses sensors and taking into account their rotatory inertia

2018 ◽  
Vol 149 ◽  
pp. 02087 ◽  
Author(s):  
A. Azrar ◽  
L. Azrar ◽  
A. A. Aljinaidi
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Methodological Approach ◽  
Nonlocal Parameter ◽  
Research Work ◽  
Mode Shapes ◽  
Attached Mass ◽  
Rotatory Inertia ◽  
Mass Sensors ◽  
Walled Carbon Nanotubes

In this research work, the transverse vibration behaviour of single-walled carbon nanotubes (SCNT) based mass sensors is studied using the Timoshenko beam and nonlocal elasticity theories. The nonlocal constitutive equations are used in the formulations and the CNT with different lengths, attached mass (viruses and bacteria) and the general boundary conditions are considered. The dimensionless frequencies and associated modes are obtained for one and two attached masses and different boundary conditions. The effects of transverse shear deformation and rotatory inertia, nonlocal parameter, length of the carbon nanotubes, and attached mass and its location are investigated in detail for each considered problem. The relationship between the frequencies and mode shapes of the sensor and the attached zeptogramme masses are obtained. The sensing devices for biological objects including viruses and bacteria can be elaborated based on the developed sensitivity and frequency shift methodological approach.

NONLOCAL ANALYTICAL FLUGGE SHELL MODEL FOR AXIAL BUCKLING OF DOUBLE-WALLED CARBON NANOTUBES WITH DIFFERENT END CONDITIONS

NANO ◽  
2012 ◽  
Vol 07 (03) ◽  
pp. 1250018 ◽  
Author(s):  
HESSAM ROUHI ◽  
REZA ANSARI
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Shell Model ◽  
Ritz Method ◽  
Nonlocal Parameter ◽  
Geometrical Parameters ◽  
Field Equations ◽  
Axial Buckling ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

In this paper, a nonlocal Flugge shell model is utilized to investigate the axial buckling behavior of double-walled carbon nanotubes (DWCNTs) under various boundary conditions. According to the nonlocal elasticity theory, the displacement field equations coupled by the van der Waals interaction are derived. The set of governing equations of motion is then solved by the Rayleigh–Ritz method. The present analysis can treat boundary conditions in a layer-wise manner. The effects of nonlocal parameter, layer-wise boundary conditions and geometrical parameters on the mechanical behavior of DWCNTs are examined. Furthermore, molecular dynamics simulations are performed to assess the validity of the results and also to predict the appropriate values of nonlocal parameter. It is found that the type of boundary conditions affects the proper value of nonlocal parameter.

scholarly journals Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of Double-Walled Carbon Nanotubes

2021 ◽  
Vol 7 (3) ◽  
pp. 61
Author(s):  
Matteo Strozzi ◽  
Oleg V. Gendelman ◽  
Isaac E. Elishakoff ◽  
Francesco Pellicano
Keyword(s):  
Carbon Nanotubes ◽  
Shell Theory ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Elastic Shell ◽  
Mode Shapes ◽  
Simplified Models ◽  
Circular Cylindrical Shells ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

The applicability and limitations of simplified models of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.

Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) With a Gap in Mindlin Plates or Panels

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Bending Vibrations ◽  
Joint System ◽  
Adhesive Layers ◽  
Bonded Joint ◽  
Free Bending

In this present study, the “Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap in Mindlin Plates or Panels” are theoretically analyzed and are numerically solved in some detail. The “plate adherends” and the upper and lower “doubler plates” of the “Bonded Joint” system are considered as dissimilar, orthotropic “Mindlin Plates” joined through the dissimilar upper and lower very thin adhesive layers. There is a symmetrically and centrally located “Gap” between the “plate adherends” of the joint system. In the “adherends” and the “doublers” of the “Bonded Joint” assembly, the transverse shear deformations and the transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers are assumed to be linearly elastic continua with transverse normal and shear stresses. The “damping effects” in the entire “Bonded Joint” system are neglected. The sets of the dynamic “Mindlin Plate” equations of the “plate adherends”, the “double doubler plates” and the thin adhesive layers are combined together with the orthotropic stress resultant-displacement expressions in a “special form”. This system of equations, after some further manipulations, is eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in terms of the “state vectors” of the problem. Hence, the final set of the aforementioned “Governing Systems of Equations” together with the “Continuity Conditions” and the “Boundary conditions” facilitate the present solution procedure. This is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). The present theoretical formulation and the method of solution are applied to a typical “Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap”. The effects of the relatively stiff (or “hard”) and the relatively flexible (or “soft”) adhesive properties, on the natural frequencies and mode shapes are considered in detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of boundary conditions. Also, several parametric studies of the dimensionless natural frequencies of the entire system are graphically presented. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials As Assumed Shape Functions

1991 ◽  
Author(s):  
C. Rajalingham ◽  
R. B. Bhat ◽  
G. D. Xistris
Keyword(s):  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Polar Coordinate System ◽  
Free Boundaries ◽  
Natural Modes ◽  
Polar Coordinate ◽  
Boundary Characteristic

Abstract The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in Cartesian coordinate system are transformed into the modified polar coordinate system. Boundary characteristic orthogonal polynomials in the radial direction, and trigonometric functions in the angular direction are used to express the deflection of the plate. These deflection shapes are classified into four basic categories, depending on its symmetrical or antisymmetrical property about the major and minor axes of the ellipse. The first six natural modes in each of the above categories are presented in the form of contour plots.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

scholarly journals An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

2021 ◽  
Author(s):  
Ishan Ali Khan
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Wide Range ◽  
Walled Cnts

Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

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